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NAG Toolbox: nag_opt_nlp1_sparse_solve (e04ug)
Purpose
nag_opt_nlp1_sparse_solve (e04ug) solves sparse nonlinear programming problems.
Syntax
[
a,
ns,
xs,
istate,
clamda,
miniz,
minz,
ninf,
sinf,
obj,
user,
lwsav,
iwsav,
rwsav,
ifail] = e04ug(
confun,
objfun,
n,
m,
ncnln,
nonln,
njnln,
iobj,
a,
ha,
ka,
bl,
bu,
start,
names,
ns,
xs,
istate,
clamda,
lwsav,
iwsav,
rwsav, 'nnz',
nnz, 'nname',
nname, 'leniz',
leniz, 'lenz',
lenz, 'user',
user)
[
a,
ns,
xs,
istate,
clamda,
miniz,
minz,
ninf,
sinf,
obj,
user,
lwsav,
iwsav,
rwsav,
ifail] = nag_opt_nlp1_sparse_solve(
confun,
objfun,
n,
m,
ncnln,
nonln,
njnln,
iobj,
a,
ha,
ka,
bl,
bu,
start,
names,
ns,
xs,
istate,
clamda,
lwsav,
iwsav,
rwsav, 'nnz',
nnz, 'nname',
nname, 'leniz',
leniz, 'lenz',
lenz, 'user',
user)
Before calling
nag_opt_nlp1_sparse_solve (e04ug), or
the option setting function
(e04uj),
nag_opt_init (e04wb) must be called.
Note: the interface to this routine has changed since earlier releases of the toolbox:
Mark 22:
lenz and
leniz have been made optional
.
Description
nag_opt_nlp1_sparse_solve (e04ug) is designed to solve a class of nonlinear programming problems that are assumed to be stated in the following general form:
where
x = (x_{1},x_{2}, … ,x_{n})^{T}$x={({x}_{1},{x}_{2},\dots ,{x}_{n})}^{\mathrm{T}}$ is a set of variables,
f(x)$f\left(x\right)$ is a smooth scalar objective function,
l$l$ and
u$u$ are constant lower and upper bounds,
F(x)$F\left(x\right)$ is a vector of smooth nonlinear constraint functions
{F_{i}(x)}$\left\{{F}_{i}\left(x\right)\right\}$ and
G$G$ is a
sparse matrix.
The constraints involving
F$F$ and
Gx$Gx$ are called the
general constraints. Note that upper and lower bounds are specified for all variables and constraints. This form allows full generality in specifying various types of constraint. In particular, the
j$j$th constraint can be defined as an
equality by setting
l_{j} = u_{j}${l}_{j}={u}_{j}$. If certain bounds are not present, the associated elements of
l$l$ or
u$u$ can be set to special values that will be treated as
− ∞$\infty $ or
+ ∞$+\infty $. (See the description of the optional parameter
Infinite Bound Size.)
nag_opt_nlp1_sparse_solve (e04ug) converts the upper and lower bounds on the
m$m$ elements of
F$F$ and
Gx$Gx$ to equalities by introducing a set of
slack variables
s$s$, where
s = (s_{1},s_{2}, … ,s_{m})^{T}$s={({s}_{1},{s}_{2},\dots ,{s}_{m})}^{\mathrm{T}}$. For example, the linear constraint
5 ≤ 2x_{1} + 3x_{2} ≤ + ∞$5\le 2{x}_{1}+3{x}_{2}\le +\infty $ is replaced by
2x_{1} + 3x_{2} − s_{1} = 0$2{x}_{1}+3{x}_{2}{s}_{1}=0$, together with the bounded slack
5 ≤ s_{1} ≤ + ∞$5\le {s}_{1}\le +\infty $. The problem defined by
(1) can therefore be rewritten in the following equivalent form:
Since the slack variables
s$s$ are subject to the same upper and lower bounds as the elements of
F$F$ and
Gx$Gx$, the bounds on
F$F$ and
Gx$Gx$ can simply be thought of as bounds on the combined vector
(x,s)$(x,s)$. The elements of
x$x$ and
s$s$ are partitioned into
basic,
nonbasic and
superbasic variables defined as follows:
– 
a basic variable (x_{j}${x}_{j}$ say) is the j$j$th variable associated with the j$j$th column of the basis matrix B$B$; 
– 
a nonbasic variable is a variable that is temporarily fixed at its current value (usually its upper or lower bound); 
– 
a superbasic variable is a nonbasic variable which is not at one of its bounds that is free to move in any desired direction (namely one that will improve the value of the objective function or reduce the sum of infeasibilities). 
For example, in the simplex method (see
Gill et al. (1981)) the elements of
x$x$ can be partitioned at each vertex into a set of
m$m$ basic variables (all nonnegative) and a set of
(n − m)$(nm)$ nonbasic variables (all zero). This is equivalent to partitioning the columns of the constraint matrix as
$\left(\begin{array}{cc}B& N\end{array}\right)$, where
B$B$ contains the
m$m$ columns that correspond to the basic variables and
N$N$ contains the
(n − m)$(nm)$ columns that correspond to the nonbasic variables. Note that
B$B$ is square and nonsingular.
The optional parameter
Maximize may be used to specify an alternative problem in which
f(x)$f\left(x\right)$ is maximized. If the objective function is nonlinear and all the constraints are linear,
F$F$ is absent and the problem is said to be
linearly constrained. In general, the objective and constraint functions are
structured in the sense that they are formed from sums of linear and nonlinear functions. This structure can be exploited by the function during the solution process as follows.
Consider the following nonlinear optimization problem with four variables (
u,v,z,w$u,v,z,w$):
subject to the constraints
and to the bounds
This problem has several characteristics that can be exploited by the function:
– 
the objective function is nonlinear. It is the sum of a nonlinear function of the variables (u,v,z$u,v,z$) and a linear function of the variables (z,w$z,w$); 
– 
the first two constraints are nonlinear. The third is linear; 
– 
each nonlinear constraint function is the sum of a nonlinear function of the variables (u,v$u,v$) and a linear function of the variables (z,w$z,w$). 
The nonlinear terms are defined by
objfun and
confun (see
Section [Parameters]), which involve only the appropriate subset of variables.
For the objective, we define the function
f(u,v,z) = (u + v + z)^{2}$f(u,v,z)={(u+v+z)}^{2}$ to include only the nonlinear part of the objective. The three variables (
u,v,z$u,v,z$) associated with this function are known as the
nonlinear objective variables. The number of them is given by
nonln (see
Section [Parameters]) and they are the only variables needed in
objfun. The linear part
3z + 5w$3z+5w$ of the objective is stored in row
iobj (see
Section [Parameters]) of the (constraint) Jacobian matrix
A$A$ (see below).
Thus, if
x^{′}${x}^{\prime}$ and
y^{′}${y}^{\prime}$ denote the nonlinear and linear objective variables, respectively, the objective may be rewritten in the form
where
f(x^{′})$f\left({x}^{\prime}\right)$ is the nonlinear part of the objective and
c$c$ and
d$d$ are constant vectors that form a row of
A$A$. In this example,
x^{′} = (u,v,z)${x}^{\prime}=(u,v,z)$ and
y^{′} = w${y}^{\prime}=w$.
Similarly for the constraints, we define a vector function
F(u,v)$F(u,v)$ to include just the nonlinear terms. In this example,
F_{1}(u,v) = u^{2} + v^{2}${F}_{1}(u,v)={u}^{2}+{v}^{2}$ and
F_{2}(u,v) = u^{4} + v^{4}${F}_{2}(u,v)={u}^{4}+{v}^{4}$, where the two variables (
u,v$u,v$) are known as the
nonlinear Jacobian variables. The number of them is given by
njnln (see
Section [Parameters]) and they are the only variables needed in
confun. Thus, if
x^{ ′ ′ }${x}^{\prime \prime}$ and
y^{ ′ ′ }${y}^{\prime \prime}$ denote the nonlinear and linear Jacobian variables, respectively, the constraint functions and the linear part of the objective have the form
where
x^{ ′ ′ } = (u,v)${x}^{\prime \prime}=(u,v)$ and
y^{ ′ ′ } = (z,w)${y}^{\prime \prime}=(z,w)$ in this example. This ensures that the Jacobian is of the form
where
J(x^{ ′ ′ }) = ( ∂ F(x^{ ′ ′ }))/( ∂ x)
$J\left({x}^{\prime \prime}\right)=\frac{\partial F\left({x}^{\prime \prime}\right)}{\partial x}$. Note that
J(x^{ ′ ′ })$J\left({x}^{\prime \prime}\right)$ always appears in the
top lefthand corner of
A$A$.
The inequalities
l_{1} ≤ F(x^{ ′ ′ }) + A_{2}y^{ ′ ′ } ≤ u_{1}${l}_{1}\le F\left({x}^{\prime \prime}\right)+{A}_{2}{y}^{\prime \prime}\le {u}_{1}$ and
l_{2} ≤ A_{3}x^{ ′ ′ } + A_{4}y^{ ′ ′ } ≤ u_{2}${l}_{2}\le {A}_{3}{x}^{\prime \prime}+{A}_{4}{y}^{\prime \prime}\le {u}_{2}$ implied by the constraint functions in
(3) are known as the
nonlinear and
linear constraints, respectively. The nonlinear constraint vector
F(x^{ ′ ′ })$F\left({x}^{\prime \prime}\right)$ in
(3) and (optionally) its partial derivative matrix
J(x^{ ′ ′ })$J\left({x}^{\prime \prime}\right)$ are set in
confun. The matrices
A_{2}${A}_{2}$,
A_{3}${A}_{3}$ and
A_{4}${A}_{4}$ contain any (constant) linear terms. Along with the sparsity pattern of
J(x^{ ′ ′ })$J\left({x}^{\prime \prime}\right)$ they are stored in the arrays
a,
ha and
ka (see
Section [Parameters]).
In general, the vectors x^{′}${x}^{\prime}$ and x^{ ′ ′ }${x}^{\prime \prime}$ have different dimensions, but they always overlap, in the sense that the shorter vector is always the beginning of the other. In the above example, the nonlinear Jacobian variables (u,v)$(u,v)$ are an ordered subset of the nonlinear objective variables (u,v,z)$(u,v,z)$. In other cases it could be the other way round (whichever is the most convenient), but the first way keeps J(x^{ ′ ′ })$J\left({x}^{\prime \prime}\right)$ as small as possible.
Note that the nonlinear objective function
f(x^{′})$f\left({x}^{\prime}\right)$ may involve either a subset or superset of the variables appearing in the nonlinear constraint functions
F(x^{ ′ ′ })$F\left({x}^{\prime \prime}\right)$. Thus,
nonln ≤ njnln${\mathbf{nonln}}\le {\mathbf{njnln}}$ (or viceversa). Sometimes the objective and constraints really involve
disjoint sets of nonlinear variables. In such cases the variables should be ordered so that
nonln > njnln${\mathbf{nonln}}>{\mathbf{njnln}}$ and
x^{′} = (x^{ ′ ′ },x^{ ′ ′ ′ })${x}^{\prime}=({x}^{\prime \prime},{x}^{\prime \prime \prime})$, where the objective is nonlinear in just the last vector
x^{ ′ ′ ′ }${x}^{\prime \prime \prime}$. The first
njnln elements of the gradient array
objgrd should also be set to zero in
objfun. This is illustrated in
Section [Example].
If all elements of the constraint Jacobian are known (i.e., the optional parameter
Derivative Level = 2${\mathbf{Derivative\; Level}}=2$ or
3$3$), any constant elements may be assigned their correct values in
a,
ha and
ka. The corresponding elements of the constraint Jacobian array
fjac need not be reset in
confun. This includes values that are identically zero as constraint Jacobian elements are assumed to be zero unless specified otherwise. It must be emphasized that, if
Derivative Level = 0${\mathbf{Derivative\; Level}}=0$ or
1$1$, unassigned elements of
fjac are
not treated as constant; they are estimated by finite differences, at nontrivial expense.
If there are no nonlinear constraints in
(1) and
f(x)$f\left(x\right)$ is linear or quadratic, then it may be more efficient to use
nag_opt_qpconvex2_sparse_solve (e04nq) to solve the resulting linear or quadratic programming problem, or one of
nag_opt_lp_solve (e04mf),
nag_opt_lsq_lincon_solve (e04nc) or
nag_opt_qp_dense_solve (e04nf) if
G$G$ is a
dense matrix. If the problem is dense and does have nonlinear constraints then one of
nag_opt_nlp1_rcomm (e04uf),
nag_opt_lsq_gencon_deriv (e04us) or
nag_opt_nlp2_solve (e04wd) (as appropriate) should be used instead.
You must supply an initial estimate of the solution to
(1), together with versions of
objfun and
confun that define
f(x^{′})$f\left({x}^{\prime}\right)$ and
F(x^{ ′ ′ })$F\left({x}^{\prime \prime}\right)$, respectively, and as many first partial derivatives as possible. Note that if there are any nonlinear constraints, then the
first call to
confun will precede the
first call to
objfun.
nag_opt_nlp1_sparse_solve (e04ug) is based on the SNOPT package described in
Gill et al. (2002), which in turn utilizes functions from the MINOS package (see
Murtagh and Saunders (1995)). It incorporates a sequential quadratic programming (SQP) method that obtains search directions from a sequence of quadratic programming (QP) subproblems. Each QP subproblem minimizes a quadratic model of a certain Lagrangian function subject to a linearization of the constraints. An augmented Lagrangian merit function is reduced along each search direction to ensure convergence from any starting point. Further details can be found in
Section [Algorithmic Details].
Throughout this document the symbol
ε$\epsilon $ is used to represent the
machine precision (see
nag_machine_precision (x02aj)).
References
Conn A R (1973) Constrained optimization using a nondifferentiable penalty function SIAM J. Numer. Anal. 10 760–779
Eldersveld S K (1991) Largescale sequential quadratic programming algorithms PhD Thesis Department of Operations Research, Stanford University, Stanford
Fletcher R (1984) An l_{1}${l}_{1}$ penalty method for nonlinear constraints Numerical Optimization 1984 (eds P T Boggs, R H Byrd and R B Schnabel) 26–40 SIAM Philadelphia
Fourer R (1982) Solving staircase linear programs by the simplex method Math. Programming 23 274–313
Gill P E, Murray W and Saunders M A (2002) SNOPT: An SQP Algorithm for Largescale Constrained Optimization 12 979–1006 SIAM J. Optim.
Gill P E, Murray W, Saunders M A and Wright M H (1986) Users' guide for NPSOL (Version 4.0): a Fortran package for nonlinear programming Report SOL 862 Department of Operations Research, Stanford University
Gill P E, Murray W, Saunders M A and Wright M H (1989) A practical anticycling procedure for linearly constrained optimization Math. Programming 45 437–474
Gill P E, Murray W, Saunders M A and Wright M H (1992) Some theoretical properties of an augmented Lagrangian merit function Advances in Optimization and Parallel Computing (ed P M Pardalos) 101–128 North Holland
Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press
Hock W and Schittkowski K (1981) Test Examples for Nonlinear Programming Codes. Lecture Notes in Economics and Mathematical Systems 187 Springer–Verlag
Murtagh B A and Saunders M A (1995) MINOS 5.4 users' guide Report SOL 8320R Department of Operations Research, Stanford University
Ortega J M and Rheinboldt W C (1970) Iterative Solution of Nonlinear Equations in Several Variables Academic Press
Powell M J D (1974) Introduction to constrained optimization Numerical Methods for Constrained Optimization (eds P E Gill and W Murray) 1–28 Academic Press
Parameters
Compulsory Input Parameters
 1:
confun – function handle or string containing name of mfile
confun must calculate the vector
F(x)$F\left(x\right)$ of nonlinear constraint functions and (optionally) its Jacobian
( = ( ∂ F)/( ∂ x)) $(=\frac{\partial F}{\partial x})$ for a specified
n_{1}^{ ′ ′ }${n}_{1}^{\prime \prime}$ (
≤ n$\text{}\le n$) element vector
x$x$. If there are no nonlinear constraints (i.e.,
ncnln = 0${\mathbf{ncnln}}=0$),
confun will never be called by
nag_opt_nlp1_sparse_solve (e04ug) and
confun may be the string
'e04ugm'. (
nag_opt_nlp1_sparse_dummy_confun (e04ugm) is included in the NAG Toolbox.) If there are nonlinear constraints, the first call to
confun will occur before the first call to
objfun.
[mode, f, fjac, user] = confun(mode, ncnln, njnln, nnzjac, x, fjac, nstate, user)
Input Parameters
 1:
mode – int64int32nag_int scalar
Indicates which values must be assigned during each call of
confun. Only the following values need be assigned:
 mode = 0${\mathbf{mode}}=0$
 f.
 mode = 1${\mathbf{mode}}=1$
 All available elements of fjac.
 mode = 2${\mathbf{mode}}=2$
 f and all available elements of fjac.
 2:
ncnln – int64int32nag_int scalar
n_{N}${n}_{N}$, the number of nonlinear constraints. These must be the first
ncnln constraints in the problem.
 3:
njnln – int64int32nag_int scalar
n_{1}^{ ′ ′ }${n}_{1}^{\prime \prime}$, the number of nonlinear variables. These must be the first
njnln variables in the problem.
 4:
nnzjac – int64int32nag_int scalar
The number of nonzero elements in the constraint Jacobian. Note that
nnzjac will usually be less than
ncnln × njnln${\mathbf{ncnln}}\times {\mathbf{njnln}}$.
 5:
x(njnln) – double array
x$x$, the vector of nonlinear Jacobian variables at which the nonlinear constraint functions and/or the available elements of the constraint Jacobian are to be evaluated.
 6:
fjac(nnzjac) – double array
The elements of
fjac are set to special values which enable
nag_opt_nlp1_sparse_solve (e04ug) to detect whether they are changed by
confun.
 7:
nstate – int64int32nag_int scalar
If
nstate = 1${\mathbf{nstate}}=1$, then
nag_opt_nlp1_sparse_solve (e04ug) is calling
confun for the first time. This parameter setting allows you to save computation time if certain data must be read or calculated only once.
If
nstate ≥ 2${\mathbf{nstate}}\ge 2$, then
nag_opt_nlp1_sparse_solve (e04ug) is calling
confun for the last time. This parameter setting allows you to perform some additional computation on the final solution. In general, the last call to
confun is made with
nstate = 2 + ifail${\mathbf{nstate}}=2+{\mathbf{ifail}}$ (see
Section [Error Indicators and Warnings]).
Otherwise,
nstate = 0${\mathbf{nstate}}=0$.
 8:
user – Any MATLAB object
confun is called from
nag_opt_nlp1_sparse_solve (e04ug) with the object supplied to
nag_opt_nlp1_sparse_solve (e04ug).
Output Parameters
 1:
mode – int64int32nag_int scalar
You may set to a negative value as follows:
 mode ≤ − 2${\mathbf{mode}}\le 2$
 The solution to the current problem is terminated and in this case nag_opt_nlp1_sparse_solve (e04ug) will terminate with ifail set to mode.
 mode = − 1${\mathbf{mode}}=1$
 The nonlinear constraint functions cannot be calculated at the current x$x$. nag_opt_nlp1_sparse_solve (e04ug) will then terminate with ifail = − 1${\mathbf{ifail}}=1$ unless this occurs during the linesearch; in this case, the linesearch will shorten the step and try again.
 2:
f(ncnln) – double array
If
mode = 0${\mathbf{mode}}=0$ or
2$2$,
f(i)${\mathbf{f}}\left(i\right)$ must contain the value of the
i$i$th nonlinear constraint function at
x$x$.
 3:
fjac(nnzjac) – double array
If
mode = 1${\mathbf{mode}}=1$ or
2$2$,
fjac must return the available elements of the constraint Jacobian evaluated at
x$x$. These elements must be stored in exactly the same positions as implied by the definitions of the arrays
a,
ha and
ka. If optional parameter
Derivative Level = 2${\mathbf{Derivative\; Level}}=2$ or
3$3$, the value of any constant Jacobian element not defined by
confun will be obtained directly from
a. Note that the function does not perform any internal checks for consistency (except indirectly via the optional parameter
Verify Level), so great care is essential.
 4:
user – Any MATLAB object
 2:
objfun – function handle or string containing name of mfile
objfun must calculate the nonlinear part of the objective function
f(x)$f\left(x\right)$ and (optionally) its gradient
( = ( ∂ f)/( ∂ x)) $(=\frac{\partial f}{\partial x})$ for a specified
n_{1}^{ ′ }${n}_{1}^{\prime}$ (
≤ n$\text{}\le n$) element vector
x$x$. If there are no nonlinear objective variables (i.e.,
nonln = 0${\mathbf{nonln}}=0$),
objfun will never be called by
nag_opt_nlp1_sparse_solve (e04ug) and
objfun may be the string
'e04ugn'. (
nag_opt_nlp1_sparse_dummy_objfun (e04ugn) is included in the NAG Toolbox.)
[mode, objf, objgrd, user] = objfun(mode, nonln, x, objgrd, nstate, user)
Input Parameters
 1:
mode – int64int32nag_int scalar
Indicates which values must be assigned during each call of
objfun. Only the following values need be assigned:
 mode = 0${\mathbf{mode}}=0$
 objf.
 mode = 1${\mathbf{mode}}=1$
 All available elements of objgrd.
 mode = 2${\mathbf{mode}}=2$
 objf and all available elements of objgrd.
 2:
nonln – int64int32nag_int scalar
n_{1}^{ ′ }${n}_{1}^{\prime}$, the number of nonlinear objective variables. These must be the first
nonln variables in the problem.
 3:
x(nonln) – double array
x$x$, the vector of nonlinear variables at which the nonlinear part of the objective function and/or all available elements of its gradient are to be evaluated.
 4:
objgrd(nonln) – double array
The elements of
objgrd are set to special values which enable
nag_opt_nlp1_sparse_solve (e04ug) to detect whether they are changed by
objfun.
 5:
nstate – int64int32nag_int scalar
If
nstate = 1${\mathbf{nstate}}=1$,
nag_opt_nlp1_sparse_solve (e04ug) is calling
objfun for the first time. This parameter setting allows you to save computation time if certain data must be read or calculated only once.
If
nstate ≥ 2${\mathbf{nstate}}\ge 2$,
nag_opt_nlp1_sparse_solve (e04ug) is calling
objfun for the last time. This parameter setting allows you to perform some additional computation on the final solution. In general, the last call to
objfun is made with
nstate = 2 + ifail${\mathbf{nstate}}=2+{\mathbf{ifail}}$ (see
Section [Error Indicators and Warnings]).
Otherwise,
nstate = 0${\mathbf{nstate}}=0$.
 6:
user – Any MATLAB object
objfun is called from
nag_opt_nlp1_sparse_solve (e04ug) with the object supplied to
nag_opt_nlp1_sparse_solve (e04ug).
Output Parameters
 1:
mode – int64int32nag_int scalar
You may set to a negative value as follows:
 mode ≤ − 2${\mathbf{mode}}\le 2$
 The solution to the current problem is terminated and in this case nag_opt_nlp1_sparse_solve (e04ug) will terminate with ifail set to mode.
 mode = − 1${\mathbf{mode}}=1$
 The nonlinear part of the objective function cannot be calculated at the current x$x$. nag_opt_nlp1_sparse_solve (e04ug) will then terminate with ifail = − 1${\mathbf{ifail}}=1$ unless this occurs during the linesearch; in this case, the linesearch will shorten the step and try again.
 2:
objf – double scalar
If
mode = 0${\mathbf{mode}}=0$ or
2$2$,
objf must be set to the value of the objective function at
x$x$.
 3:
objgrd(nonln) – double array
If
mode = 1${\mathbf{mode}}=1$ or
2$2$,
objgrd must return the available elements of the gradient evaluated at
x$x$.
 4:
user – Any MATLAB object
 3:
n – int64int32nag_int scalar
n$n$, the number of variables (excluding slacks). This is the number of columns in the full Jacobian matrix A$A$.
Constraint:
n ≥ 1${\mathbf{n}}\ge 1$.
 4:
m – int64int32nag_int scalar
m$m$, the number of general constraints (or slacks). This is the number of rows in
A$A$, including the free row (if any; see
iobj). Note that
A$A$ must contain at least one row. If your problem has no constraints, or only upper and lower bounds on the variables, then you must include a dummy ‘free’ row consisting of a single (zero) element subject to ‘infinite’ upper and lower bounds. Further details can be found under the descriptions for
iobj,
nnz,
a,
ha,
ka,
bl and
bu.
Constraint:
m ≥ 1${\mathbf{m}}\ge 1$.
 5:
ncnln – int64int32nag_int scalar
n_{N}${n}_{N}$, the number of nonlinear constraints.
Constraint:
0 ≤ ncnln ≤ m$0\le {\mathbf{ncnln}}\le {\mathbf{m}}$.
 6:
nonln – int64int32nag_int scalar
n_{1}^{ ′ }${n}_{1}^{\prime}$, the number of nonlinear objective variables. If the objective function is nonlinear, the leading
n_{1}^{ ′ }${n}_{1}^{\prime}$ columns of
A$A$ belong to the nonlinear objective variables. (See also the description for
njnln.)
Constraint:
0 ≤ nonln ≤ n$0\le {\mathbf{nonln}}\le {\mathbf{n}}$.
 7:
njnln – int64int32nag_int scalar
n_{1}^{ ′ ′ }${n}_{1}^{\prime \prime}$, the number of nonlinear Jacobian variables. If there are any nonlinear constraints, the leading n_{1}^{ ′ ′ }${n}_{1}^{\prime \prime}$ columns of A$A$ belong to the nonlinear Jacobian variables. If n_{1}^{ ′ } > 0${n}_{1}^{\prime}>0$ and n_{1}^{ ′ ′ } > 0${n}_{1}^{\prime \prime}>0$, the nonlinear objective and Jacobian variables overlap. The total number of nonlinear variables is given by n = max (n_{1}^{ ′ },n_{1}^{ ′ ′ })$\stackrel{}{n}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}({n}_{1}^{\prime},{n}_{1}^{\prime \prime})$.
Constraints:
 if ncnln = 0${\mathbf{ncnln}}=0$, njnln = 0${\mathbf{njnln}}=0$;
 if ncnln > 0${\mathbf{ncnln}}>0$, 1 ≤ njnln ≤ n$1\le {\mathbf{njnln}}\le {\mathbf{n}}$.
 8:
iobj – int64int32nag_int scalar
If
iobj > ncnln${\mathbf{iobj}}>{\mathbf{ncnln}}$, row
iobj of
A$A$ is a free row containing the nonzero elements of the linear part of the objective function.
 iobj = 0${\mathbf{iobj}}=0$
 There is no free row.
 iobj = − 1${\mathbf{iobj}}=1$
 There is a dummy ‘free’ row.
Constraints:
 if iobj > 0${\mathbf{iobj}}>0$, ncnln < iobj ≤ m${\mathbf{ncnln}}<{\mathbf{iobj}}\le {\mathbf{m}}$;
 otherwise iobj ≥ − 1${\mathbf{iobj}}\ge 1$.
 9:
a(nnz) – double array
nnz, the dimension of the array, must satisfy the constraint
1 ≤ nnz ≤ n × m$1\le {\mathbf{nnz}}\le {\mathbf{n}}\times {\mathbf{m}}$.
The nonzero elements of the Jacobian matrix
A$A$, ordered by increasing column index. Since the constraint Jacobian matrix
J(x^{ ′ ′ })$J\left({x}^{\prime \prime}\right)$ must always appear in the top lefthand corner of
A$A$, those elements in a column associated with any nonlinear constraints must come before any elements belonging to the linear constraint matrix
G$G$ and the free row (if any; see
iobj).
In general,
a is partitioned into a nonlinear part and a linear part corresponding to the nonlinear variables and linear variables in the problem. Elements in the nonlinear part may be set to any value (e.g., zero) because they are initialized at the first point that satisfies the linear constraints and the upper and lower bounds.
If
Derivative Level = 2${\mathbf{Derivative\; Level}}=2$ or
3$3$, the nonlinear part may also be used to store any constant Jacobian elements. Note that if
confun does not define the constant Jacobian element
fjac(i)${\mathbf{fjac}}\left(i\right)$ then the missing value will be obtained directly from
a(j)${\mathbf{a}}\left(j\right)$ for some
j ≥ i$j\ge i$.
If
Derivative Level = 0${\mathbf{Derivative\; Level}}=0$ or
1$1$, unassigned elements of
fjac are
not treated as constant; they are estimated by finite differences, at nontrivial expense.
The linear part must contain the nonzero elements of
G$G$ and the free row (if any). If
iobj = − 1${\mathbf{iobj}}=1$, set
a(1) = 0${\mathbf{a}}\left(1\right)=0$. Elements with the same row and column indices are not allowed. (See also the descriptions for
ha and
ka.)
 10:
ha(nnz) – int64int32nag_int array
nnz, the dimension of the array, must satisfy the constraint
1 ≤ nnz ≤ n × m$1\le {\mathbf{nnz}}\le {\mathbf{n}}\times {\mathbf{m}}$.
ha(i)${\mathbf{ha}}\left(\mathit{i}\right)$ must contain the row index of the nonzero element stored in
a(i)${\mathbf{a}}\left(\mathit{i}\right)$, for
i = 1,2, … ,nnz$\mathit{i}=1,2,\dots ,{\mathbf{nnz}}$. The row indices for a column may be supplied in any order subject to the condition that those elements in a column associated with any nonlinear constraints must appear before those elements associated with any linear constraints (including the free row, if any). Note that
confun must define the Jacobian elements in the same order. If
iobj = − 1${\mathbf{iobj}}=1$, set
ha(1) = 1${\mathbf{ha}}\left(1\right)=1$.
Constraint:
1 ≤ ha(i) ≤ m$1\le {\mathbf{ha}}\left(\mathit{i}\right)\le {\mathbf{m}}$, for
i = 1,2, … ,nnz$\mathit{i}=1,2,\dots ,{\mathbf{nnz}}$.
 11:
ka(n + 1${\mathbf{n}}+1$) – int64int32nag_int array
ka(j)${\mathbf{ka}}\left(\mathit{j}\right)$ must contain the index in
a of the start of the
j$\mathit{j}$th column, for
j = 1,2, … ,n$\mathit{j}=1,2,\dots ,{\mathbf{n}}$. To specify the
j$\mathit{j}$th column as empty, set
ka(j) = ka(j + 1)${\mathbf{ka}}\left(\mathit{j}\right)={\mathbf{ka}}\left(\mathit{j}+1\right)$. Note that the first and last elements of
ka must be such that
ka(1) = 1${\mathbf{ka}}\left(1\right)=1$ and
ka(n + 1) = nnz + 1${\mathbf{ka}}\left({\mathbf{n}}+1\right)={\mathbf{nnz}}+1$. If
iobj = − 1${\mathbf{iobj}}=1$, set
ka(j) = 2${\mathbf{ka}}\left(\mathit{j}\right)=2$, for
j = 2,3, … ,n$\mathit{j}=2,3,\dots ,{\mathbf{n}}$.
Constraints:
 ka(1) = 1${\mathbf{ka}}\left(1\right)=1$;
 ka(j) ≥ 1${\mathbf{ka}}\left(\mathit{j}\right)\ge 1$, for j = 2,3, … ,n$\mathit{j}=2,3,\dots ,{\mathbf{n}}$;
 ka(n + 1) = nnz + 1${\mathbf{ka}}\left({\mathbf{n}}+1\right)={\mathbf{nnz}}+1$;
 0 ≤ ka(j + 1) − ka(j) ≤ m$0\le {\mathbf{ka}}\left(\mathit{j}+1\right){\mathbf{ka}}\left(\mathit{j}\right)\le {\mathbf{m}}$, for j = 1,2, … ,n$\mathit{j}=1,2,\dots ,{\mathbf{n}}$.
 12:
bl(n + m${\mathbf{n}}+{\mathbf{m}}$) – double array
l$l$, the lower bounds for all the variables and general constraints, in the following order. The first
n elements of
bl must contain the bounds on the variables
x$x$, the next
ncnln elements the bounds for the nonlinear constraints
F(x)$F\left(x\right)$ (if any) and the next (
m − ncnln${\mathbf{m}}{\mathbf{ncnln}}$) elements the bounds for the linear constraints
Gx$Gx$ and the free row (if any). To specify a nonexistent lower bound (i.e.,
l_{j} = − ∞${l}_{j}=\infty $), set
bl(j) ≤ − bigbnd${\mathbf{bl}}\left(j\right)\le \mathit{bigbnd}$. To specify the
j$j$th constraint as an
equality, set
bl(j) = bu(j) = β${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)=\beta $, say, where
β < bigbnd$\left\beta \right<\mathit{bigbnd}$. If
iobj = − 1${\mathbf{iobj}}=1$, set
bl(n + abs(iobj)) ≤ − bigbnd${\mathbf{bl}}\left({\mathbf{n}}+\mathrm{abs}\left({\mathbf{iobj}}\right)\right)\le \mathit{bigbnd}$.
Constraint:
if
ncnln < iobj ≤ m${\mathbf{ncnln}}<{\mathbf{iobj}}\le {\mathbf{m}}$ or
iobj = − 1${\mathbf{iobj}}=1$,
bl(n + abs(iobj)) ≤ − bigbnd${\mathbf{bl}}\left({\mathbf{n}}+\mathrm{abs}\left({\mathbf{iobj}}\right)\right)\le \mathit{bigbnd}$(See also the description for
bu.)
 13:
bu(n + m${\mathbf{n}}+{\mathbf{m}}$) – double array
u$u$, the upper bounds for all the variables and general constraints, in the following order. The first
n elements of
bu must contain the bounds on the variables
x$x$, the next
ncnln elements the bounds for the nonlinear constraints
F(x)$F\left(x\right)$ (if any) and the next (
m − ncnln${\mathbf{m}}{\mathbf{ncnln}}$) elements the bounds for the linear constraints
Gx$Gx$ and the free row (if any). To specify a nonexistent upper bound (i.e.,
u_{j} = + ∞${u}_{j}=+\infty $), set
bu(j) ≥ bigbnd${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$. To specify the
j$j$th constraint as an
equality, set
bu(j) = bl(j) = β${\mathbf{bu}}\left(j\right)={\mathbf{bl}}\left(j\right)=\beta $, say, where
β < bigbnd$\left\beta \right<\mathit{bigbnd}$. If
iobj = − 1${\mathbf{iobj}}=1$, set
bu(n + abs(iobj)) ≥ bigbnd${\mathbf{bu}}\left({\mathbf{n}}+\mathrm{abs}\left({\mathbf{iobj}}\right)\right)\ge \mathit{bigbnd}$.
Constraints:
 if ncnln < iobj ≤ m${\mathbf{ncnln}}<{\mathbf{iobj}}\le {\mathbf{m}}$ or iobj = − 1${\mathbf{iobj}}=1$, bu(n + abs(iobj)) ≥ bigbnd${\mathbf{bu}}\left({\mathbf{n}}+\mathrm{abs}\left({\mathbf{iobj}}\right)\right)\ge \mathit{bigbnd}$;
 bl(j) ≤ bu(j)${\mathbf{bl}}\left(\mathit{j}\right)\le {\mathbf{bu}}\left(\mathit{j}\right)$, for j = 1,2, … ,n + m$\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{m}}$;
 if bl(j) = bu(j) = β${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)=\beta $, β < bigbnd$\left\beta \right<\mathit{bigbnd}$.
 14:
start – string (length ≥ 1)
Indicates how a starting basis is to be obtained.
 start = 'C'${\mathbf{start}}=\text{'C'}$
 An internal Crash procedure will be used to choose an initial basis.
 start = 'W'${\mathbf{start}}=\text{'W'}$
 A basis is already defined in istate and ns (probably from a previous call).
Constraint:
start = 'C'${\mathbf{start}}=\text{'C'}$ or
'W'$\text{'W'}$.
 15:
names(nname) – cell array of strings
nname, the dimension of the array, must satisfy the constraint
nname = 1${\mathbf{nname}}=1$ or
n + m${\mathbf{n}}+{\mathbf{m}}$.
Specifies the column and row names to be used in the printed output.
If
nname = 1${\mathbf{nname}}=1$,
names is not referenced and the printed output will use default names for the columns and rows.
If
nname = n + m${\mathbf{nname}}={\mathbf{n}}+{\mathbf{m}}$, the first
n elements must contain the names for the columns, the next
ncnln elements must contain the names for the nonlinear rows (if any) and the next
(m − ncnln)$({\mathbf{m}}{\mathbf{ncnln}})$ elements must contain the names for the linear rows (if any) to be used in the printed output. Note that the name for the free row or dummy ‘free’ row must be stored in
names(n + abs(iobj))${\mathbf{names}}\left({\mathbf{n}}+\mathrm{abs}\left({\mathbf{iobj}}\right)\right)$.
 16:
ns – int64int32nag_int scalar
n_{S}${n}_{S}$, the number of superbasics. It need not be specified if
start = 'C'${\mathbf{start}}=\text{'C'}$, but must retain its value from a previous call when
start = 'W'${\mathbf{start}}=\text{'W'}$.
 17:
xs(n + m${\mathbf{n}}+{\mathbf{m}}$) – double array
The initial values of the variables and slacks
(x,s)$(x,s)$. (See the description for
istate.)
 18:
istate(n + m${\mathbf{n}}+{\mathbf{m}}$) – int64int32nag_int array
If
start = 'C'${\mathbf{start}}=\text{'C'}$, the first
n elements of
istate and
xs must specify the initial states and values, respectively, of the variables
x$x$. (The slacks
s$s$ need not be initialized.) An internal Crash procedure is then used to select an initial basis matrix
B$B$. The initial basis matrix will be triangular (neglecting certain small elements in each column). It is chosen from various rows and columns of
$\left(\begin{array}{cc}A& I\end{array}\right)$. Possible values for
istate(j)${\mathbf{istate}}\left(j\right)$ are as follows:
istate(j)${\mathbf{istate}}\left(j\right)$  State of xs(j)${\mathbf{xs}}\left(j\right)$ during Crash procedure 
0$0$ or 1$1$  Eligible for the basis 
2$2$  Ignored 
3$3$  Eligible for the basis (given preference over 0$0$ or 1$1$) 
4$4$ or 5$5$  Ignored 
If nothing special is known about the problem, or there is no wish to provide special information, you may set
istate(j) = 0${\mathbf{istate}}\left(\mathit{j}\right)=0$ and
xs(j) = 0.0${\mathbf{xs}}\left(\mathit{j}\right)=0.0$, for
j = 1,2, … ,n$\mathit{j}=1,2,\dots ,{\mathbf{n}}$. All variables will then be eligible for the initial basis. Less trivially, to say that the
j$j$th variable will probably be equal to one of its bounds, set
istate(j) = 4${\mathbf{istate}}\left(j\right)=4$ and
xs(j) = bl(j)${\mathbf{xs}}\left(j\right)={\mathbf{bl}}\left(j\right)$ or
istate(j) = 5${\mathbf{istate}}\left(j\right)=5$ and
xs(j) = bu(j)${\mathbf{xs}}\left(j\right)={\mathbf{bu}}\left(j\right)$ as appropriate.
Following the Crash procedure, variables for which
istate(j) = 2${\mathbf{istate}}\left(j\right)=2$ are made superbasic. Other variables not selected for the basis are then made nonbasic at the value
xs(j)${\mathbf{xs}}\left(j\right)$ if
bl(j) ≤ xs(j) ≤ bu(j)${\mathbf{bl}}\left(j\right)\le {\mathbf{xs}}\left(j\right)\le {\mathbf{bu}}\left(j\right)$, or at the value
bl(j)${\mathbf{bl}}\left(j\right)$ or
bu(j)${\mathbf{bu}}\left(j\right)$ closest to
xs(j)${\mathbf{xs}}\left(j\right)$.
If
start = 'W'${\mathbf{start}}=\text{'W'}$,
istate and
xs must specify the initial states and values, respectively, of the variables and slacks
(x,s)$(x,s)$. If the function has been called previously with the same values of
n and
m,
istate already contains satisfactory information.
Constraints:
 if start = 'C'${\mathbf{start}}=\text{'C'}$, 0 ≤ istate(j) ≤ 5$0\le {\mathbf{istate}}\left(\mathit{j}\right)\le 5$, for j = 1,2, … ,n$\mathit{j}=1,2,\dots ,{\mathbf{n}}$;
 if start = 'W'${\mathbf{start}}=\text{'W'}$, 0 ≤ istate(j) ≤ 3$0\le {\mathbf{istate}}\left(\mathit{j}\right)\le 3$, for j = 1,2, … ,n + m$\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{m}}$.
 19:
clamda(n + m${\mathbf{n}}+{\mathbf{m}}$) – double array
If
ncnln > 0${\mathbf{ncnln}}>0$,
clamda(j)${\mathbf{clamda}}\left(\mathit{j}\right)$ must contain a Lagrange multiplier estimate for the
j$\mathit{j}$th nonlinear constraint
F_{j}(x)${F}_{\mathit{j}}\left(x\right)$, for
j = n + 1, … ,n + ncnln$\mathit{j}={\mathbf{n}}+1,\dots ,{\mathbf{n}}+{\mathbf{ncnln}}$. If nothing special is known about the problem, or there is no wish to provide special information, you may set
clamda(j) = 0.0${\mathbf{clamda}}\left(j\right)=0.0$. The remaining elements need not be set.
 20:
lwsav(20$20$) – logical array
 21:
iwsav(550$550$) – int64int32nag_int array
 22:
rwsav(550$550$) – double array
The arrays
lwsav,
iwsav and
rwsav must not be altered between calls to any of the functions
nag_opt_nlp1_sparse_solve (e04ug),
(e04uh) or
(e04uj).
Optional Input Parameters
 1:
nnz – int64int32nag_int scalar
Default:
The dimension of the arrays
a,
ha. (An error is raised if these dimensions are not equal.)
The number of nonzero elements in
A$A$ (including the Jacobian for any nonlinear constraints). If
iobj = − 1${\mathbf{iobj}}=1$, set
nnz = 1${\mathbf{nnz}}=1$.
Constraint:
1 ≤ nnz ≤ n × m$1\le {\mathbf{nnz}}\le {\mathbf{n}}\times {\mathbf{m}}$.
 2:
nname – int64int32nag_int scalar
Default:
The dimension of the array
names.
The number of column (i.e., variable) and row (i.e., constraint) names supplied in
names.
 nname = 1${\mathbf{nname}}=1$
 There are no names. Default names will be used in the printed output.
 nname = n + m${\mathbf{nname}}={\mathbf{n}}+{\mathbf{m}}$
 All names must be supplied.
Constraint:
nname = 1${\mathbf{nname}}=1$ or
n + m${\mathbf{n}}+{\mathbf{m}}$.
 3:
leniz – int64int32nag_int scalar
Default:
max (500,n + m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(500,{\mathbf{n}}+{\mathbf{m}})$ The dimension of the array
iz as declared in the (sub)program from which
nag_opt_nlp1_sparse_solve (e04ug) is called.
Constraint:
leniz ≥ max (500,n + m)${\mathbf{leniz}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(500,{\mathbf{n}}+{\mathbf{m}})$.
 4:
lenz – int64int32nag_int scalar
The dimension of the array
z as declared in the (sub)program from which
nag_opt_nlp1_sparse_solve (e04ug) is called.
Constraint:
lenz ≥ 500${\mathbf{lenz}}\ge 500$.
The amounts of workspace provided (i.e.,
leniz and
lenz) and required (i.e.,
miniz and
minz) are (by default) output on the current advisory message unit (as defined by
nag_file_set_unit_advisory (x04ab)). Since the minimum values of
leniz and
lenz required to start solving the problem are returned in
miniz and
minz respectively, you may prefer to obtain appropriate values from the output of a preliminary run with
leniz set to
max (500,n + m)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(500,{\mathbf{n}}+{\mathbf{m}})$ and/or
lenz set to
500$500$. (
nag_opt_nlp1_sparse_solve (e04ug) will then terminate with
ifail = 15${\mathbf{ifail}}={\mathbf{15}}$ or
16${\mathbf{16}}$.)
 5:
user – Any MATLAB object
user is not used by
nag_opt_nlp1_sparse_solve (e04ug), but is passed to
confun and
objfun. Note that for large objects it may be more efficient to use a global variable which is accessible from the mfiles than to use
user.
Input Parameters Omitted from the MATLAB Interface
 iz z iuser ruser
Output Parameters
 1:
a(nnz) – double array
Elements in the nonlinear part corresponding to nonlinear Jacobian variables are overwritten.
 2:
ns – int64int32nag_int scalar
The final number of superbasics.
 3:
xs(n + m${\mathbf{n}}+{\mathbf{m}}$) – double array
The final values of the variables and slacks (x,s)$(x,s)$.
 4:
istate(n + m${\mathbf{n}}+{\mathbf{m}}$) – int64int32nag_int array
The final states of the variables and slacks
(x,s)$(x,s)$. The significance of each possible value of
istate(j)${\mathbf{istate}}\left(j\right)$ is as follows:
istate(j)${\mathbf{istate}}\left(j\right)$  State of variable j$j$  Normal value of xs(j)${\mathbf{xs}}\left(j\right)$ 
0$0$  Nonbasic  bl(j)${\mathbf{bl}}\left(j\right)$ 
1$1$  Nonbasic  bu(j)${\mathbf{bu}}\left(j\right)$ 
2$2$  Superbasic  Between bl(j)${\mathbf{bl}}\left(j\right)$ and bu(j)${\mathbf{bu}}\left(j\right)$ 
3$3$  Basic  Between bl(j)${\mathbf{bl}}\left(j\right)$ and bu(j)${\mathbf{bu}}\left(j\right)$ 
If
ninf = 0${\mathbf{ninf}}=0$, basic and superbasic variables may be outside their bounds by as much as the value of the optional parameter
Minor Feasibility Tolerance. Note that if scaling is specified, the optional parameter
Minor Feasibility Tolerance applies to the variables of the
scaled problem. In this case, the variables of the original problem may be as much as
0.1$0.1$ outside their bounds, but this is unlikely unless the problem is very badly scaled.
Very occasionally some nonbasic variables may be outside their bounds by as much as the optional parameter
Minor Feasibility Tolerance and there may be some nonbasic variables for which
xs(j)${\mathbf{xs}}\left(j\right)$ lies strictly between its bounds.
If
ninf > 0${\mathbf{ninf}}>0$, some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by
sinf if scaling was not used).
 5:
clamda(n + m${\mathbf{n}}+{\mathbf{m}}$) – double array
A set of Lagrange multipliers for the bounds on the variables (
reduced costs) and the general constraints (
shadow costs). More precisely, the first
n elements contain the multipliers for the bounds on the variables, the next
ncnln elements contain the multipliers for the nonlinear constraints
F(x)$F\left(x\right)$ (if any) and the next (
m − ncnln${\mathbf{m}}{\mathbf{ncnln}}$) elements contain the multipliers for the linear constraints
Gx$Gx$ and the free row (if any).
 6:
miniz – int64int32nag_int scalar
The minimum value of
leniz required to start solving the problem. If
ifail = 12${\mathbf{ifail}}={\mathbf{12}}$,
nag_opt_nlp1_sparse_solve (e04ug) may be called again with
leniz suitably larger than
miniz. (The bigger the better, since it is not certain how much workspace the basis factors need.)
 7:
minz – int64int32nag_int scalar
The minimum value of
lenz required to start solving the problem. If
ifail = 13${\mathbf{ifail}}={\mathbf{13}}$,
nag_opt_nlp1_sparse_solve (e04ug) may be called again with
lenz suitably larger than
minz. (The bigger the better, since it is not certain how much workspace the basis factors need.)
 8:
ninf – int64int32nag_int scalar
The number of constraints that lie outside their bounds by more than the value of the optional parameter
Minor Feasibility Tolerance.
If the
linear constraints are infeasible, the sum of the infeasibilities of the linear constraints is minimized subject to the upper and lower bounds being satisfied. In this case,
ninf contains the number of elements of
Gx$Gx$ that lie outside their upper or lower bounds. Note that the nonlinear constraints are not evaluated.
Otherwise, the sum of the infeasibilities of the
nonlinear constraints is minimized subject to the linear constraints and the upper and lower bounds being satisfied. In this case,
ninf contains the number of elements of
F(x)$F\left(x\right)$ that lie outside their upper or lower bounds.
 9:
sinf – double scalar
The sum of the infeasibilities of constraints that lie outside their bounds by more than the value of the optional parameter
Minor Feasibility Tolerance.
 10:
obj – double scalar
The value of the objective function.
 11:
user – Any MATLAB object
 12:
lwsav(20$20$) – logical array
 13:
iwsav(550$550$) – int64int32nag_int array
 14:
rwsav(550$550$) – double array
 15:
ifail – int64int32nag_int scalar
ifail = 0${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see
[Error Indicators and Warnings]).
nag_opt_nlp1_sparse_solve (e04ug) returns with
ifail = 0${\mathbf{ifail}}={\mathbf{0}}$ if the iterates have converged to a point
x$x$ that satisfies the firstorder Kuhn–Karesh–Tucker conditions (see
Section [Major Iteration Printout]) to the accuracy requested by the optional parameters
Major Feasibility Tolerance (
default value = sqrt(ε)$\text{default value}=\sqrt{\epsilon}$) and
Major Optimality Tolerance (
default value = sqrt(ε)$\text{default value}=\sqrt{\epsilon}$).
Error Indicators and Warnings
Note: nag_opt_nlp1_sparse_solve (e04ug) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
Cases prefixed with W are classified as warnings and
do not generate an error of type NAG:error_n. See nag_issue_warnings.
 W ifail < 0${\mathbf{ifail}}<0$
A negative value of
ifail indicates an exit from
nag_opt_nlp1_sparse_solve (e04ug) because you set
mode < 0${\mathbf{mode}}<0$ in
objfun or
confun. The value of
ifail will be the same as your setting of
mode.
 W ifail = 1${\mathbf{ifail}}=1$
The problem is infeasible. The general constraints cannot all be satisfied simultaneously to within the values of the optional parameters
Major Feasibility Tolerance (
default value = sqrt(ε)$\text{default value}=\sqrt{\epsilon}$) and
Minor Feasibility Tolerance (
default value = sqrt(ε)$\text{default value}=\sqrt{\epsilon}$).
 W ifail = 2${\mathbf{ifail}}=2$
The problem is unbounded (or badly scaled). The objective function is not bounded below (or above in the case of maximization) in the feasible region because a nonbasic variable can apparently be increased or decreased by an arbitrary amount without causing a basic variable to violate a bound. Add an upper or lower bound to the variable (whose index is printed by default by nag_opt_nlp1_sparse_solve (e04ug)) and rerun nag_opt_nlp1_sparse_solve (e04ug).
 W ifail = 3${\mathbf{ifail}}=3$
The problem may be unbounded. Check that the values of the optional parameters
Unbounded Objective (
default value = 10^{15}$\text{default value}={10}^{15}$) and
Unbounded Step Size (
default value = max (bigbnd,10^{20})$\text{default value}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}(\mathit{bigbnd},{10}^{20})$) are not too small. This exit also implies that the objective function is not bounded below (or above in the case of maximization) in the feasible region defined by expanding the bounds by the value of the optional parameter
Violation Limit (
default value = 10.0$\text{default value}=10.0$).
 ifail = 4${\mathbf{ifail}}=4$
Too many iterations. The values of the optional parameters
Major Iteration Limit (
default value = 1000$\text{default value}=1000$) and/or
Iteration Limit (
default value = 10000$\text{default value}=10000$) are too small.
 W ifail = 5${\mathbf{ifail}}=5$
Feasible solution found, but requested accuracy could not be achieved. Check that the value of the optional parameter
Major Optimality Tolerance (
default value = sqrt(ε)$\text{default value}=\sqrt{\epsilon}$) is not too small (say,
< ε$\text{}<\epsilon $).
 ifail = 6${\mathbf{ifail}}=6$
The value of the optional parameter
Superbasics Limit (
default value = min (500,n + 1)$\text{default value}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(500,\stackrel{}{n}+1)$) is too small.
 ifail = 7${\mathbf{ifail}}=7$

An input parameter is invalid.
 ifail = 8${\mathbf{ifail}}=8$
The usersupplied derivatives of the objective function computed by
objfun appear to be incorrect. Check that
objfun has been coded correctly and that all relevant elements of the objective gradient have been assigned their correct values.
 ifail = 9${\mathbf{ifail}}=9$
The usersupplied derivatives of the nonlinear constraint functions computed by
confun appear to be incorrect. Check that
confun has been coded correctly and that all relevant elements of the nonlinear constraint Jacobian have been assigned their correct values.
 W ifail = 10${\mathbf{ifail}}=10$
The current point cannot be improved upon. Check that
objfun and
confun have been coded correctly and that they are consistent with the value of the optional parameter
Derivative Level (
default value = 3$\text{default value}=3$).
 ifail = 11${\mathbf{ifail}}=11$
Numerical error in trying to satisfy the linear constraints (or the linearized nonlinear constraints). The basis is very illconditioned.
 ifail = 12${\mathbf{ifail}}=12$
Not enough integer workspace for the basis factors. Increase
leniz and rerun
nag_opt_nlp1_sparse_solve (e04ug).
 ifail = 13${\mathbf{ifail}}=13$
Not enough real workspace for the basis factors. Increase
lenz and rerun
nag_opt_nlp1_sparse_solve (e04ug).
 W ifail = 14${\mathbf{ifail}}=14$
The basis is singular after
15$15$ attempts to factorize it (and adding slacks where necessary). Either the problem is badly scaled or the value of the optional parameter
LU Factor Tolerance (
default value = 5.0$\text{default value}=5.0$ or
100.0$100.0$) is too large.
 W ifail = 15${\mathbf{ifail}}=15$
Not enough integer workspace to start solving the problem. Increase
leniz to at least
miniz and rerun
nag_opt_nlp1_sparse_solve (e04ug).
 W ifail = 16${\mathbf{ifail}}=16$
Not enough real workspace to start solving the problem. Increase
lenz to at least
minz and rerun
nag_opt_nlp1_sparse_solve (e04ug).
 ifail = 17${\mathbf{ifail}}=17$
An unexpected error has occurred. Please contact
NAG.
Accuracy
If the value of the optional parameter
Major Optimality Tolerance is set to
10^{ − d}${10}^{d}$ (
default value = sqrt(ε)$\text{default value}=\sqrt{\epsilon}$) and
ifail = 0${\mathbf{ifail}}={\mathbf{0}}$ on exit, then the final value of
f(x)$f\left(x\right)$ should have approximately
d$d$ correct significant digits.
Further Comments
This section contains a description of the printed output.
Major Iteration Printout
This section describes the intermediate printout and final printout produced by the major iterations of
nag_opt_nlp1_sparse_solve (e04ug). The intermediate printout is a subset of the monitoring information produced by the function at every iteration (see
Section [Description of Monitoring Information]). You can control the level of printed output (see the description of the optional parameter
Major Print Level).
Note that the intermediate printout and final printout are produced only if
Major Print Level ≥ 10${\mathbf{Major\; Print\; Level}}\ge 10$ (the default for
nag_opt_nlp1_sparse_solve (e04ug), by default no output is produced by
nag_opt_nlp1_sparse_solve (e04ug)).
The following line of summary output (
< 80$\text{}<80$ characters) is produced at every major iteration. In all cases, the values of the quantities printed are those in effect
on completion of the given iteration.
Maj 
is the major iteration count.

Mnr 
is the number of minor iterations required by the feasibility and optimality phases of the QP subproblem. Generally, Mnr will be 1$1$ in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution
(see Section [Algorithmic Details]).
Note that Mnr may be greater than the optional parameter Minor Iteration Limit if some iterations are required for the feasibility phase.

Step 
is the step α_{k}${\alpha}_{k}$ taken along the computed search direction. On reasonably wellbehaved problems, the unit step (i.e., α_{k} = 1${\alpha}_{k}=1$) will be taken as the solution is approached.

Merit Function 
is the value of the augmented Lagrangian merit function (6) at the current iterate. This function will decrease at each iteration unless it was necessary to increase the penalty parameters
(see Section [Major Iteration Printout]).
As the solution is approached, Merit Function will converge to the value of the objective function at the solution.
In elastic mode (see Section [Treatment of Constraint Infeasibilities]) then the merit function is a composite function involving the constraint violations weighted by the value of the optional parameter Elastic Weight. If there are no nonlinear constraints present then this entry contains Objective, the value of the objective function f(x)$f\left(x\right)$. In this case, f(x)$f\left(x\right)$ will decrease monotonically to its optimal value.

Feasibl 
is the value of rowerr, the largest element of the scaled nonlinear constraint residual vector defined in the description of the optional parameter Major Feasibility Tolerance. The solution is regarded as ‘feasible’ if Feasibl is less than (or equal to) the optional parameter Major Feasibility Tolerance. Feasibl will be approximately zero in the neighbourhood of a solution. If there are no nonlinear constraints present, all iterates are feasible and this entry is not printed.

Optimal 
is the value of maxgap, the largest element of the maximum complementarity gap vector defined in the description of the optional parameter Major Optimality Tolerance. The Lagrange multipliers are regarded as ‘optimal’ if Optimal is less than (or equal to) the optional parameter Major Optimality Tolerance. Optimal will be approximately zero in the neighbourhood of a solution.

Cond Hz 
is an estimate of the condition number of the reduced Hessian of the Lagrangian (not printed if ncnln and nonln are both zero). It is the square of the ratio between the largest and smallest diagonal elements of the upper triangular matrix R$R$. This constitutes a lower bound on the condition number of the matrix R^{T}R${R}^{\mathrm{T}}R$ that approximates the reduced Hessian. The larger this number, the more difficult the problem.

PD 
is a twoletter indication of the status of the convergence tests involving the feasibility and optimality of the iterates defined in the descriptions of the optional parameters Major Feasibility Tolerance and Major Optimality Tolerance. Each letter is T if the test is satisfied and F otherwise. The tests indicate whether the values of Feasibl and Optimal are sufficiently small. For example, TF or TT is printed if there are no nonlinear constraints present (since all iterates are feasible). If either indicator is F when nag_opt_nlp1_sparse_solve (e04ug) terminates with ifail = 0${\mathbf{ifail}}={\mathbf{0}}$, you should check the solution carefully.

M 
is printed if an extra evaluation of usersupplied functions objfun and confun was needed in order to define an acceptable positive definite quasiNewton update to the Hessian of the Lagrangian. This modification is only performed when there are nonlinear constraints present.

m 
is printed if, in addition, it was also necessary to modify the update to include an augmented Lagrangian term.

s 
is printed if a selfscaled BFGS (Broyden–Fletcher–Goldfarb–Shanno) update was performed. This update is always used when the Hessian approximation is diagonal and hence always follows a Hessian reset.

S 
is printed if, in addition, it was also necessary to modify the selfscaled update in order to maintain positivedefiniteness.

n 
is printed if no positive definite BFGS update could be found, in which case the approximate Hessian is unchanged from the previous iteration.

r 
is printed if the approximate Hessian was reset after 10$10$ consecutive major iterations in which no BFGS update could be made. The diagonal elements of the approximate Hessian are retained if at least one update has been performed since the last reset. Otherwise, the approximate Hessian is reset to the identity matrix.

R 
is printed if the approximate Hessian has been reset by discarding all but its diagonal elements. This reset will be forced periodically by the values of the optional parameters Hessian Frequency and Hessian Updates. However, it may also be necessary to reset an illconditioned Hessian from time to time.

l 
is printed if the change in the norm of the variables was greater than the value defined by the optional parameter Major Step Limit. If this output occurs frequently during later iterations, it may be worthwhile increasing the value of Major Step Limit.

c 
is printed if central differences have been used to compute the unknown elements of the objective and constraint gradients. A switch to central differences is made if either the linesearch gives a small step, or x$x$ is close to being optimal. In some cases, it may be necessary to resolve the QP subproblem with the central difference gradient and Jacobian.

u 
is printed if the QP subproblem was unbounded.

t 
is printed if the minor iterations were terminated after the number of iterations specified by the value of the optional parameter Minor Iteration Limit was reached.

i 
is printed if the QP subproblem was infeasible when the function was not in elastic mode. This event triggers the start of nonlinear elastic mode, which remains in effect for all subsequent iterations. Once in elastic mode, the QP subproblems are associated with the elastic problem (8) (see Section [Treatment of Constraint Infeasibilities]). It is also printed if the minimizer of the elastic subproblem does not satisfy the linearized constraints when the function is already in elastic mode. (In this case, a feasible point for the usual QP subproblem may or may not exist.)

w 
is printed if a weak solution of the QP subproblem was found.

The final printout includes a listing of the status of every variable and constraint.
The following describes the printout for each variable. A full stop (.) is printed for any numerical value that is zero.
Variable 
gives the name of the variable. If nname = 1${\mathbf{nname}}=1$, a default name is assigned to the j$\mathit{j}$th variable, for j = 1,2, … ,n$\mathit{j}=1,2,\dots ,n$. If nname = n + m${\mathbf{nname}}={\mathbf{n}}+{\mathbf{m}}$, the name supplied in names(j)${\mathbf{names}}\left(\mathit{j}\right)$ is assigned to the j$\mathit{j}$th variable.

State 
gives the state of the variable (LL if nonbasic on its lower bound, UL if nonbasic on its upper bound, EQ if nonbasic and fixed, FR if nonbasic and strictly between its bounds, BS if basic and SBS if superbasic).
A key is sometimes printed before State.
Note that unless the optional parameter Scale Option = 0${\mathbf{Scale\; Option}}=0$ is specified, the tests for assigning a key are applied to the variables of the scaled problem.
A 
Alternative optimum possible. The variable is nonbasic, but its reduced gradient is essentially zero. This means that if the variable were allowed to start moving away from its current value, there would be no change in the value of the objective function. The values of the basic and superbasic variables might change, giving a genuine alternative solution. The values of the Lagrange multipliers might also change.

D 
Degenerate. The variable is basic, but it is equal to (or very close to) one of its bounds.

I 
Infeasible. The variable is basic and is currently violating one of its bounds by more than the value of the optional parameter Minor Feasibility Tolerance.

N 
Not precisely optimal. The variable is nonbasic. Its reduced gradient is larger than the value of the optional parameter Major Feasibility Tolerance.


Value 
is the value of the variable at the final iteration.

Lower Bound 
is the lower bound specified for the variable. None indicates that bl(j) ≤ − bigbnd${\mathbf{bl}}\left(j\right)\le \mathit{bigbnd}$.

Upper Bound 
is the upper bound specified for the variable. None indicates that bu(j) ≥ bigbnd${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$.

Lagr Mult 
is the Lagrange multiplier for the associated bound. This will be zero if State is FR. If x$x$ is optimal, the multiplier should be nonnegative if State is LL, nonpositive if State is UL and zero if State is BS or SBS.

Residual 
is the difference between the variable Value and the nearer of its (finite) bounds bl(j)${\mathbf{bl}}\left(j\right)$ and bu(j)${\mathbf{bu}}\left(j\right)$. A blank entry indicates that the associated variable is not bounded (i.e., bl(j) ≤ − bigbnd${\mathbf{bl}}\left(j\right)\le \mathit{bigbnd}$ and bu(j) ≥ bigbnd${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$).

The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’,
n$n$ replaced by
m$m$,
names(j)${\mathbf{names}}\left(j\right)$ replaced by
names(n + j)${\mathbf{names}}\left(n+j\right)$,
bl(j)${\mathbf{bl}}\left(j\right)$ and
bu(j)${\mathbf{bu}}\left(j\right)$ are replaced by
bl(n + j)${\mathbf{bl}}\left(n+j\right)$ and
bu(n + j)${\mathbf{bu}}\left(n+j\right)$ respectively. The heading is changed as follows:
Constrnt 
gives the name of the general constraint.

Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.
Minor Iteration Printout
This section describes the printout produced by the minor iterations of
nag_opt_nlp1_sparse_solve (e04ug), which involve solving a QP subproblem at every major iteration. (Further details can be found in
Section [Major Iteration Printout].) The printout is a subset of the monitoring information produced by the function at every iteration (see
Section [Description of Monitoring Information]). You can control the level of printed output (see the description of the optional parameter
Minor Print Level). Note that the printout is produced only if
Minor Print Level ≥ 1${\mathbf{Minor\; Print\; Level}}\ge 1$ (
default value = 0$\text{default value}=0$, which produces no output).
The following line of summary output (
< 80$\text{}<80$ characters) is produced at every minor iteration. In all cases, the values of the quantities printed are those in effect
on completion of the given iteration of the QP subproblem.
Itn 
is the iteration count.

Step 
is the step taken along the computed search direction.

Ninf 
is the number of infeasibilities. This will not increase unless the iterations are in elastic mode. Ninf will be zero during the optimality phase.

Sinf 
is the value of the sum of infeasibilities if Ninf is nonzero. This will be zero during the optimality phase.

Objective 
is the value of the current QP objective function when Ninf is zero and the iterations are not in elastic mode. The switch to elastic mode is indicated by a change in the heading to Composite Obj.

Composite Obj 
is the value of the composite objective function (9) when the iterations are in elastic mode. This function will decrease monotonically at each iteration.

Norm rg 
is the Euclidean norm of the reduced gradient of the QP objective function. During the optimality phase, this norm will be approximately zero after a unit step.

Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.
Example
Open in the MATLAB editor:
nag_opt_nlp1_sparse_solve_example
function nag_opt_nlp1_sparse_solve_example
n = int64(4);
m = int64(6);
ncnln = int64(3);
nonln = int64(4);
njnln = int64(2);
iobj = int64(6);
a = [1e25;1e25;1e25;1;1;1e25;1e25;1e25;1;1;3;1;1;2];
ha = [int64(1);2;3;5;4;1;2;3;5;4;6;1;2;6];
ka = [int64(1);6;11;13;15];
bl = [0.55;0.55;0;0;894.8;894.8;1294.8;0.55;0.55;1e25];
bu = [0.55;0.55;1200;1200;894.8;894.8;1294.8;1e25;1e25;1e25];
start = 'C';
names = {'Varble 1'; 'Varble 2'; 'Varble 3'; 'Varble 4'; 'NlnCon 1'; ...
'NlnCon 2'; 'NlnCon 3'; 'LinCon 1'; 'LinCon 2'; 'Free Row'};
ns = int64(0);
xs = [0;0;0;0;0;0;0;0;0;0];
istate = zeros(10, 1, 'int64');
clamda = [0;0;0;0;0;0;0;0;0;0];
leniz = int64(1000);
lenz = int64(1000);
[cwsav,lwsav,iwsav,rwsav,ifail] = nag_opt_init('nag_opt_nlp1_sparse_solve');
disp('First call to nag_opt_nlp1_sparse_solve')
[aOut, nsOut, xsOut, istateOut, clamdaOut, miniz, minz, ninf, sinf, obj, ...
user, lwsavOut, iwsavOut, rwsavOut, ifail] = ...
nag_opt_nlp1_sparse_solve(@confun, @objfun, n, m, ncnln, nonln, njnln, ...
iobj, a, ha, ka, bl, bu, start, names, ns, xs, istate, ...
clamda, lwsav, iwsav, rwsav);
if (ifail == 15  ifail == 16)
disp('Second call to nag_opt_nlp1_sparse_solve')
[aOut, nsOut, xsOut, istateOut, clamdaOut, miniz, minz, ninf, sinf, obj, ...
user, lwsavOut, iwsavOut, rwsavOut, ifail] = ...
nag_opt_nlp1_sparse_solve(@confun, @objfun, n, m, ncnln, nonln, njnln, ...
iobj, a, ha, ka, bl, bu, start, names, ns, xs, istate, ...
clamda, lwsav, iwsav, rwsav, 'lenz', int64(10*double(minz)), ...
'leniz', int64(10*double(miniz)));
obj
ifail
end
function [mode, f, fjac, user] = ...
confun(mode, ncnln, njnln, nnzjac, x, fjac, nstate, user)
f = zeros(ncnln, 1);
if (mode == 0  mode == 2)
f(1) = 1000*sin(x(1)0.25) + 1000*sin(x(2)0.25);
f(2) = 1000*sin(x(1)0.25) + 1000*sin(x(1)x(2)0.25);
f(3) = 1000*sin(x(2)x(1)0.25) + 1000*sin(x(2)0.25);
end
if (mode == 1  mode == 2)
fjac(1) = 1000*cos(x(1)0.25);
fjac(2) = 1000*cos(x(1)0.25) + 1000*cos(x(1)x(2)0.25);
fjac(3) = 1000*cos(x(2)x(1)0.25);
fjac(4) = 1000*cos(x(2)0.25);
fjac(5) = 1000*cos(x(1)x(2)0.25);
fjac(6) = 1000*cos(x(2)x(1)0.25) + 1000*cos(x(2) 0.25);
end
function [mode, objf, objgrd, user] = ...
objfun(mode, nonln, x, objgrd, nstate, user)
if (mode == 0  mode == 2)
objf = 1.0e6*x(3)^3 + 2.0e6*x(4)^3/3;
end
if (mode == 1  mode == 2)
objgrd(1) = 0;
objgrd(2) = 0;
objgrd(3) = 3.0e6*x(3)^2;
objgrd(4) = 2.0e6*x(4)^2;
end
First call to nag_opt_nlp1_sparse_solve
Warning: nag_opt_nlp1_sparse_solve (e04ug) returned a warning indicator (15)
Second call to nag_opt_nlp1_sparse_solve
obj =
5.1265e+03
ifail =
0
Open in the MATLAB editor:
e04ug_example
function e04ug_example
n = int64(4);
m = int64(6);
ncnln = int64(3);
nonln = int64(4);
njnln = int64(2);
iobj = int64(6);
a = [1e25;
1e25;
1e25;
1;
1;
1e25;
1e25;
1e25;
1;
1;
3;
1;
1;
2];
ha = [int64(1);2;3;5;4;1;2;3;5;4;6;1;2;6];
ka = [int64(1);6;11;13;15];
bl = [0.55;0.55;0;0;894.8;894.8;1294.8;0.55;0.55;1e25];
bu = [0.55;0.55;1200;1200;894.8;894.8;1294.8;1e25;1e25;1e25];
start = 'C';
names = {'Varble 1'; 'Varble 2'; 'Varble 3'; 'Varble 4'; 'NlnCon 1'; ...
'NlnCon 2'; 'NlnCon 3'; 'LinCon 1'; 'LinCon 2'; 'Free Row'};
ns = int64(0);
xs = [0;0;0;0;0;0;0;0;0;0];
istate = zeros(10, 1, 'int64');
clamda = [0;0;0;0;0;0;0;0;0;0];
leniz = int64(1000);
lenz = int64(1000);
[cwsav,lwsav,iwsav,rwsav,ifail] = e04wb('e04ug');
disp('First call to e04ug')
[aOut, nsOut, xsOut, istateOut, clamdaOut, miniz, minz, ninf, sinf, obj, ...
user, lwsavOut, iwsavOut, rwsavOut, ifail] = ...
e04ug(@confun, @objfun, n, m, ncnln, nonln, njnln, ...
iobj, a, ha, ka, bl, bu, start, names, ns, xs, istate, ...
clamda, lwsav, iwsav, rwsav);
if (ifail == 15  ifail == 16)
disp('Second call to e04ug')
[aOut, nsOut, xsOut, istateOut, clamdaOut, miniz, minz, ninf, sinf, obj, ...
user, lwsavOut, iwsavOut, rwsavOut, ifail] = ...
e04ug(@confun, @objfun, n, m, ncnln, nonln, njnln, ...
iobj, a, ha, ka, bl, bu, start, names, ns, xs, istate, clamda, ...
lwsav, iwsav, rwsav, 'lenz', int64(10*double(minz)), 'leniz', int64(10*double(miniz)));
obj
ifail
end
function [mode, f, fjac, user] = ...
confun(mode, ncnln, njnln, nnzjac, x, fjac, nstate, user)
f = zeros(ncnln, 1);
if (mode == 0  mode == 2)
f(1) = 1000*sin(x(1)0.25) + 1000*sin(x(2)0.25);
f(2) = 1000*sin(x(1)0.25) + 1000*sin(x(1)x(2)0.25);
f(3) = 1000*sin(x(2)x(1)0.25) + 1000*sin(x(2)0.25);
end
if (mode == 1  mode == 2)
fjac(1) = 1000*cos(x(1)0.25);
fjac(2) = 1000*cos(x(1)0.25) + 1000*cos(x(1)x(2)0.25);
fjac(3) = 1000*cos(x(2)x(1)0.25);
fjac(4) = 1000*cos(x(2)0.25);
fjac(5) = 1000*cos(x(1)x(2)0.25);
fjac(6) = 1000*cos(x(2)x(1)0.25) + 1000*cos(x(2) 0.25);
end
function [mode, objf, objgrd, user] = objfun(mode, nonln, x, objgrd, nstate, user)
if (mode == 0  mode == 2)
objf = 1.0e6*x(3)^3 + 2.0e6*x(4)^3/3;
end
if (mode == 1  mode == 2)
objgrd(1) = 0;
objgrd(2) = 0;
objgrd(3) = 3.0e6*x(3)^2;
objgrd(4) = 2.0e6*x(4)^2;
end
First call to e04ug
Warning: nag_opt_nlp1_sparse_solve (e04ug) returned a warning indicator (15)
Second call to e04ug
obj =
5.1265e+03
ifail =
0
the remainder of this document is intended for more advanced users. Section [Algorithmic Details] contains a detailed description of the algorithm which may be needed in order to understand Sections [Optional Parameters] and [Description of Monitoring Information]. Section [Optional Parameters] describes the optional parameters which may be set by calls to nag_opt_nlp1_sparse_option_string (e04uj). Section [Description of Monitoring Information] describes the quantities which can be requested to monitor the course of the computation.
Algorithmic Details
This section contains a detailed description of the method used by nag_opt_nlp1_sparse_solve (e04ug).
Overview
Here we briefly summarise the main features of the method and introduce some terminology. Where possible, explicit reference is made to the names of variables that are parameters of the function or appear in the printed output. Further details can be found in
Gill et al. (2002).
At a solution of
(1), some of the constraints will be
active, i.e., satisfied exactly. Let
and
G$\mathcal{G}$ denote the set of indices of
r(x)$r\left(x\right)$ corresponding to active constraints at an arbitrary point
x$x$. Let
r_{j}^{ ′ }(x)${r}_{j}^{\prime}\left(x\right)$ denote the usual
derivative of
r_{j}(x)${r}_{j}\left(x\right)$, which is the row vector of first partial derivatives of
r_{j}(x)${r}_{j}\left(x\right)$ (see
Ortega and Rheinboldt (1970)). The vector
r_{j}^{ ′ }(x)${r}_{j}^{\prime}\left(x\right)$ comprises the
j$j$th row of
r^{′}(x)${r}^{\prime}\left(x\right)$ so that
where
J(x)$J\left(x\right)$ is the Jacobian of
F(x)$F\left(x\right)$.
A point
x$x$ is a
firstorder Kuhn–Karesh–Tucker (KKT) point for
(1) (see
Powell (1974)) if the following conditions hold:
(a) 
x$x$ is feasible; 
(b) 
there exists a vector λ$\lambda $ (the Lagrange multiplier vector for the bound and general constraints) such that
where g$g$ is the gradient of f$f$ evaluated at x$x$; 
(c) 
the Lagrange multiplier λ_{j}${\lambda}_{j}$ associated with the j$j$th constraint satisfies λ_{j} = 0${\lambda}_{j}=0$ if l_{j} < r_{j}(x) < u_{j}${l}_{j}<{r}_{j}\left(x\right)<{u}_{j}$; λ_{j} ≥ 0${\lambda}_{j}\ge 0$ if l_{j} = r_{j}(x)${l}_{j}={r}_{j}\left(x\right)$; λ_{j} ≤ 0${\lambda}_{j}\le 0$ if r_{j}(x) = u_{j}${r}_{j}\left(x\right)={u}_{j}$; and λ_{j}${\lambda}_{j}$ can have any value if l_{j} = u_{j}${l}_{j}={u}_{j}$. 
An equivalent statement of the condition
(4) is
where
Z$Z$ is a matrix defined as follows. Consider the set
N$N$ of vectors orthogonal to the gradients of the active constraints, i.e.,
The columns of
Z$Z$ may then be taken as any basis for the vector space
N$N$. The vector
Z^{T}g${Z}^{\mathrm{T}}g$ is termed the
reduced gradient of
f$f$ at
x$x$. Certain additional conditions must be satisfied in order for a firstorder KKT point to be a solution of
(1) (see
Powell (1974)).
The basic structure of
nag_opt_nlp1_sparse_solve (e04ug) involves
major and
minor iterations. The major iterations generate a sequence of iterates
{x_{k}}$\left\{{x}_{k}\right\}$ that satisfy the linear constraints and converge to a point
x^{*}${x}^{*}$ that satisfies the firstorder KKT optimality conditions. At each iterate a QP subproblem is used to generate a search direction towards the next iterate (
x_{k + 1}${x}_{k+1}$). The constraints of the subproblem are formed from the linear constraints
Gx − s_{L} = 0$Gx{s}_{L}=0$ and the nonlinear constraint linearization
where
F^{′}(x_{k})${F}^{\prime}\left({x}_{k}\right)$ denotes the
Jacobian matrix, whose rows are the first partial derivatives of
F(x)$F\left(x\right)$ evaluated at the point
x_{k}${x}_{k}$. The QP constraints therefore comprise the
m$m$ linear constraints
where
x$x$ and
s = (s_{N},s_{L})^{T}$s={({s}_{N},{s}_{L})}^{\mathrm{T}}$ are bounded above and below by
u$u$ and
l$l$ as before. If the
m$m$ by
n$n$ matrix
A$A$ and
m$m$ element vector
b$b$ are defined as
then the QP subproblem can be written as
where
q(x)$q\left(x\right)$ is a quadratic approximation to a modified Lagrangian function (see
Gill et al. (2002)).
The linear constraint matrix
A$A$ is stored in the arrays
a,
ha and
ka (see
Section [Parameters]). This allows you to specify the sparsity pattern of nonzero elements in
F^{′}(x)${F}^{\prime}\left(x\right)$ and
G$G$ and to identify any nonzero elements that remain constant throughout the minimization.
Solving the QP subproblem is itself an iterative procedure, with the
minor iterations of an SQP method being the iterations of the QP method. At each minor iteration, the constraints
Ax − s = b$Axs=b$ are (conceptually) partitioned into the form
where the
basis matrix
B$B$ is square and nonsingular. The elements of
x_{B}${x}_{B}$,
x_{S}${x}_{S}$ and
x_{N}${x}_{N}$ are called the
basic,
superbasic and
nonbasic variables respectively; they are a permutation of the elements of
x$x$ and
s$s$. At a QP solution, the basic and superbasic variables will lie somewhere between their bounds, while the nonbasic variables will be equal to one of their upper or lower bounds. At each minor iteration,
x_{S}${x}_{S}$ is regarded as a set of independent variables that are free to move in any desired direction, namely one that will improve the value of the QP objective function
q(x)$q\left(x\right)$ or sum of infeasibilities (as appropriate). The basic variables are then adjusted in order to ensure that (
x,s$x,s$) continues to satisfy
Ax − s = b$Axs=b$. The number of superbasic variables (
n_{S}${n}_{S}$ say) therefore indicates the number of degrees of freedom remaining after the constraints have been satisfied. In broad terms,
n_{S}${n}_{S}$ is a measure of
how nonlinear the problem is. In particular,
n_{S}${n}_{S}$ will always be zero if there are no nonlinear constraints in
(1) and
f(x)$f\left(x\right)$ is linear.
If it appears that no improvement can be made with the current definition of B$B$, S$S$ and N$N$, a nonbasic variable is selected to be added to S$S$ and the process is repeated with the value of n_{S}${n}_{S}$ increased by one. At all stages, if a basic or superbasic variable encounters one of its bounds, the variable is made nonbasic and the value of n_{S}${n}_{S}$ decreased by one.
Associated with each of the
m$m$ equality constraints
Ax − s = b$Axs=b$ is a
dual variable
π_{i}${\pi}_{i}$. Similarly, each variable in
(x,s)$(x,s)$ has an associated
reduced gradient
d_{j}${d}_{j}$ (also known as a
reduced cost). The reduced gradients for the variables
x$x$ are the quantities
g − A^{T}π$g{A}^{\mathrm{T}}\pi $, where
g$g$ is the gradient of the QP objective function
q(x)$q\left(x\right)$; the reduced gradients for the slack variables
s$s$ are the dual variables
π$\pi $. The QP subproblem
(5) is optimal if
d_{j} ≥ 0${d}_{j}\ge 0$ for all nonbasic variables at their lower bounds,
d_{j} ≤ 0${d}_{j}\le 0$ for all nonbasic variables at their upper bounds and
d_{j} = 0${d}_{j}=0$ for other variables (including superbasics). In practice, an
approximate QP solution is found by slightly relaxing these conditions on
d_{j}${d}_{j}$ (see the description of the optional parameter
Minor Optimality Tolerance).
After a QP subproblem has been solved, new estimates of the solution to
(1) are computed using a linesearch on the augmented Lagrangian merit function
where
D$D$ is a diagonal matrix of penalty parameters. If (
x_{k},s_{k},π_{k}${x}_{k},{s}_{k},{\pi}_{k}$) denotes the current estimate of the solution and (
x̂,ŝ,π̂$\hat{x},\hat{s},\hat{\pi}$) denotes the optimal QP solution, the linesearch determines a step
α_{k}${\alpha}_{k}$ (where
0 < α_{k} ≤ 1$0<{\alpha}_{k}\le 1$) such that the new point
produces a
sufficient decrease in the merit function
(6). When necessary, the penalties in
D$D$ are increased by the minimumnorm perturbation that ensures descent for
M$\mathcal{M}$ (see
Gill et al. (1992)). As in
nag_opt_nlp2_solve (e04wd),
s_{N}${s}_{N}$ is adjusted to minimize the merit function as a function of
s$s$ before the solution of the QP subproblem. Further details can be found in
Eldersveld (1991) and
Gill et al. (1986).
Treatment of Constraint Infeasibilities
nag_opt_nlp1_sparse_solve (e04ug) makes explicit allowance for infeasible constraints. Infeasible linear constraints are detected first by solving a problem of the form
where
e = (1,1, … ,1)^{T}$e={(1,1,\dots ,1)}^{\mathrm{T}}$. This is equivalent to minimizing the sum of the general linear constraint violations subject to the simple bounds. (In the linear programming literature, the approach is often called
elastic programming.)
If the linear constraints are infeasible (i.e., v ≠ 0$v\ne 0$ or w ≠ 0$w\ne 0$), the function terminates without computing the nonlinear functions.
If the linear constraints are feasible, all subsequent iterates will satisfy the linear constraints. (Such a strategy allows linear constraints to be used to define a region in which
f(x)$f\left(x\right)$ and
F(x)$F\left(x\right)$ can be safely evaluated.) The function then proceeds to solve
(1) as given, using search directions obtained from a sequence of QP subproblems
(5). Each QP subproblem minimizes a quadratic model of a certain Lagrangian function subject to linearized constraints. An augmented Lagrangian merit function
(6) is reduced along each search direction to ensure convergence from any starting point.
The function enters ‘elastic’ mode if the QP subproblem proves to be infeasible or unbounded (or if the dual variables
π$\pi $ for the nonlinear constraints become ‘large’) by solving a problem of the form
where
is called a
composite objective and
γ$\gamma $ is a nonnegative parameter (the
elastic weight). If
γ$\gamma $ is sufficiently large, this is equivalent to minimizing the sum of the nonlinear constraint violations subject to the linear constraints and bounds. A similar
l_{1}${l}_{1}$ formulation of
(1) is fundamental to the
Sl_{1}${\mathrm{Sl}}_{1}$QP algorithm of
Fletcher (1984). See also
Conn (1973).
Optional Parameters
Several optional parameters in nag_opt_nlp1_sparse_solve (e04ug) define choices in the problem specification or the algorithm logic. In order to reduce the number of formal parameters of nag_opt_nlp1_sparse_solve (e04ug) these optional parameters have associated default values that are appropriate for most problems. Therefore, you need only specify those optional parameters whose values are to be different from their default values.
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in
Section [Description of the optional parameters].
Optional parameters may be specified by calling
nag_opt_nlp1_sparse_option_string (e04uj) before a call to
nag_opt_nlp1_sparse_solve (e04ug).
nag_opt_nlp1_sparse_option_string (e04uj) can be called to supply options directly, one call being necessary for each optional parameter. For example,
[lwsav, iwsav, rwsav, inform] = e04uj('Print Level = 5', lwsav, iwsav, rwsav);
nag_opt_nlp1_sparse_option_string (e04uj) should be consulted for a full description of this method of supplying optional parameters.
All optional parameters not specified by you are set to their default values. Optional parameters specified by you are unaltered by nag_opt_nlp1_sparse_solve (e04ug) (unless they define invalid values) and so remain in effect for subsequent calls to nag_opt_nlp1_sparse_solve (e04ug) from the calling program (unless altered by you).
Description of the Optional Parameters
For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
 the keywords, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted);
 a parameter value,
where the letters a$a$, i and r$i\text{ and}r$ denote options that take character, integer and real values respectively;
 the default value is used whenever the condition i ≥ 100000000$\lefti\right\ge 100000000$ is satisfied and where the symbol ε$\epsilon $ is a generic notation for machine precision (see nag_machine_precision (x02aj)).
Keywords and character values are case and white space insensitive.
Central Difference Interval r$r$Default = root(3,Function Precision)$\text{}=\sqrt[3]{{\mathbf{Function\; Precision}}}$Note that this option does not apply when
Derivative Level = 3${\mathbf{Derivative\; Level}}=3$.
The value of
r$r$ is used near an optimal solution in order to obtain more accurate (but more expensive) estimates of gradients. This requires twice as many function evaluations as compared to using forward differences (see optional parameter
Forward Difference Interval). The interval used for the
j$j$th variable is
h_{j} = r(1 + x_{j})${h}_{j}=r(1+\left{x}_{j}\right)$. The resulting gradient estimates should be accurate to
O(r^{2})$\mathit{O}\left({r}^{2}\right)$, unless the functions are badly scaled. The switch to central differences is indicated by
c at the end of each line of intermediate printout produced by the major iterations (see
Section [Major Iteration Printout]). See
Gill et al. (1981) for a discussion of the accuracy in finite difference approximations.
If r ≤ 0$r\le 0$, the default value is used.
Check Frequency i$i$Default = 60$\text{}=60$Every i$i$th minor iteration after the most recent basis factorization, a numerical test is made to see if the current solution (x,s)$(x,s)$ satisfies the general linear constraints (including any linearized nonlinear constraints). The constraints are of the form Ax − s = b$Axs=b$, where s$s$ is the set of slack variables. If the largest element of the residual vector r = b − Ax + s$r=bAx+s$ is judged to be too large, the current basis is refactorized and the basic variables recomputed to satisfy the general constraints more accurately.
If i < 0$i<0$, the default value is used. If i = 0$i=0$, the value i = 99999999$i=99999999$ is used and effectively no checks are made.
Crash Option i$i$Default = 0 or 3$\text{}=0\text{ or}3$The default value of
i$i$ is
0$0$ if there are any nonlinear constraints and
3$3$ otherwise. Note that this option does not apply when
start = 'W'${\mathbf{start}}=\text{'W'}$ (see
Section [Parameters]).
If
start = 'C'${\mathbf{start}}=\text{'C'}$, an internal Crash procedure is used to select an initial basis from various rows and columns of the constraint matrix
$\left(\begin{array}{cc}A& I\end{array}\right)$. The value of
i$i$ determines which rows and columns of
A$A$ are initially eligible for the basis and how many times the Crash procedure is called. Columns of
− I$I$ are used to pad the basis where necessary. The possible choices for
i$i$ are the following.
i$i$ 
Meaning 
0 
The initial basis contains only slack variables: B = I$B=I$. 
1 
The Crash procedure is called once (looking for a triangular basis in all rows and columns of A$A$). 
2 
The Crash procedure is called twice (if there are any nonlinear constraints). The first call looks for a triangular basis in linear rows and the iteration proceeds with simplex iterations until the linear constraints are satisfied. The Jacobian is then evaluated for the first major iteration and the Crash procedure is called again to find a triangular basis in the nonlinear rows (whilst retaining the current basis for linear rows). 
3 
The Crash procedure is called up to three times (if there are any nonlinear constraints). The first two calls treat linear equality constraints and linear inequality constraints separately. The Jacobian is then evaluated for the first major iteration and the Crash procedure is called again to find a triangular basis in the nonlinear rows (whilst retaining the current basis for linear rows). 
If i < 0$i<0$ or i > 3$i>3$, the default value is used.
If i ≥ 1$i\ge 1$, certain slacks on inequality rows are selected for the basis first. (If i ≥ 2$i\ge 2$, numerical values are used to exclude slacks that are close to a bound.) The Crash procedure then makes several passes through the columns of A$A$, searching for a basis matrix that is essentially triangular. A column is assigned to ‘pivot’ on a particular row if the column contains a suitably large element in a row that has not yet been assigned. (The pivot elements ultimately form the diagonals of the triangular basis.) For remaining unassigned rows, slack variables are inserted to complete the basis.
Crash Tolerance r$r$Default = 0.1$\text{}=0.1$The value r$r$ (0 ≤ r < 1$0\le r<1$) allows the Crash procedure to ignore certain ‘small’ nonzero elements in the columns of A$A$ while searching for a triangular basis. If a_{max}${a}_{\mathrm{max}}$ is the largest element in the j$j$th column, other nonzeros a_{ij}${a}_{ij}$ in the column are ignored if a_{ij} ≤ a_{max} × r$\left{a}_{ij}\right\le {a}_{\mathrm{max}}\times r$.
When r > 0$r>0$, the basis obtained by the Crash procedure may not be strictly triangular, but it is likely to be nonsingular and almost triangular. The intention is to obtain a starting basis containing more columns of A$A$ and fewer (arbitrary) slacks. A feasible solution may be reached earlier on some problems.
If r < 0$r<0$ or r ≥ 1$r\ge 1$, the default value is used.
Defaults This special keyword may be used to reset all optional parameters to their default values.
Derivative Level i$i$Default = 3$\text{}=3$This parameter indicates which nonlinear function gradients are provided in usersupplied functions
objfun and
confun. The possible choices for
i$i$ are the following.
i$i$ 
Meaning 
3 
All elements of the objective gradient and the constraint Jacobian are provided. 
2 
All elements of the constraint Jacobian are provided, but some (or all) elements of the objective gradient are not specified. 
1 
All elements of the objective gradient are provided, but some (or all) elements of the constraint Jacobian are not specified. 
0 
Some (or all) elements of both the objective gradient and the constraint Jacobian are not specified. 
The default value i = 3$i=3$ should be used whenever possible. It is the most reliable and will usually be the most efficient.
If
i = 0 or 2$i=0\text{ or}2$,
nag_opt_nlp1_sparse_solve (e04ug) will
estimate the unspecified elements of the objective gradient, using finite differences. This may simplify the coding of
objfun. However, the computation of finite difference approximations usually increases the total runtime substantially (since a call to
objfun is required for each unspecified element) and there is less assurance that an acceptable solution will be found.
If
i = 0 or 1$i=0\text{ or}1$,
nag_opt_nlp1_sparse_solve (e04ug) will approximate unspecified elements of the constraint Jacobian. For each column of the Jacobian, one call to
confun is needed to estimate all unspecified elements in that column (if any). For example, if the sparsity pattern of the Jacobian has the form
where ‘
*$*$’ indicates an element provided and ‘?’ indicates an unspecified element,
nag_opt_nlp1_sparse_solve (e04ug) will call
confun twice: once to estimate the missing element in column
2$2$ and again to estimate the two missing elements in column
3$3$. (Since columns
1$1$ and
4$4$ are known, they require no calls to
confun.)
At times, central differences are used rather than forward differences, in which case twice as many calls to
objfun and
confun are needed. (The switch to central differences is not under your control.)
If i < 0$i<0$ or i > 3$i>3$, the default value is used.
Derivative Linesearch DefaultNonderivative Linesearch At each major iteration, a linesearch is used to improve the value of the Lagrangian merit function
(6). The default linesearch uses safeguarded cubic interpolation and requires both function and gradient values in order to compute estimates of the step
α_{k}${\alpha}_{k}$. If some analytic derivatives are not provided or optional parameter
Nonderivative Linesearch is specified, a linesearch based upon safeguarded quadratic interpolation (which does not require the evaluation or approximation of any gradients) is used instead.
A nonderivative linesearch can be slightly less robust on difficult problems and it is recommended that the default be used if the functions and their derivatives can be computed at approximately the same cost. If the gradients are very expensive to compute relative to the functions however, a nonderivative linesearch may result in a significant decrease in the total runtime.
If optional parameter
Nonderivative Linesearch is selected,
nag_opt_nlp1_sparse_solve (e04ug) signals the evaluation of the linesearch by calling
objfun and
confun with
mode = 0${\mathbf{mode}}=0$. Once the linesearch is complete, the nonlinear functions are reevaluated with
mode = 2${\mathbf{mode}}=2$. If the potential savings offered by a nonderivative linesearch are to be fully realised, it is essential that
objfun and
confun be coded so that no derivatives are computed when
mode = 0${\mathbf{mode}}=0$.
Elastic Weight r$r$Default = 1.0$\text{}=1.0$ or 100.0$100.0$The default value of r$r$ is 100.0$100.0$ if there are any nonlinear constraints and 1.0$1.0$ otherwise.
This option defines the initial weight
γ$\gamma $ associated with problem
(8).
At any given major iteration k$k$, elastic mode is entered if the QP subproblem is infeasible or the QP dual variables (Lagrange multipliers) are larger in magnitude than r × (1 + ‖g(x_{k})‖_{2})$r\times (1+{\Vert g\left({x}_{k}\right)\Vert}_{2})$, where g$g$ is the objective gradient. In either case, the QP subproblem is resolved in elastic mode with γ = r × (1 + ‖g(x_{k})‖_{2})$\gamma =r\times (1+{\Vert g\left({x}_{k}\right)\Vert}_{2})$.
Thereafter,
γ$\gamma $ is increased (subject to a maximum allowable value) at any point that is optimal for problem
(8), but not feasible for problem
(1). After the
p$p$th increase,
γ = r × 10^{p} × (1 + ‖g(x_{k1})‖_{2})$\gamma =r\times {10}^{p}\times (1+{\Vert g\left({x}_{{k}_{1}}\right)\Vert}_{2})$, where
x_{k1}${x}_{{k}_{1}}$ is the iterate at which
γ$\gamma $ was first needed.
If r < 0$r<0$, the default value is used.
Expand Frequency i$i$Default = 10000$\text{}=10000$This option is part of the EXPAND anticycling procedure due to
Gill et al. (1989), which is designed to make progress even on highly degenerate problems.
For linear models, the strategy is to force a positive step at every iteration, at the expense of violating the constraints by a small amount. Suppose that the value of optional parameter
Minor Feasibility Tolerance is
δ$\delta $. Over a period of
i$i$ iterations, the feasibility tolerance actually used by
nag_opt_nlp1_sparse_solve (e04ug) (i.e., the
working feasibility tolerance) increases from
0.5δ$0.5\delta $ to
δ$\delta $ (in steps of
0.5δ / i$0.5\delta /i$).
For nonlinear models, the same procedure is used for iterations in which there is only one superbasic variable. (Cycling can only occur when the current solution is at a vertex of the feasible region.) Thus, zero steps are allowed if there is more than one superbasic variable, but otherwise positive steps are enforced.
Increasing the value of
i$i$ helps reduce the number of slightly infeasible nonbasic basic variables (most of which are eliminated during the resetting procedure). However, it also diminishes the freedom to choose a large pivot element (see optional parameter
Pivot Tolerance).
If i < 0$i<0$, the default value is used. If i = 0$i=0$, the value i = 99999999$i=99999999$ is used and effectively no anticycling procedure is invoked.
Factorization Frequency i$i$Default = 50 or 100$\text{}=50\text{ or}100$The default value of i$i$ is 50$50$ if there are any nonlinear constraints and 100$100$ otherwise.
If i > 0$i>0$, at most i$i$ basis changes will occur between factorizations of the basis matrix.
For linear problems, the basis factors are usually updated at every iteration. The default value i = 100$i=100$ is reasonable for typical problems, particularly those that are extremely sparse and wellscaled.
When the objective function is nonlinear, fewer basis updates will occur as the solution is approached. The number of iterations between basis factorizations will therefore increase. During these iterations a test is made regularly according to the value of optional parameter
Check Frequency to ensure that the general constraints are satisfied. If necessary, the basis will be refactorized before the limit of
i$i$ updates is reached.
If i ≤ 0$i\le 0$, the default value is used.
Infeasible Exit DefaultFeasible Exit Note that this option is ignored if the value of optional parameter
Major Iteration Limit is exceeded, or the linear constraints are infeasible.
If termination is about to occur at a point that does not satisfy the nonlinear constraints and optional parameter
Feasible Exit is selected, this option requests that additional iterations be performed in order to find a feasible point (if any) for the nonlinear constraints. This involves solving a feasible point problem in which the objective function is omitted.
Otherwise, this option requests no additional iterations be performed.
Minimize DefaultMaximize Feasible Point If optional parameter
Feasible Point is selected, this option attempts to find a feasible point (if any) for the nonlinear constraints by omitting the objective function. It can also be used to check whether the nonlinear constraints are feasible.
Otherwise, this option specifies the required direction of the optimization. It applies to both linear and nonlinear terms (if any) in the objective function. Note that if two problems are the same except that one minimizes f(x)$f\left(x\right)$ and the other maximizes − f(x)$f\left(x\right)$, their solutions will be the same but the signs of the dual variables π_{i}${\pi}_{i}$ and the reduced gradients d_{j}${d}_{j}$ will be reversed.
Forward Difference Interval r$r$Default = sqrt(Function Precision)$\text{}=\sqrt{{\mathbf{Function\; Precision}}}$This option defines an interval used to estimate derivatives by forward differences in the following circumstances:
(a) 
For verifying the objective and/or constraint gradients (see the description of the optional parameter Verify Level). 
(b) 
For estimating unspecified elements of the objective gradient and/or the constraint Jacobian. 
A derivative with respect to
x_{j}${x}_{j}$ is estimated by perturbing that element of
x$x$ to the value
x_{j} + r(1 + x_{j})${x}_{j}+r(1+\left{x}_{j}\right)$ and then evaluating
f(x)$f\left(x\right)$ and/or
F(x)$F\left(x\right)$ (as appropriate) at the perturbed point. The resulting gradient estimates should be accurate to
O(r)$\mathit{O}\left(r\right)$, unless the functions are badly scaled. Judicious alteration of
r$r$ may sometimes lead to greater accuracy. See
Gill et al. (1981) for a discussion of the accuracy in finite difference approximations.
If r ≤ 0$r\le 0$, the default value is used.
Function Precision r$r$Default = ε^{0.8}$\text{}={\epsilon}^{0.8}$This parameter defines the relative function precision ε_{r}${\epsilon}_{r}$, which is intended to be a measure of the relative accuracy with which the nonlinear functions can be computed. For example, if f(x)$f\left(x\right)$ (or F_{i}(x)${F}_{i}\left(x\right)$) is computed as 1000.56789$1000.56789$ for some relevant x$x$ and the first 6$6$ significant digits are known to be correct then the appropriate value for ε_{r}${\epsilon}_{r}$ would be 10^{ − 6}${10}^{6}$.
Ideally the functions f(x)$f\left(x\right)$ or F_{i}(x)${F}_{i}\left(x\right)$ should have magnitude of order 1$1$. If all functions are substantially less than 1$1$ in magnitude, ε_{r}${\epsilon}_{r}$ should be the absolute precision. For example, if f(x)$f\left(x\right)$ (or F_{i}(x)${F}_{i}\left(x\right)$) is computed as 1.23456789 × 10^{ − 4}$1.23456789\times {10}^{4}$ for some relevant x$x$ and the first 6$6$ significant digits are known to be correct then the appropriate value for ε_{r}${\epsilon}_{r}$ would be 10^{ − 10}${10}^{10}$.
The choice of
ε_{r}${\epsilon}_{r}$ can be quite complicated for badly scaled problems; see Chapter 8 of
Gill et al. (1981) for a discussion of scaling techniques. The default value is appropriate for most simple functions that are computed with full accuracy.
In some cases the function values will be the result of extensive computation, possibly involving an iterative procedure that can provide few digits of precision at reasonable cost. Specifying an appropriate value of r$r$ may therefore lead to savings, by allowing the linesearch procedure to terminate when the difference between function values along the search direction becomes as small as the absolute error in the values.
If r < ε$r<\epsilon $ or r ≥ 1$r\ge 1$, the default value is used.
Hessian Frequency i$i$Default = 99999999$\text{}=99999999$This option forces the approximate Hessian formed from
i$i$ BFGS updates to be reset to the identity matrix upon completion of a major iteration. It is intended to be used in conjunction with optional parameter
Hessian Full Memory.
If i ≤ 0$i\le 0$, the default value is used and effectively no resets occur.
Hessian Full Memory Default when n < 75$\stackrel{}{n}<75$Hessian Limited Memory Default when n ≥ 75$\stackrel{}{n}\ge 75$These options specify the method for storing and updating the quasiNewton approximation to the Hessian of the Lagrangian function.
If
Hessian Full Memory is specified, the approximate Hessian is treated as a dense matrix and BFGS quasiNewton updates are applied explicitly. This is most efficient when the total number of nonlinear variables is not too large (say,
n < 75$\stackrel{}{n}<75$). In this case, the storage requirement is fixed and you can expect
1$1$step Qsuperlinear convergence to the solution.
Hessian Limited Memory should only be specified when
n$\stackrel{}{n}$ is very large. In this case a limited memory procedure is used to update a diagonal Hessian approximation
H_{r}${H}_{r}$ a limited number of times. (Updates are accumulated as a list of vector pairs. They are discarded at regular intervals after
H_{r}${H}_{r}$ has been reset to their diagonal.)
Note that if
Hessian Frequency = 20${\mathbf{Hessian\; Frequency}}=20$ is used in conjunction with
Hessian Full Memory, the effect will be similar to using
Hessian Limited Memory in conjunction with
Hessian Updates = 20${\mathbf{Hessian\; Updates}}=20$, except that the latter will retain the current diagonal during resets.
Hessian Updates i$i$Default = 20 or 99999999$\text{}=20\text{ or}99999999$The default value of
i$i$ is
20$20$ when
Hessian Limited Memory is in effect and
99999999$99999999$ when
Hessian Full Memory is in effect, in which case no updates are performed.
If
Hessian Limited Memory is in effect, this option defines the maximum number of pairs of Hessian update vectors that are to be used to define the quasiNewton approximate Hessian. Once the limit of
i$i$ updates is reached, all but the diagonal elements of the accumulated updates are discarded and the process starts again. Broadly speaking, the more updates that are stored, the better the quality of the approximate Hessian. On the other hand, the more vectors that are stored, the greater the cost of each QP iteration.
The default value of i$i$ is likely to give a robust algorithm without significant expense, but faster convergence may be obtained with far fewer updates (e.g., i = 5$i=5$).
If i < 0$i<0$, the default value is used.
Infinite Bound Size r$r$Default = 10^{20}$\text{}={10}^{20}$If r > 0$r>0$, r$r$ defines the ‘infinite’ bound bigbnd$\mathit{bigbnd}$ in the definition of the problem constraints. Any upper bound greater than or equal to bigbnd$\mathit{bigbnd}$ will be regarded as + ∞$+\infty $ (and similarly any lower bound less than or equal to − bigbnd$\mathit{bigbnd}$ will be regarded as − ∞$\infty $).
If r ≤ 0$r\le 0$, the default value is used.
Iteration Limit i$i$Default = 10000$\text{}=10000$The value of
i$i$ specifies the maximum number of minor iterations allowed (i.e., iterations of the simplex method or the QP algorithm), summed over all major iterations. (See also the description of the optional parameters
Major Iteration Limit and
Minor Iteration Limit.)
If i < 0$i<0$, the default value is used.
Linesearch Tolerance r$r$Default = 0.9$\text{}=0.9$This option controls the accuracy with which a step length will be located along the direction of search at each iteration. At the start of each linesearch a target directional derivative for the Lagrangian merit function is identified. The value of r$r$ therefore determines the accuracy to which this target value is approximated.
The default value r = 0.9$r=0.9$ requests an inaccurate search and is appropriate for most problems, particularly those with any nonlinear constraints.
If the nonlinear functions are cheap to evaluate, a more accurate search may be appropriate; try r = 0.1,0.01 or 0.001$r=0.1,0.01\text{ or}0.001$. The number of major iterations required to solve the problem might decrease.
If the nonlinear functions are expensive to evaluate, a less accurate search may be appropriate. If
Derivative Level = 3${\mathbf{Derivative\; Level}}=3$, try
r = 0.99$r=0.99$. (The number of major iterations required to solve the problem might increase, but the total number of function evaluations may decrease enough to compensate.)
If
Derivative Level < 3${\mathbf{Derivative\; Level}}<3$, a moderately accurate search may be appropriate; try
r = 0.5$r=0.5$. Each search will (typically) require only
1 − 5$15$ function values, but many function calls will then be needed to estimate the missing gradients for the next iteration.
If r < 0$r<0$ or r ≥ 1$r\ge 1$, the default value is used.
List Default for e04ug = List$\text{e04ug}={\mathbf{List}}$Nolist Default for e04ug = Nolist$\text{e04ug}={\mathbf{Nolist}}$Normally each optional parameter specification is printed as it is supplied. Optional parameter
Nolist may be used to suppress the printing and optional parameter
List may be used to restore printing.
LU Density Tolerance r_{1}${r}_{1}$Default = 0.6$\text{}=0.6$LU Singularity Tolerance r_{2}${r}_{2}$Default = ε^{0.67}$\text{}={\epsilon}^{0.67}$If r_{1} > 0${r}_{1}>0$, r_{1}${r}_{1}$ defines the density tolerance used during the LU$LU$ factorization of the basis matrix. Columns of L$L$ and rows of U$U$ are formed one at a time and the remaining rows and columns of the basis are altered appropriately. At any stage, if the density of the remaining matrix exceeds r_{1}${r}_{1}$, the Markowitz strategy for choosing pivots is terminated. The remaining matrix is then factorized using a dense LU$LU$ procedure. Increasing the value of r_{1}${r}_{1}$ towards unity may give slightly sparser LU$LU$ factors, with a slight increase in factorization time. If r_{1} ≤ 0${r}_{1}\le 0$, the default value is used.
If
r_{2} > 0${r}_{2}>0$,
r_{2}${r}_{2}$ defines the singularity tolerance used to guard against illconditioned basis matrices. Whenever the basis is refactorized, the diagonal elements of
U$U$ are tested as follows. If
u_{jj} ≤ r_{2}$\left{u}_{jj}\right\le {r}_{2}$ or
u_{jj} < r_{2} × max_{i} u_{ij}$\left{u}_{jj}\right<{r}_{2}\times {\displaystyle \underset{i}{\mathrm{max}}}\phantom{\rule{0.25em}{0ex}}\left{u}_{ij}\right$, the
j$j$th column of the basis is replaced by the corresponding slack variable. This is most likely to occur when
start = 'W'${\mathbf{start}}=\text{'W'}$ (see
Section [Parameters]), or at the start of a major iteration. If
r_{2} ≤ 0${r}_{2}\le 0$, the default value is used.
In some cases, the Jacobian matrix may converge to values that make the basis exactly singular (e.g., a whole row of the Jacobian matrix could be zero at an optimal solution). Before exact singularity occurs, the basis could become very illconditioned and the optimization could progress very slowly (if at all). Setting r_{2} = 0.00001${r}_{2}=0.00001$ (say) may therefore help cause a judicious change of basis in such situations.
LU Factor Tolerance r_{1}${r}_{1}$Default = 5.0$\text{}=5.0$ or 100.0$100.0$LU Update Tolerance r_{2}${r}_{2}$Default = 5.0$\text{}=5.0$ or 10.0$10.0$The default value of r_{1}${r}_{1}$ is 5.0$5.0$ if there are any nonlinear constraints and 100.0$100.0$ otherwise. The default value of r_{2}${r}_{2}$ is 5.0$5.0$ if there are any nonlinear constraints and 10.0$10.0$ otherwise.
If
r_{1} ≥ 1${r}_{1}\ge 1$ and
r_{2} ≥ 1${r}_{2}\ge 1$, the values of
r_{1}${r}_{1}$ and
r_{2}${r}_{2}$ affect the stability and sparsity of the basis factorization
B = LU$B=LU$, during refactorization and updating, respectively. The lower triangular matrix
L$L$ is a product of matrices of the form
where the multipliers
μ$\mu $ satisfy
μ ≤ r_{i}$\left\mu \right\le {r}_{i}$. Smaller values of
r_{i}${r}_{i}$ favour stability, while larger values favour sparsity. The default values of
r_{1}${r}_{1}$ and
r_{2}${r}_{2}$ usually strike a good compromise. For large and relatively dense problems, setting
r_{1} = 10.0 or 5.0${r}_{1}=10.0\text{ or}5.0$ (say) may give a marked improvement in sparsity without impairing stability to a serious degree. Note that for problems involving band matrices, it may be necessary to reduce
r_{1}${r}_{1}$ and/or
r_{2}${r}_{2}$ in order to achieve stability.
If r_{1} < 1${r}_{1}<1$ or r_{2} < 1${r}_{2}<1$, the appropriate default value is used.
Major Feasibility Tolerance r$r$Default = sqrt(ε)$\text{}=\sqrt{\epsilon}$This option specifies how accurately the nonlinear constraints should be satisfied. The default value is appropriate when the linear and nonlinear constraints contain data to approximately that accuracy. A larger value may be appropriate if some of the problem functions are known to be of low accuracy.
Let
rowerr be defined as the maximum nonlinear constraint violation normalized by the size of the solution. It is required to satisfy
where
viol_{i}${\mathit{viol}}_{i}$ is the violation of the
i$i$th nonlinear constraint.
If r ≤ ε$r\le \epsilon $, the default value is used.
Major Iteration Limit i$i$Default = 1000$\text{}=1000$The value of
i$i$ specifies the maximum number of major iterations allowed before termination. It is intended to guard against an excessive number of linearizations of the nonlinear constraints. Setting
i = 0$i=0$ and
Major Print Level > 0${\mathbf{Major\; Print\; Level}}>0$ means that the objective and constraint gradients will be checked if
Verify Level > 0${\mathbf{Verify\; Level}}>0$ and the workspace needed to start solving the problem will be computed and printed, but no iterations will be performed.
If i < 0$i<0$, the default value is used.
Major Optimality Tolerance r$r$Default = sqrt(ε)$\text{}=\sqrt{\epsilon}$Optimality Tolerance r$r$This option specifies the final accuracy of the dual variables. If
nag_opt_nlp1_sparse_solve (e04ug) terminates with
ifail = 0${\mathbf{ifail}}={\mathbf{0}}$, a primal and dual solution (
x,s,π$x,s,\pi $) will have been computed such that
where
gap_{j}${\mathit{gap}}_{j}$ is an estimate of the complementarity gap for the
j$j$th variable and
‖π‖$\Vert \pi \Vert $ is a measure of the size of the QP dual variables (or Lagrange multipliers) given by
It is included to make the tests independent of a scale factor on the objective function. Specifically,
gap_{j}${\mathit{gap}}_{j}$ is computed from the final QP solution using the reduced gradients
d_{j} = g_{j} − π^{T}a_{j}${d}_{j}={g}_{j}{\pi}^{\mathrm{T}}{a}_{j}$, where
g_{j}${g}_{j}$ is the
j$j$th element of the objective gradient and
a_{j}${a}_{j}$ is the associated column of the constraint matrix
$\left(\begin{array}{cc}A& I\end{array}\right)$:
If r ≤ 0$r\le 0$, the default value is used.
Major Print Level i$i$Print Level The value of
i$i$ controls the amount of printout produced by the major iterations of
nag_opt_nlp1_sparse_solve (e04ug), as indicated below. A detailed description of the printed output is given in
Section [Major Iteration Printout] (summary output at each major iteration and the final solution) and
Section [Description of Monitoring Information] (monitoring information at each major iteration). (See also the description of
Minor Print Level.)
The following printout is sent to the current advisory message unit (as defined by
nag_file_set_unit_advisory (x04ab)):
i$i$ 
Output 
≥ 00$\phantom{\ge 0}0$ 
No output. 
≥ 01$\phantom{\ge 0}1$ 
The final solution only. 
≥ 05$\phantom{\ge 0}5$ 
One line of summary output ( < 80$\text{}<80$ characters; see Section [Major Iteration Printout]) for each major iteration (no printout of the final solution). 
≥ 10$\text{}\ge 10$ 
The final solution and one line of summary output for each major iteration. 
The following printout is sent to the logical unit number defined by the optional parameter
Monitoring File:
i$i$ 
Output 
≥ 00$\phantom{\ge 0}0$ 
No output. 
≥ 01$\phantom{\ge 0}1$ 
The final solution only. 
≥ 05$\phantom{\ge 0}5$ 
One long line of output ( < 120$\text{}<120$ characters; see Section [Description of Monitoring Information]) for each major iteration (no printout of the final solution). 
≥ 10$\text{}\ge 10$ 
The final solution and one long line of output for each major iteration. 
≥ 20$\text{}\ge 20$ 
The final solution, one long line of output for each major iteration, matrix statistics (initial status of rows and columns, number of elements, density, biggest and smallest elements, etc.), details of the scale factors resulting from the scaling procedure (if Scale Option = 1${\mathbf{Scale\; Option}}=1$ or 2$2$), basis factorization statistics and details of the initial basis resulting from the Crash procedure (if start = 'C'${\mathbf{start}}=\text{'C'}$; see Section [Parameters]). 
If
Major Print Level ≥ 5${\mathbf{Major\; Print\; Level}}\ge 5$ and the unit number defined by the optional parameter
Monitoring File is the same as that defined by
nag_file_set_unit_advisory (x04ab) then the summary output for each major iteration is suppressed.
Major Step Limit r$r$Default = 2.0$\text{}=2.0$If r > 0,r$r>0,r$ limits the change in x$x$ during a linesearch. It applies to all nonlinear problems once a ‘feasible solution’ or ‘feasible subproblem’ has been found.
A linesearch determines a step α$\alpha $ in the interval 0 < α ≤ β$0<\alpha \le \beta $, where β = 1$\beta =1$ if there are any nonlinear constraints, or the step to the nearest upper or lower bound on x$x$ if all the constraints are linear. Normally, the first step attempted is α_{1} = min (1,β)${\alpha}_{1}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(1,\beta )$.
In some cases, such as
f(x) = ae^{bx}$f\left(x\right)=a{e}^{bx}$ or
f(x) = ax^{b}$f\left(x\right)=a{x}^{b}$, even a moderate change in the elements of
x$x$ can lead to floating point overflow. The parameter
r$r$ is therefore used to define a step limit
β$\stackrel{}{\beta}$ given by
where
p$p$ is the search direction and the first evaluation of
f(x)$f\left(x\right)$ is made at the (potentially) smaller step length
α_{1} = min (1,β,β)${\alpha}_{1}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(1,\stackrel{}{\beta},\beta )$.
Wherever possible, upper and lower bounds on x$x$ should be used to prevent evaluation of nonlinear functions at meaningless points. The default value r = 2.0$r=2.0$ should not affect progress on wellbehaved functions, but values such as r = 0.1 or 0.01$r=0.1\text{ or}0.01$ may be helpful when rapidly varying functions are present. If a small value of r$r$ is selected, a ‘good’ starting point may be required. An important application is to the class of nonlinear least squares problems.
If r ≤ 0$r\le 0$, the default value is used.
Minor Feasibility Tolerance r$r$Default = sqrt(ε)$\text{}=\sqrt{\epsilon}$Feasibility Tolerance r$r$This option attempts to ensure that all variables eventually satisfy their upper and lower bounds to within the tolerance
r$r$. Since this includes slack variables, general linear constraints should also be satisfied to within
r$r$. Note that feasibility with respect to nonlinear constraints is judged by the value of optional parameter
Major Feasibility Tolerance and not by
r$r$.
If the bounds and linear constraints cannot be satisfied to within r$r$, the problem is declared infeasible. Let Sinf be the corresponding sum of infeasibilities. If Sinf is quite small, it may be appropriate to raise r$r$ by a factor of 10$10$ or 100$100$. Otherwise, some error in the data should be suspected.
If
Scale Option ≥ 1${\mathbf{Scale\; Option}}\ge 1$, feasibility is defined in terms of the
scaled problem (since it is more likely to be meaningful).
Nonlinear functions will only be evaluated at points that satisfy the bounds and linear constraints. If there are regions where a function is undefined, every effort should be made to eliminate these regions from the problem. For example, if f(x_{1},x_{2}) = sqrt(x_{1}) + log(x_{2})$f({x}_{1},{x}_{2})=\sqrt{{x}_{1}}+\mathrm{log}\left({x}_{2}\right)$, it is essential to place lower bounds on both x_{1}${x}_{1}$ and x_{2}${x}_{2}$. If the value r = 10^{ − 6}$r={10}^{6}$ is used, the bounds x_{1} ≥ 10^{ − 5}${x}_{1}\ge {10}^{5}$ and x_{2} ≥ 10^{ − 4}${x}_{2}\ge {10}^{4}$ might be appropriate. (The log singularity is more serious; in general, you should attempt to keep x$x$ as far away from singularities as possible.)
In reality,
r$r$ is used as a feasibility tolerance for satisfying the bounds on
x$x$ and
s$s$ in each QP subproblem. If the sum of infeasibilities cannot be reduced to zero, the QP subproblem is declared infeasible and the function is then in
elastic mode thereafter (with only the linearized nonlinear constraints defined to be elastic). (See also the description of
Elastic Weight.)
If r ≤ ε$r\le \epsilon $, the default value is used.
Minor Iteration Limit i$i$Default = 500$\text{}=500$The value of
i$i$ specifies the maximum number of iterations allowed between successive linearizations of the nonlinear constraints. A value in the range
10 ≤ i ≤ 50$10\le i\le 50$ prevents excessive effort being expended on early major iterations, but allows later QP subproblems to be solved to completion. Note that an extra
m$m$ minor iterations are allowed if the first QP subproblem to be solved starts with the allslack basis
B = I$B=I$. (See the description of the optional parameter
Crash Option.)
In general, it is unsafe to specify values as small as i = 1 or 2$i=1\text{ or}2$ (because even when an optimal solution has been reached, a few minor iterations may be needed for the corresponding QP subproblem to be recognized as optimal).
If i ≤ 0$i\le 0$, the default value is used.
Minor Optimality Tolerance r$r$Default = sqrt(ε)$\text{}=\sqrt{\epsilon}$This option is used to judge optimality for each QP subproblem. Let the QP reduced gradients be d_{j} = g_{j} − π^{T}a_{j}${d}_{j}={g}_{j}{\pi}^{\mathrm{T}}{a}_{j}$, where g_{j}${g}_{j}$ is the j$j$th element of the QP gradient, a_{j}${a}_{j}$ is the associated column of the QP constraint matrix and π$\pi $ is the set of QP dual variables.
By construction, the reduced gradients for basic variables are always zero. The QP subproblem will be declared optimal if the reduced gradients for nonbasic variables at their upper or lower bounds satisfy
respectively, and if
(d_{j})/(‖π‖) ≤ r$\frac{\left{d}_{j}\right}{\Vert \pi \Vert}\le r$ for superbasic variables.
Note that
‖π‖$\Vert \pi \Vert $ is a measure of the size of the dual variables. It is included to make the tests independent of a scale factor on the objective function. (The value of
‖π‖$\Vert \pi \Vert $ actually used is defined in the description for optional parameter
Major Optimality Tolerance.)
If the objective is scaled down to be very small, the optimality test reduces to comparing d_{j}${d}_{j}$ against r$r$.
If r ≤ 0$r\le 0$, the default value is used.
Minor Print Level i$i$Default = 0$\text{}=0$The value of
i$i$ controls the amount of printout produced by the minor iterations of
nag_opt_nlp1_sparse_solve (e04ug) (i.e., the iterations of the quadratic programming algorithm), as indicated below. A detailed description of the printed output is given in
Section [Minor Iteration Printout] (summary output at each minor iteration) and
Section [Description of Monitoring Information] (monitoring information at each minor iteration). (See also the description of the optional parameter
Major Print Level.)
The following printout is sent to the current advisory message unit (as defined by
nag_file_set_unit_advisory (x04ab)):
i$i$ 
Output 
≥ 0$\phantom{\ge}0$ 
No output. 
≥ 1$\text{}\ge 1$ 
One line of summary output ( < 80$\text{}<80$ characters; see Section [Minor Iteration Printout]) for each minor iteration. 
The following printout is sent to the logical unit number defined by the optional parameter
Monitoring File:
If
Minor Print Level ≥ 1${\mathbf{Minor\; Print\; Level}}\ge 1$ and the unit number defined by the optional parameter
Monitoring File is the same as that defined by
nag_file_set_unit_advisory (x04ab) then the summary output for each minor iteration is suppressed.
Monitoring File i$i$Default = − 1$\text{}=1$If
i ≥ 0$i\ge 0$ and
Major Print Level ≥ 5${\mathbf{Major\; Print\; Level}}\ge 5$ or
i ≥ 0$i\ge 0$ and
Minor Print Level ≥ 1${\mathbf{Minor\; Print\; Level}}\ge 1$ then monitoring information is produced by
nag_opt_nlp1_sparse_solve (e04ug) at every iteration is sent to a file with logical unit number
i$i$. If
i < 0$i<0$ and/or
Major Print Level < 5${\mathbf{Major\; Print\; Level}}<5$ and
Minor Print Level < 1${\mathbf{Minor\; Print\; Level}}<1$ then no monitoring information is produced.
Partial Price i$i$Default = 1 or 10$\text{}=1\text{ or}10$The default value of i$i$ is 1$1$ if there are any nonlinear constraints and 10$10$ otherwise.
This option is recommended for large problems that have significantly more variables than constraints (i.e.,
n ≫ m$n\gg m$). It reduces the work required for each ‘pricing’ operation (i.e., when a nonbasic variable is selected to become superbasic). The possible choices for
i$i$ are the following.
i$i$ 
Meaning 
≥ 1$\phantom{\ge}1$ 
All columns of the constraint matrix $\left(\begin{array}{cc}A& I\end{array}\right)$ are searched. 
≥ 2$\text{}\ge 2$ 
Both A$A$ and I$I$ are partitioned to give i$i$ roughly equal segments A_{j},I_{j}${A}_{\mathit{j}},{I}_{\mathit{j}}$, for j = 1,2, … ,p$\mathit{j}=1,2,\dots ,p$ (modulo p$p$). If the previous pricing search was successful on A_{j},I_{j}${A}_{j},{I}_{j}$, the next search begins on the segments A_{j + 1},I_{j + 1}${A}_{j+1},{I}_{j+1}$. If a reduced gradient is found that is larger than some dynamic tolerance, the variable with the largest such reduced gradient (of appropriate sign) is selected to enter the basis. If nothing is found, the search continues on the next segments A_{j + 2},I_{j + 2}${A}_{j+2},{I}_{j+2}$ and so on. 
If i ≤ 0$i\le 0$, the default value is used.
Pivot Tolerance r$r$Default = ε^{0.67}$\text{}={\epsilon}^{0.67}$If r > 0$r>0$, r$r$ is used during the solution of QP subproblems to prevent columns entering the basis if they would cause the basis to become almost singular.
When x$x$ changes to x + αp$x+\alpha p$ for some specified search direction p$p$, a ‘ratio test’ is used to determine which element of x$x$ reaches an upper or lower bound first. The corresponding element of p$p$ is called the pivot element. Elements of p$p$ are ignored (and therefore cannot be pivot elements) if they are smaller than r$r$.
It is common in practice for two (or more) variables to reach a bound at essentially the same time. In such cases, the
Minor Feasibility Tolerance provides some freedom to maximize the pivot element and thereby improve numerical stability. Excessively
small values of
Minor Feasibility Tolerance should therefore not be specified. To a lesser extent, the
Expand Frequency also provides some freedom to maximize the pivot element. Excessively
large values of
Expand Frequency should therefore not be specified.
If r ≤ 0$r\le 0$, the default value is used.
Scale Option i$i$Default = 1 or 2$\text{}=1\text{ or}2$The default value of i$i$ is 1$1$ if there are any nonlinear constraints and 2$2$ otherwise.
This option enables you to scale the variables and constraints using an iterative procedure due to
Fourer (1982), which attempts to compute row scales
r_{i}${r}_{i}$ and column scales
c_{j}${c}_{j}$ such that the scaled matrix coefficients
a_{ij} = a_{ij} × (c_{j} / r_{i})${\stackrel{}{a}}_{ij}={a}_{ij}\times ({c}_{j}/{r}_{i})$ are as close as possible to unity. (The lower and upper bounds on the variables and slacks for the scaled problem are redefined as
l_{j} = l_{j} / c_{j}${\stackrel{}{l}}_{j}={l}_{j}/{c}_{j}$ and
u_{j} = u_{j} / c_{j}${\stackrel{}{u}}_{j}={u}_{j}/{c}_{j}$ respectively, where
c_{j} ≡ r_{j − n}${c}_{j}\equiv {r}_{jn}$ if
j > n$j>n$.) The possible choices for
i$i$ are the following.
i$i$ 
Meaning 
0 
No scaling is performed. This is recommended if it is known that the elements of x$x$ and the constraint matrix A$A$ (along with its Jacobian) never become large (say, > 1000$\text{}>1000$). 
1 
All linear constraints and variables are scaled. This may improve the overall efficiency of the function on some problems. 
2 
All constraints and variables are scaled. Also, an additional scaling is performed that takes into account columns of $\left(\begin{array}{cc}A& I\end{array}\right)$ that are fixed or have positive lower bounds or negative upper bounds. 
If there are any nonlinear constraints present, the scale factors depend on the Jacobian at the first point that satisfies the linear constraints and the upper and lower bounds. The setting i = 2$i=2$ should therefore be used only if a ‘good’ starting point is available and the problem is not highly nonlinear.
If i < 0$i<0$ or i > 2$i>2$, the default value is used.
Scale Tolerance r$r$Default = 0.9$\text{}=0.9$Note that this option does not apply when
Scale Option = 0${\mathbf{Scale\; Option}}=0$.
The value r$r$ (0 < r < 1$0<r<1$) is used to control the number of scaling passes to be made through the constraint matrix A$A$. At least 3$3$ (and at most 10$10$) passes will be made. More precisely, let a_{p}${a}_{p}$ denote the largest column ratio (i.e., ('biggest' element)/('smallest' element)
$\frac{\text{'biggest'}\text{ element}}{\text{'smallest'}\text{ element}}$ in some sense) after the p$p$th scaling pass through A$A$. The scaling procedure is terminated if a_{p} ≥ a_{p − 1} × r${a}_{p}\ge {a}_{p1}\times r$ for some p ≥ 3$p\ge 3$. Thus, increasing the value of r$r$ from 0.9$0.9$ to 0.99$0.99$ (say) will probably increase the number of passes through A$A$.
If r ≤ 0$r\le 0$ or r ≥ 1$r\ge 1$, the default value is used.
Start Objective Check At Column i_{1}${i}_{1}$Default = 1$\text{}=1$Stop Objective Check At Column i_{2}${i}_{2}$Default = n_{1}^{ ′ }$\text{}={n}_{1}^{\prime}$Start Constraint Check At Column i_{3}${i}_{3}$Default = 1$\text{}=1$Stop Constraint Check At Column i_{4}${i}_{4}$Default = n_{1}^{ ′ ′ }$\text{}={n}_{1}^{\prime \prime}$These keywords take effect only if
Verify Level > 0${\mathbf{Verify\; Level}}>0$. They may be used to control the verification of gradient elements computed by
objfun and/or Jacobian elements computed by
confun. For example, if the first
30$30$ elements of the objective gradient appeared to be correct in an earlier run, so that only element
31$31$ remains questionable then it is reasonable to specify
Start Objective Check At Column = 31${\mathbf{Start\; Objective\; Check\; At\; Column}}=31$. Similarly for columns of the Jacobian. If the first
30$30$ variables occur nonlinearly in the constraints but the remaining variables are nonlinear only in the objective, then
objfun must set the first
30$30$ elements of the array
objgrd to zero, but these hardly need to be verified. Again it is reasonable to specify
Start Objective Check At Column = 31${\mathbf{Start\; Objective\; Check\; At\; Column}}=31$.
If i_{2} ≤ 0${i}_{2}\le 0$ or i_{2} > n_{1}^{ ′ }${i}_{2}>{n}_{1}^{\prime}$, the default value is used.
If i_{1} ≤ 0${i}_{1}\le 0$ or i_{1} > min (n_{1}^{ ′ },i_{2})${i}_{1}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}({n}_{1}^{\prime},{i}_{2})$, the default value is used.
If i_{4} ≤ 0${i}_{4}\le 0$ or i_{4} > n_{1}^{ ′ ′ }${i}_{4}>{n}_{1}^{\prime \prime}$, the default value is used.
If i_{3} ≤ 0${i}_{3}\le 0$ or i_{3} > min (n_{1}^{ ′ ′ },i_{4})${i}_{3}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}({n}_{1}^{\prime \prime},{i}_{4})$, the default value is used.
Superbasics Limit i$i$Default = min (500,n + 1)$\text{}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}(500,\stackrel{}{n}+1)$Note that this option does not apply to linear problems.
It places a limit on the storage allocated for superbasic variables. Ideally, the value of i$i$ should be set slightly larger than the ‘number of degrees of freedom’ expected at the solution.
For nonlinear problems, the number of degrees of freedom is often called the ‘number of independent variables’. Normally, the value of i$i$ need not be greater than n + 1$\stackrel{}{n}+1$, but for many problems it may be considerably smaller. (This will save storage if n$\stackrel{}{n}$ is very large.)
If i ≤ 0$i\le 0$, the default value is used.
Unbounded Objective r_{1}${r}_{1}$Default = 10^{15}$\text{}={10}^{15}$Unbounded Step Size r_{2}${r}_{2}$Default = max (bigbnd,10^{20})$\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}(\mathit{bigbnd},{10}^{20})$These options are intended to detect unboundedness in nonlinear problems. During the linesearch, the objective function
f$f$ is evaluated at points of the form
x + αp$x+\alpha p$, where
x$x$ and
p$p$ are fixed and
α$\alpha $ varies. If
f$\leftf\right$ exceeds
r_{1}${r}_{1}$ or
α$\alpha $ exceeds
r_{2}${r}_{2}$, the iterations are terminated and the function returns with
ifail = 3${\mathbf{ifail}}={\mathbf{3}}$.
If singularities are present, unboundedness in f(x)$f\left(x\right)$ may manifest itself by a floating point overflow during the evaluation of f(x + αp)$f(x+\alpha p)$, before the test against r_{1}${r}_{1}$ can be made.
Unboundedness in x$x$ is best avoided by placing finite upper and lower bounds on the variables.
If r_{1} ≤ 0${r}_{1}\le 0$ or r_{2} ≤ 0${r}_{2}\le 0$, the appropriate default value is used.
Verify Level i$i$Default = 0$\text{}=0$This option refers to finite difference checks on the gradient elements computed by
objfun and
confun. Gradients are verified at the first point that satisfies the linear constraints and the upper and lower bounds. Unspecified gradient elements are not checked and hence they result in no overhead. The possible choices for
i$i$ are the following.
i$i$ 
Meaning 
− 1$1$ 
No checks are performed. 
− 0$\phantom{}0$ 
Only a ‘cheap’ test will be performed, requiring three calls to objfun and two calls to confun. Note that no checks are carried out if every column of the constraint gradients (Jacobian) contains a missing element. 
− 1$\phantom{}1$ 
Individual objective gradient elements will be checked using a reliable (but more expensive) test. If Major Print Level > 0${\mathbf{Major\; Print\; Level}}>0$, a key of the form OK or BAD? indicates whether or not each element appears to be correct. 
− 2$\phantom{}2$ 
Individual columns of the constraint gradients (Jacobian) will be checked using a reliable (but more expensive) test. If Major Print Level > 0${\mathbf{Major\; Print\; Level}}>0$, a key of the form OK or BAD? indicates whether or not each element appears to be correct. 
− 3$\phantom{}3$ 
Check both constraint and objective gradients (in that order) as described above for i = 2$i=2$ and i = 1$i=1$ respectively. 
The value
i = 3$i=3$ should be used whenever a new function function is being developed. The
Start Objective Check At Column and
Stop Objective Check At Column keywords may be used to limit the number of nonlinear variables to be checked.
If i < − 1$i<1$ or i > 3$i>3$, the default value is used.
Violation Limit r$r$Default = 10.0$\text{}=10.0$This option defines an absolute limit on the magnitude of the maximum constraint violation after the linesearch. Upon completion of the linesearch, the new iterate
x_{k + 1}${x}_{k+1}$ satisfies the condition
where
x_{0}${x}_{0}$ is the point at which the nonlinear constraints are first evaluated and
v_{i}(x)${v}_{i}\left(x\right)$ is the
i$i$th nonlinear constraint violation
v_{i}(x) =
max (0,l_{i} − F_{i}(x),F_{i}(x) − u_{i})${v}_{i}\left(x\right)=\mathrm{max}\phantom{\rule{0.125em}{0ex}}(0,{l}_{i}{F}_{i}\left(x\right),{F}_{i}\left(x\right){u}_{i})$.
The effect of the violation limit is to restrict the iterates to lie in an expanded feasible region whose size depends on the magnitude of r$r$. This makes it possible to keep the iterates within a region where the objective function is expected to be welldefined and bounded below (or above in the case of maximization). If the objective function is bounded below (or above in the case of maximization) for all values of the variables, then r$r$ may be any large positive value.
If r ≤ 0$r\le 0$, the default value is used.
Description of Monitoring Information
This section describes the intermediate printout and final printout which constitutes the monitoring information produced by
nag_opt_nlp1_sparse_solve (e04ug). (See also the description of the optional parameters
Monitoring File,
Major Print Level and
Minor Print Level.) You can control the level of printed output.
When
Major Print Level ≥ 20${\mathbf{Major\; Print\; Level}}\ge 20$ and
Monitoring File ≥ 0${\mathbf{Monitoring\; File}}\ge 0$, the following line of intermediate printout (
< 120$\text{}<120$ characters) is produced at every major iteration on the unit number specified by optional parameter
Monitoring File. Unless stated otherwise, the values of the quantities printed are those in effect
on completion of the given iteration.
Major 
is the major iteration count.

Minor 
is the number of minor iterations required by the feasibility and optimality phases of the QP subproblem. Generally, Minor will be 1$1$ in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution (see Section [Algorithmic Details]).

Step 
is the step α_{k}${\alpha}_{k}$ taken along the computed search direction. On reasonably wellbehaved problems, the unit step (i.e., α_{k} = 1${\alpha}_{k}=1$) will be taken as the solution is approached.

nObj 
is the number of times objfun has been called to evaluate the nonlinear part of the objective function. Evaluations needed for the estimation of the gradients by finite differences are not included. nObj is printed as a guide to the amount of work required for the linesearch.

nCon 
is the number of times confun has been called to evaluate the nonlinear constraint functions (not printed if ncnln is zero).

Merit 
is the value of the augmented Lagrangian merit function (6) at the current iterate. This function will decrease at each iteration unless it was necessary to increase the penalty parameters (see Section [Major Iteration Printout]). As the solution is approached, Merit will converge to the value of the objective function at the solution.In elastic mode (see Section [Treatment of Constraint Infeasibilities]), the merit function is a composite function involving the constraint violations weighted by the value of the optional parameter Elastic Weight. If there are no nonlinear constraints present, this entry contains Objective, the value of the objective function f(x)$f\left(x\right)$. In this case, f(x)$f\left(x\right)$ will decrease monotonically to its optimal value.

Feasibl 
is the value of rowerr, the largest element of the scaled nonlinear constraint residual vector defined in the description of the optional parameter Major Feasibility Tolerance. The solution is regarded as ‘feasible’ if Feasibl is less than (or equal to) the optional parameter Major Feasibility Tolerance. Feasibl will be approximately zero in the neighbourhood of a solution. If there are no nonlinear constraints present, all iterates are feasible and this entry is not printed.

Optimal 
is the value of maxgap, the largest element of the maximum complementarity gap vector defined in the description of the optional parameter Major Optimality Tolerance. The Lagrange multipliers are regarded as ‘optimal’ if Optimal is less than (or equal to) the optional parameter Major Optimality Tolerance. Optimal will be approximately zero in the neighbourhood of a solution.

nS 
is the current number of superbasic variables.

Penalty 
is the Euclidean norm of the vector of penalty parameters used in the augmented Lagrangian merit function (not printed if ncnln is zero).

LU 
is the number of nonzeros representing the basis factors L$L$ and U$U$ on completion of the QP subproblem. If there are nonlinear constraints present, the basis factorization B = LU$B=LU$ is computed at the start of the first minor iteration. At this stage, LU = lenL + lenU$\mathtt{LU}=\mathtt{lenL}+\mathtt{lenU}$, where lenL is the number of subdiagonal elements in the columns of a lower triangular matrix and lenU is the number of diagonal and superdiagonal elements in the rows of an upper triangular matrix. As columns of B$B$ are replaced during the minor iterations, the value of LU may fluctuate up or down (but in general will tend to increase). As the solution is approached and the number of minor iterations required to solve each QP subproblem decreases towards zero, LU will reflect the number of nonzeros in the LU$LU$ factors at the start of each QP subproblem. If there are no nonlinear constraints present, refactorization is subject only to the value of the optional parameter Factorization Frequency and hence LU will tend to increase between factorizations.

Swp 
is the number of columns of the basis matrix B$B$ that were swapped with columns of S$S$ in order to improve the condition number of B$B$ (not printed if ncnln is zero). The swaps are determined by an LU$LU$ factorization of the rectangular matrix
BS
=
(BS)
${B}_{S}={\left(\begin{array}{cc}B& S\end{array}\right)}^{\mathrm{T}}$, with stability being favoured more than sparsity.

Cond Hz 
is an estimate of the condition number of the reduced Hessian of the Lagrangian (not printed if ncnln and nonln are both zero). It is the square of the ratio between the largest and smallest diagonal elements of the upper triangular matrix R$R$. This constitutes a lower bound on the condition number of the matrix R^{T}R${R}^{\mathrm{T}}R$ that approximates the reduced Hessian. The larger this number, the more difficult the problem.

PD 
is a twoletter indication of the status of the convergence tests involving the feasibility and optimality of the iterates defined in the descriptions of the optional parameters Major Feasibility Tolerance and Major Optimality Tolerance. Each letter is T if the test is satisfied and F otherwise. The tests indicate whether the values of Feasibl and Optimal are sufficiently small. For example, TF or TT is printed if there are no nonlinear constraints present (since all iterates are feasible). If either indicator is F when nag_opt_nlp1_sparse_solve (e04ug) terminates with ifail = 0${\mathbf{ifail}}={\mathbf{0}}$, you should check the solution carefully.

M 
is printed if an extra evaluation of usersupplied functions objfun and confun was needed in order to define an acceptable positive definite quasiNewton update to the Hessian of the Lagrangian. This modification is only performed when there are nonlinear constraints present.

m 
is printed if, in addition, it was also necessary to modify the update to include an augmented Lagrangian term.

s 
is printed if a selfscaled BFGS (Broyden–Fletcher–Goldfarb–Shanno) update was performed. This update is always used when the Hessian approximation is diagonal and hence always follows a Hessian reset.

S 
is printed if, in addition, it was also necessary to modify the selfscaled update in order to maintain positivedefiniteness.

n 
is printed if no positive definite BFGS update could be found, in which case the approximate Hessian is unchanged from the previous iteration.

r 
is printed if the approximate Hessian was reset after 10$10$ consecutive major iterations in which no BFGS update could be made. The diagonal elements of the approximate Hessian are retained if at least one update has been performed since the last reset. Otherwise, the approximate Hessian is reset to the identity matrix.

R 
is printed if the approximate Hessian has been reset by discarding all but its diagonal elements. This reset will be forced periodically by the values of the optional parameters Hessian Frequency and Hessian Updates. However, it may also be necessary to reset an illconditioned Hessian from time to time.

l 
is printed if the change in the norm of the variables was greater than the value defined by the optional parameter Major Step Limit. If this output occurs frequently during later iterations, it may be worthwhile increasing the value of Major Step Limit.

c 
is printed if central differences have been used to compute the unknown elements of the objective and constraint gradients. A switch to central differences is made if either the linesearch gives a small step, or x$x$ is close to being optimal. In some cases, it may be necessary to resolve the QP subproblem with the central difference gradient and Jacobian.

u 
is printed if the QP subproblem was unbounded.

t 
is printed if the minor iterations were terminated after the number of iterations specified by the value of the optional parameter Minor Iteration Limit was reached.

i 
is printed if the QP subproblem was infeasible when the function was not in elastic mode. This event triggers the start of nonlinear elastic mode, which remains in effect for all subsequent iterations. Once in elastic mode, the QP subproblems are associated with the elastic problem (8) (see Section [Treatment of Constraint Infeasibilities]). It is also printed if the minimizer of the elastic subproblem does not satisfy the linearized constraints when the function is already in elastic mode. (In this case, a feasible point for the usual QP subproblem may or may not exist.)

w 
is printed if a weak solution of the QP subproblem was found.

When
Minor Print Level ≥ 1${\mathbf{Minor\; Print\; Level}}\ge 1$ and
Monitoring File ≥ 0${\mathbf{Monitoring\; File}}\ge 0$, the following line of intermediate printout (
< 120$\text{}<120$ characters) is produced at every minor iteration on the unit number specified by optional parameter
Monitoring File. Unless stated otherwise, the values of the quantities printed are those in effect
on completion of the given iteration.
In the description below, a ‘pricing’ operation is defined to be the process by which a nonbasic variable is selected to become superbasic (in addition to those already in the superbasic set). If the problem is purely linear, the variable selected will usually become basic immediately (unless it happens to reach its opposite bound and return to the nonbasic set).
Itn 
is the iteration count.

pp 
is the partial price indicator. The variable selected by the last pricing operation came from the ppth partition of A$A$ and − I$I$. Note that pp is reset to zero whenever the basis is refactorized.

dj 
is the value of the reduced gradient (or reduced cost) for the variable selected by the pricing operation at the start of the current iteration.

+SBS 
is the variable selected by the pricing operation to be added to the superbasic set.

SBS 
is the variable chosen to leave the superbasic set. It has become basic if the entry under B is nonzero; otherwise it has become nonbasic.

BS 
is the variable removed from the basis (if any) to become nonbasic.

B 
is the variable removed from the basis (if any) to swap with a slack variable made superbasic by the latest pricing operation. The swap is done to ensure that there are no superbasic slacks.

Step 
is the value of the step length α$\alpha $ taken along the current search direction p$p$. The variables x$x$ have just been changed to x + αp$x+\alpha p$. If a variable is made superbasic during the current iteration (i.e., +SBS is positive), Step will be the step to the nearest bound. During the optimality phase, the step can be greater than unity only if the reduced Hessian is not positive definite.

Pivot 
is the r$r$th element of a vector y$y$ satisfying By = a_{q}$By={a}_{q}$ whenever a_{q}${a}_{q}$ (the q$q$th column of the constraint matrix
$\left(\begin{array}{cc}A& I\end{array}\right)$) replaces the r$r$th column of the basis matrix B$B$. Wherever possible, Step is chosen so as to avoid extremely small values of Pivot (since they may cause the basis to be nearly singular). In extreme cases, it may be necessary to increase the value of the optional parameter Pivot Tolerance to exclude very small elements of y$y$ from consideration during the computation of Step.

Ninf 
is the number of infeasibilities. This will not increase unless the iterations are in elastic mode. Ninf will be zero during the optimality phase.

Sinf/Objective 
is the value of the current objective function. If x$x$ is infeasible, Sinf gives the value of the sum of infeasibilities at the start of the current iteration. It will usually decrease at each nonzero value of Step, but may occasionally increase if the value of Ninf decreases by a factor of 2$2$ or more. However, in elastic mode this entry gives the value of the composite objective function (9), which will decrease monotonically at each iteration. If x$x$ is feasible, Objective is the value of the current QP objective function.

L 
is the number of nonzeros in the basis factor L$L$. Immediately after a basis factorization B = LU$B=LU$, this entry contains lenL. Further nonzeros are added to L when various columns of B$B$ are later replaced. (Thus, L increases monotonically.)

U 
is the number of nonzeros in the basis factor U$U$. Immediately after a basis factorization B = LU$B=LU$, this entry contains lenU. As columns of B$B$ are replaced, the matrix U$U$ is maintained explicitly (in sparse form). The value of U may fluctuate up or down; in general, it will tend to increase.

Ncp 
is the number of compressions required to recover workspace in the data structure for U$U$. This includes the number of compressions needed during the previous basis factorization. Normally, Ncp should increase very slowly. If it does not, increase leniz and lenz by at least L + U$\mathtt{L}+\mathtt{U}$ and rerun nag_opt_nlp1_sparse_solve (e04ug) (possibly using start = 'W'${\mathbf{start}}=\text{'W'}$; see Section [Parameters]).

The following items are printed only if the problem is nonlinear or the superbasic set is nonempty (i.e., if the current solution is nonbasic).
Norm rg 
is the Euclidean norm of the reduced gradient of the QP objective function. During the optimality phase, this norm will be approximately zero after a unit step.

nS 
is the current number of superbasic variables.

Cond Hz 
is an estimate of the condition number of the reduced Hessian of the Lagrangian (not printed if ncnln and nonln are both zero). It is the square of the ratio between the largest and smallest diagonal elements of the upper triangular matrix R$R$. This constitutes a lower bound on the condition number of the matrix R^{T}R${R}^{\mathrm{T}}R$ that approximates the reduced Hessian. The larger this number, the more difficult the problem.

When
Major Print Level ≥ 20${\mathbf{Major\; Print\; Level}}\ge 20$ and
Monitoring File ≥ 0${\mathbf{Monitoring\; File}}\ge 0$, the following lines of intermediate printout (
< 120$\text{}<120$ characters) are produced on the unit number specified by optional parameter
Monitoring File whenever the matrix
B$B$ or
B_{S} = (B S)^{T}${B}_{S}={\left(B\text{\hspace{1em}}S\right)}^{\mathrm{T}}$ is factorized before solving the next QP subproblem. Gaussian elimination is used to compute a sparse
LU$LU$ factorization of
B$B$ or
B_{S}${B}_{S}$, where
PLP^{T}$PL{P}^{\mathrm{T}}$ is a lower triangular matrix and
PUQ$PUQ$ is an upper triangular matrix for some permutation matrices
P$P$ and
Q$Q$. The factorization is stabilized in the manner described under the optional parameter
LU Factor Tolerance (
default value = 5.0$\text{default value}=5.0$ or
100.0$100.0$).
Note that
B_{S}${B}_{S}$ may be factorized at the beginning of just some of the major iterations. It is immediately followed by a factorization of
B$B$ itself.
Factorize 
is the factorization count.

Iteration 
is the iteration count.

Nonlinear 
is the number of nonlinear variables in the current basis B$B$ (not printed if B_{S}${B}_{S}$ is factorized).

Linear 
is the number of linear variables in B$B$ (not printed if B_{S}${B}_{S}$ is factorized).

Slacks 
is the number of slack variables in B$B$ (not printed if B_{S}${B}_{S}$ is factorized).

Elems 
is the number of nonzeros in B$B$ (not printed if B_{S}${B}_{S}$ is factorized).

Density 
is the percentage nonzero density of B$B$ (not printed if B_{S}${B}_{S}$ is factorized). More precisely, Density = 100 × Elems / (Nonlinear + Linear + Slacks)^{2}$\mathtt{Density}=100\times \mathtt{Elems}/{(\mathtt{Nonlinear}+\mathtt{Linear}+\mathtt{Slacks})}^{2}$.

Compressns 
is the number of times the data structure holding the partially factorized matrix needed to be compressed, in order to recover unused workspace. Ideally, it should be zero. If it is more than 3$3$ or 4$4$, increase leniz and lenz and rerun nag_opt_nlp1_sparse_solve (e04ug) (possibly using start = 'W'${\mathbf{start}}=\text{'W'}$; see Section [Parameters]).

Merit 
is the average Markowitz merit count for the elements chosen to be the diagonals of PUQ$PUQ$. Each merit count is defined to be (c − 1)(r − 1)$(c1)(r1)$, where c$c$ and r$r$ are the number of nonzeros in the column and row containing the element at the time it is selected to be the next diagonal. Merit is the average of m such quantities. It gives an indication of how much work was required to preserve sparsity during the factorization.

lenL 
is the number of nonzeros in L$L$.

lenU 
is the number of nonzeros in U$U$.

Increase 
is the percentage increase in the number of nonzeros in L$L$ and U$U$ relative to the number of nonzeros in B$B$. More precisely, Increase = 100 × (lenL + lenU − Elems) / Elems$\mathtt{Increase}=100\times (\mathtt{lenL}+\mathtt{lenU}\mathtt{Elems})/\mathtt{Elems}$.

m 
is the number of rows in the problem. Note that m = Ut + Lt + bp$\mathtt{m}=\mathtt{Ut}+\mathtt{Lt}+\mathtt{bp}$.

Ut 
is the number of triangular rows of B$B$ at the top of U$U$.

d1 
is the number of columns remaining when the density of the basis matrix being factorized reached 0.3$0.3$.

Lmax 
is the maximum subdiagonal element in the columns of L$L$. This will not exceed the value of the optional parameter LU Factor Tolerance.

Bmax 
is the maximum nonzero element in B$B$ (not printed if B_{S}${B}_{S}$ is factorized).

BSmax 
is the maximum nonzero element in B_{S}${B}_{S}$ (not printed if B$B$ is factorized).

Umax 
is the maximum nonzero element in U$U$, excluding elements of B$B$ that remain in U$U$ unchanged. (For example, if a slack variable is in the basis, the corresponding row of B$B$ will become a row of U$U$ without modification. Elements in such rows will not contribute to Umax. If the basis is strictly triangular then none of the elements of B$B$ will contribute and Umax will be zero.)Ideally, Umax should not be significantly larger than Bmax. If it is several orders of magnitude larger, it may be advisable to reset the optional parameter LU Factor Tolerance to some value nearer unity. Umax is not printed if B_{S}${B}_{S}$ is factorized.

Umin 
is the magnitude of the smallest diagonal element of PUQ$PUQ$.

Growth 
is the value of the ratio Umax/Bmax, which should not be too large. Providing Lmax is not large (say, < 10.0$\text{}<10.0$), the ratio max (Bmax,Umax) / Umin$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(\mathtt{Bmax},\mathtt{Umax})/\mathtt{Umin}$ is an estimate of the condition number of B$B$. If this number is extremely large, the basis is nearly singular and some numerical difficulties might occur. (However, an effort is made to avoid nearsingularity by using slacks to replace columns of B$B$ that would have made Umin extremely small and the modified basis is refactorized.)

Lt 
is the number of triangular columns of B$B$ at the left of L$L$.

bp 
is the size of the ‘bump’ or block to be factorized nontrivially after the triangular rows and columns of B$B$ have been removed.

d2 
is the number of columns remaining when the density of the basis matrix being factorized has reached 0.6$0.6$.

When
Major Print Level ≥ 20${\mathbf{Major\; Print\; Level}}\ge 20$,
Monitoring File ≥ 0${\mathbf{Monitoring\; File}}\ge 0$ and
Crash Option > 0${\mathbf{Crash\; Option}}>0$ (
default value = 0 or 3$\text{default value}=0\text{ or}3$), the following lines of intermediate printout (
< 80$\text{}<80$ characters) are produced on the unit number specified by optional parameter
Monitoring File whenever
start = 'C'${\mathbf{start}}=\text{'C'}$ (see
Section [Parameters]). They refer to the number of columns selected by the Crash procedure during each of several passes through
A$A$ while searching for a triangular basis matrix.
Slacks 
is the number of slacks selected initially.

Free cols 
is the number of free columns in the basis, including those whose bounds are rather far apart.

Preferred 
is the number of ‘preferred’ columns in the basis (i.e., istate(j) = 3${\mathbf{istate}}\left(j\right)=3$ for some j ≤ n$j\le n$). It will be a subset of the columns for which istate(j) = 3${\mathbf{istate}}\left(j\right)=3$ was specified.

Unit 
is the number of unit columns in the basis.

Double 
is the number of columns in the basis containing two nonzeros.

Triangle 
is the number of triangular columns in the basis with three (or more) nonzeros.

Pad 
is the number of slacks used to pad the basis (to make it a nonsingular triangle).

When
Major Print Level = 1${\mathbf{Major\; Print\; Level}}=1$ or
≥ 10$\text{}\ge 10$ and
Monitoring File ≥ 0${\mathbf{Monitoring\; File}}\ge 0$, the following lines of final printout (
< 120$\text{}<120$ characters) are produced on the unit number specified by optional parameter
Monitoring File.
Let x_{j}${x}_{\mathit{j}}$ denote the j$\mathit{j}$th ‘column variable’, for j = 1,2, … ,n$\mathit{j}=1,2,\dots ,n$. We assume that a typical variable x_{j}${x}_{j}$ has bounds α ≤ x_{j} ≤ β$\alpha \le {x}_{j}\le \beta $.
The following describes the printout for each column (or variable). A full stop (.) is printed for any numerical value that is zero.
Number 
is the column number j$j$. (This is used internally to refer to x_{j}${x}_{j}$ in the intermediate output.)

Column 
gives the name of x_{j}${x}_{j}$.

State 
gives the state of x_{j}${x}_{j}$ relative to the bounds α$\alpha $ and β$\beta $.
The various possible states are as follows:
LL 
x_{j}${x}_{j}$ is nonbasic at its lower limit, α$\alpha $.

UL 
x_{j}${x}_{j}$ is nonbasic at its upper limit, β$\beta $.

EQ 
x_{j}${x}_{j}$ is nonbasic and fixed at the value α = β$\alpha =\beta $.

FR 
x_{j}${x}_{j}$ is nonbasic at some value strictly between its bounds: α < x_{j} < β$\alpha <{x}_{j}<\beta $.

BS 
x_{j}${x}_{j}$ is basic. Usually α < x_{j} < β$\alpha <{x}_{j}<\beta $.

A key is sometimes printed before State.
Note that unless the optional parameter Scale Option = 0${\mathbf{Scale\; Option}}=0$ is specified, the tests for assigning a key are applied to the variables of the scaled problem.
A 
Alternative optimum possible. The variable is nonbasic, but its reduced gradient is essentially zero. This means that if the variable were allowed to start moving away from its current value, there would be no change in the value of the objective function. The values of the basic and superbasic variables might change, giving a genuine alternative solution. The values of the Lagrange multipliers might also change.

D 
Degenerate. The variable is basic, but it is equal to (or very close to) one of its bounds.

I 
Infeasible. The variable is basic and is currently violating one of its bounds by more than the value of the optional parameter Minor Feasibility Tolerance.

N 
Not precisely optimal. The variable is nonbasic. Its reduced gradient is larger than the value of the optional parameter Major Feasibility Tolerance.


Activity 
is the value of x_{j}${x}_{j}$ at the final iterate.

Obj Gradient 
is the value of g_{j}${g}_{j}$ at the final iterate. (If any x_{j}${x}_{j}$ is infeasible, g_{j}${g}_{j}$ is the gradient of the sum of infeasibilities.)

Lower Bound 
is the lower bound specified for the variable. None indicates that bl(j) ≤ − bigbnd${\mathbf{bl}}\left(j\right)\le \mathit{bigbnd}$.

Upper Bound 
is the upper bound specified for the variable. None indicates that bu(j) ≥ bigbnd${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$.

Reduced Gradnt 
is the value of d_{j}${d}_{j}$ at the final iterate.

m + j 
is the value of m + j$m+j$.

General linear constraints take the form l ≤ Ax ≤ u$l\le Ax\le u$. The i$i$th constraint is therefore of the form α ≤ a_{i}^{T}x ≤ β$\alpha \le {a}_{i}^{\mathrm{T}}x\le \beta $ and the value of a_{i}^{T}x${a}_{i}^{\mathrm{T}}x$ is called the row activity. Internally, the linear constraints take the form Ax − s = 0$Axs=0$, where the slack variables s$s$ should satisfy the bounds l ≤ s ≤ u$l\le s\le u$. For the i$i$th ‘row’, it is the slack variable s_{i}${s}_{i}$ that is directly available and it is sometimes convenient to refer to its state. Slacks may be basic or nonbasic (but not superbasic).
Nonlinear constraints α ≤ F_{i}(x) + a_{i}^{T}x ≤ β$\alpha \le {F}_{i}\left(x\right)+{a}_{i}^{\mathrm{T}}x\le \beta $ are treated similarly, except that the row activity and degree of infeasibility are computed directly from F_{i}(x) + a_{i}^{T}x${F}_{i}\left(x\right)+{a}_{i}^{\mathrm{T}}x$ rather than from s_{i}${s}_{i}$.
The following describes the printout for each row (or constraint). A full stop (.) is printed for any numerical value that is zero.
Number 
is the value of n + i$n+i$. (This is used internally to refer to s_{i}${s}_{i}$ in the intermediate output.)

Row 
gives the name of the i$i$th row.

State 
gives the state of the i$i$th row relative to the bounds α$\alpha $ and β$\beta $.
The various possible states are as follows:
LL 
The row is at its lower limit, α$\alpha $.

UL 
The row is at its upper limit, β$\beta $.

EQ 
The limits are the same (α = β)$(\alpha =\beta )$.

BS 
The constraint is not binding. s_{i}${s}_{i}$ is basic.

A key is sometimes printed before State.
Note that unless the optional parameter Scale Option = 0${\mathbf{Scale\; Option}}=0$ is specified, the tests for assigning a key are applied to the variables of the scaled problem.
A 
Alternative optimum possible. The variable is nonbasic, but its reduced gradient is essentially zero. This means that if the variable were allowed to start moving away from its current value, there would be no change in the value of the objective function. The values of the basic and superbasic variables might change, giving a genuine alternative solution. The values of the Lagrange multipliers might also change.

D 
Degenerate. The variable is basic, but it is equal to (or very close to) one of its bounds.

I 
Infeasible. The variable is basic and is currently violating one of its bounds by more than the value of the optional parameter Minor Feasibility Tolerance.

N 
Not precisely optimal. The variable is nonbasic. Its reduced gradient is larger than the value of the optional parameter Major Feasibility Tolerance.


Activity 
is the value of a_{i}^{T}x${a}_{i}^{\mathrm{T}}x$ (or F_{i}(x) + a_{i}^{T}x${F}_{i}\left(x\right)+{a}_{i}^{\mathrm{T}}x$ for nonlinear rows) at the final iterate.

Slack Activity 
is the value by which the row differs from its nearest bound. (For the free row (if any), it is set to Activity.)

Lower Bound 
is α$\alpha $, the lower bound specified for the i$i$th row. None indicates that bl(n + i) ≤ − bigbnd${\mathbf{bl}}\left(n+i\right)\le \mathit{bigbnd}$.

Upper Bound 
is β$\beta $, the upper bound specified for the i$i$th row. None indicates that bu(n + i) ≥ bigbnd${\mathbf{bu}}\left(n+i\right)\ge \mathit{bigbnd}$.

Dual Activity 
is the value of the dual variable π_{i}${\pi}_{i}$.

i 
gives the index i$i$ of the i$i$th row.

Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.
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© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013