PDF version (NAG web site
, 64bit version, 64bit version)
NAG Toolbox: nag_opt_nlp1_rcomm (e04uf)
Purpose
nag_opt_nlp1_rcomm (e04uf) is designed to minimize an arbitrary smooth function subject to constraints (which may include simple bounds on the variables, linear constraints and smooth nonlinear constraints) using a sequential quadratic programming (SQP) method. You should supply as many first derivatives as possible; any unspecified derivatives are approximated by finite differences. It is not intended for large sparse problems.
nag_opt_nlp1_rcomm (e04uf) may also be used for unconstrained, boundconstrained and linearly constrained optimization.
nag_opt_nlp1_rcomm (e04uf) uses reverse communication for evaluating the objective function, the nonlinear constraint functions and any of their derivatives.
Syntax
[
irevcm,
iter,
istate,
c,
cjac,
clamda,
objf,
objgrd,
r,
x,
needc,
iwork,
work,
cwsav,
lwsav,
iwsav,
rwsav,
ifail] = e04uf(
irevcm,
a,
bl,
bu,
iter,
istate,
c,
cjac,
clamda,
objf,
objgrd,
r,
x,
iwork,
work,
cwsav,
lwsav,
iwsav,
rwsav, 'n',
n, 'nclin',
nclin, 'ncnln',
ncnln)
[
irevcm,
iter,
istate,
c,
cjac,
clamda,
objf,
objgrd,
r,
x,
needc,
iwork,
work,
cwsav,
lwsav,
iwsav,
rwsav,
ifail] = nag_opt_nlp1_rcomm(
irevcm,
a,
bl,
bu,
iter,
istate,
c,
cjac,
clamda,
objf,
objgrd,
r,
x,
iwork,
work,
cwsav,
lwsav,
iwsav,
rwsav, 'n',
n, 'nclin',
nclin, 'ncnln',
ncnln)
Before calling
nag_opt_nlp1_rcomm (e04uf), or either of the option setting functions
(e04ue),
nag_opt_init (e04wb) must be called.
Note: the interface to this routine has changed since earlier releases of the toolbox:
.
Description
nag_opt_nlp1_rcomm (e04uf) is designed to solve the nonlinear programming problem – the minimization of a smooth nonlinear function subject to a set of constraints on the variables. The problem is assumed to be stated in the following form:
where
F(x)$F\left(x\right)$ (the
objective function) is a nonlinear function,
A_{L}${A}_{L}$ is an
n_{L}${n}_{L}$ by
n$n$ constant matrix, and
c(x)$c\left(x\right)$ is an
n_{N}${n}_{N}$ element vector of nonlinear constraint functions. (The matrix
A_{L}${A}_{L}$ and the vector
c(x)$c\left(x\right)$ may be empty.) The objective function and the constraint functions are assumed to be smooth, i.e., at least twicecontinuously differentiable. (The method of
nag_opt_nlp1_rcomm (e04uf) will usually solve
(1) if there are only isolated discontinuities away from the solution.)
Note that although the bounds on the variables could be included in the definition of the linear constraints, we prefer to distinguish between them for reasons of computational efficiency. For the same reason, the linear constraints should
not be included in the definition of the nonlinear constraints. Upper and lower bounds are specified for all the variables and for all the constraints. An
equality constraint can be specified by setting
l_{i} = u_{i}${l}_{i}={u}_{i}$. 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 $. (See the description of the optional parameter
Infinite Bound Size.)
If there are no nonlinear constraints in
(1) and
F$F$ is linear or quadratic then it will generally be more efficient to use one of
nag_opt_lp_solve (e04mf),
nag_opt_lsq_lincon_solve (e04nc) or
nag_opt_qp_dense_solve (e04nf), or
nag_opt_qpconvex2_sparse_solve (e04nq) if the problem is large and sparse. If the problem is large and sparse and does have nonlinear constraints,
nag_opt_nlp1_sparse_solve (e04ug) should be used, since
nag_opt_nlp1_rcomm (e04uf) treats all matrices as dense.
nag_opt_nlp1_rcomm (e04uf) uses reverse communication for evaluating
F(x)$F\left(x\right)$,
c(x)$c\left(x\right)$ and as many of their first partial derivatives as possible; any remaining derivatives are approximated by finite differences. See the description of the optional parameter
Derivative Level.
On initial entry, you must supply an initial estimate of the solution to
(1).
On intermediate exits, the calling program must compute appropriate values for the objective function, the nonlinear constraints or their derivatives, as specified by the parameter
irevcm, and then reenter the function.
For maximum reliability, it is preferable to provide all partial derivatives (see Chapter 8 of
Gill et al. (1981), for a detailed discussion). If they cannot all be provided, it is advisable to provide as many as possible. While developing code to evaluate the objective function and the constraints, the optional parameter
Verify should be used to check the calculation of any known derivatives.
The method used by
nag_opt_nlp1_rcomm (e04uf) is described in detail in
Section [Algorithmic Details].
nag_opt_nlp2_solve (e04wd) is an alternative function which uses a similar method, but with
forward communication:
that is, the objective and constraint functions are evaluated by functions, supplied as parameters to the function.
References
Dennis J E Jr and Moré J J (1977) QuasiNewton methods, motivation and theory SIAM Rev. 19 46–89
Dennis J E Jr and Schnabel R B (1981) A new derivation of symmetric positivedefinite secant updates nonlinear programming (eds O L Mangasarian, R R Meyer and S M Robinson) 4 167–199 Academic Press
Dennis J E Jr and Schnabel R B (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations Prentice–Hall
Fletcher R (1987) Practical Methods of Optimization (2nd Edition) Wiley
Gill P E, Hammarling S, Murray W, Saunders M A and Wright M H (1986) Users' guide for LSSOL (Version 1.0) Report SOL 861 Department of Operations Research, Stanford University
Gill P E, Murray W, Saunders M A and Wright M H (1984a) Procedures for optimization problems with a mixture of bounds and general linear constraints ACM Trans. Math. Software 10 282–298
Gill P E, Murray W, Saunders M A and Wright M H (1984b) Users' guide for SOL/QPSOL version 3.2 Report SOL 84–5 Department of Operations Research, Stanford University
Gill P E, Murray W, Saunders M A and Wright M H (1986a) Some theoretical properties of an augmented Lagrangian merit function Report SOL 86–6R Department of Operations Research, Stanford University
Gill P E, Murray W, Saunders M A and Wright M H (1986b) 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 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 (1983) MINOS 5.0 user's guide Report SOL 8320 Department of Operations Research, Stanford University
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
Powell M J D (1983) Variable metric methods in constrained optimization Mathematical Programming: the State of the Art (eds A Bachem, M Grötschel and B Korte) 288–311 Springer–Verlag
Parameters
Note: this function uses
reverse communication. Its use involves an initial entry, intermediate exits and reentries, and a final exit, as indicated by the
parameter irevcm. Between intermediate exits and reentries,
all parameters must remain unchanged except those specified by the value of irevcm.
Compulsory Input Parameters
 1:
irevcm – int64int32nag_int scalar
On initial entry: must be set to 0$0$.
On intermediate reentry:
must remain unchanged, unless you wish to terminate the solution to the current problem. In this case
irevcm may be set to a negative value and then
nag_opt_nlp1_rcomm (e04uf) will take a final exit with
ifail set to this value of
irevcm.
Constraint:
irevcm ≤ 6${\mathbf{irevcm}}\le 6$.
 2:
a(lda, : $:$) – double array

The first dimension of the array
a must be at least
max (1,nclin)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{nclin}})$The second dimension of the array must be at least
n${\mathbf{n}}$ if
nclin > 0${\mathbf{nclin}}>0$, and at least
1$1$ otherwise
On initial entry: the
i$i$th row of the matrix
A_{L}${A}_{L}$ of general linear constraints in
(1) must be stored in
a(i,j)${\mathbf{a}}\left(\mathit{i},\mathit{j}\right)$, for
i = 1,2, … ,nclin$\mathit{i}=1,2,\dots ,{\mathbf{nclin}}$ and
j = 1,2, … ,n$\mathit{j}=1,2,\dots ,{\mathbf{n}}$.
That is, the
i$\mathit{i}$th row contains the coefficients of the
i$\mathit{i}$th general linear constraint, for
i = 1,2, … ,nclin$\mathit{i}=1,2,\dots ,{\mathbf{nclin}}$.
If
nclin = 0${\mathbf{nclin}}=0$, the array
a is not referenced.
 3:
bl(n + nclin + ncnln${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnln}}$) – double array
 4:
bu(n + nclin + ncnln${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnln}}$) – double array
On initial entry:
bl must contain the lower bounds and
bu the upper bounds, for all the constraints in the following order. The first
n$n$ elements of each array must contain the bounds on the variables, the next
n_{L}${n}_{L}$ elements the bounds for the general linear constraints (if any) and the next
n_{N}${n}_{N}$ elements the bounds for the general nonlinear constraints (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}$, and to specify a nonexistent upper bound (i.e.,
u_{j} = + ∞${u}_{j}=+\infty $), set
bu(j) ≥ bigbnd${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$; the default value of
bigbnd$\mathit{bigbnd}$ is
10^{20}${10}^{20}$, but this may be changed by the optional parameter
Infinite Bound Size. 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}$.
Constraints:
 bl(j) ≤ bu(j)${\mathbf{bl}}\left(\mathit{j}\right)\le {\mathbf{bu}}\left(\mathit{j}\right)$, for j = 1,2, … ,n + nclin + ncnln$\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnln}}$;
 if bl(j) = bu(j) = β${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)=\beta $, β < bigbnd$\left\beta \right<\mathit{bigbnd}$.
 5:
iter – int64int32nag_int scalar
On intermediate reentry: must remain unchanged from a previous call to nag_opt_nlp1_rcomm (e04uf).
 6:
istate(n + nclin + ncnln${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnln}}$) – int64int32nag_int array
On initial entry: need not be set if the (default) optional parameter
Cold Start is used.
If the optional parameter
Warm Start has been chosen, the elements of
istate corresponding to the bounds and linear constraints define the initial working set for the procedure that finds a feasible point for the linear constraints and bounds. The active set at the conclusion of this procedure and the elements of
istate corresponding to nonlinear constraints then define the initial working set for the first QP subproblem. More precisely, the first
n$n$ elements of
istate refer to the upper and lower bounds on the variables, the next
n_{L}${n}_{L}$ elements refer to the upper and lower bounds on
A_{L}x${A}_{L}x$, and the next
n_{N}${n}_{N}$ elements refer to the upper and lower bounds on
c(x)$c\left(x\right)$. Possible values for
istate(j)${\mathbf{istate}}\left(j\right)$ are as follows:
istate(j)${\mathbf{istate}}\left(j\right)$  Meaning 
0  The corresponding constraint is not in the initial QP working set. 
1  This inequality constraint should be in the working set at its lower bound. 
2  This inequality constraint should be in the working set at its upper bound. 
3  This equality constraint should be in the initial working set. This value must not be specified unless bl(j) = bu(j)${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)$. 
The values
− 2$2$,
− 1$1$ and
4$4$ are also acceptable but will be modified by the function. If
nag_opt_nlp1_rcomm (e04uf) has been called previously with the same values of
n,
nclin and
ncnln,
istate already contains satisfactory information. (See also the description of the optional parameter
Warm Start.) The function also adjusts (if necessary) the values supplied in
x to be consistent with
istate.
Constraint:
− 2 ≤ istate(j) ≤ 4$2\le {\mathbf{istate}}\left(\mathit{j}\right)\le 4$, for
j = 1,2, … ,n + nclin + ncnln$\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnln}}$.
 7:
c( : $:$) – double array

Note: the dimension of the array
c
must be at least
max (1,ncnln)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{ncnln}})$.
On initial entry: need not be set.
On intermediate reentry: if
irevcm = 4${\mathbf{irevcm}}=4$ or
6$6$ and
needc(i) > 0${\mathbf{needc}}\left(i\right)>0$,
c(i)${\mathbf{c}}\left(i\right)$ must contain the value of the
i$i$th constraint at
x$x$. The remaining elements of
c, corresponding to the nonpositive elements of
needc, are ignored.
 8:
cjac(ldcj, : $:$) – double array

The first dimension of the array
cjac must be at least
max (1,ncnln)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{ncnln}})$The second dimension of the array must be at least
n${\mathbf{n}}$ if
ncnln > 0${\mathbf{ncnln}}>0$, and at least
1$1$ otherwise
On initial entry: in general,
cjac need not be initialized before the call to
nag_opt_nlp1_rcomm (e04uf). However, if the optional parameter
Derivative Level = 2${\mathbf{Derivative\; Level}}=2$ or
3$3$, you may optionally set the constant elements of
cjac. Such constant elements need not be reassigned on subsequent intermediate exits.
If all elements of the constraint Jacobian are known (i.e.,
Derivative Level = 2${\mathbf{Derivative\; Level}}=2$ or
3$3$), any constant elements may be assigned to
cjac one time only at the start of the optimization. An element of
cjac that is not subsequently assigned during an intermediate exit will retain its initial value throughout. Constant elements may be loaded into
cjac either before the call to
nag_opt_nlp1_rcomm (e04uf) or during the first intermediate exit. The ability to preload constants is useful when many Jacobian elements are identically zero, in which case
cjac may be initialized to zero and nonzero elements may be reset during intermediate exits.
On intermediate reentry: if
irevcm = 5${\mathbf{irevcm}}=5$ or
6$6$ and
needc(i) > 0${\mathbf{needc}}\left(i\right)>0$, the
i$i$th row of
cjac must contain the available elements of the vector
∇c_{i}$\nabla {c}_{i}$ given by
where
( ∂ c_{i})/( ∂ x_{j}) $\frac{\partial {c}_{i}}{\partial {x}_{j}}$ is the partial derivative of the
i$i$th constraint with respect to the
j$j$th variable, evaluated at the point
x$x$. The remaining rows of
cjac, corresponding to nonpositive elements of
needc, are ignored. The
i$\mathit{i}$th row of the Jacobian should be stored in elements
cjac(i,j)${\mathbf{cjac}}\left(\mathit{i},\mathit{j}\right)$, for
i = 1,2, … ,ncnln$\mathit{i}=1,2,\dots ,{\mathbf{ncnln}}$ and
j = 1,2, … ,n$\mathit{j}=1,2,\dots ,{\mathbf{n}}$.
Note that constant nonzero elements do affect the values of the constraints. Thus, if
cjac(i,j)${\mathbf{cjac}}\left(i,j\right)$ is set to a constant value, it need not be reset during subsequent intermediate exits, but the value
cjac(i,j) × x(j)${\mathbf{cjac}}\left(i,j\right)\times {\mathbf{x}}\left(j\right)$ must nonetheless be added to
c(i)${\mathbf{c}}\left(i\right)$. For example, if
cjac(1,1) = 2${\mathbf{cjac}}\left(1,1\right)=2$ and
cjac(1,2) = − 5${\mathbf{cjac}}\left(1,2\right)=5$, then the term
2 × x(1) − 5 × x(2)$2\times {\mathbf{x}}\left(1\right)5\times {\mathbf{x}}\left(2\right)$ must be included in the definition of
c(1)${\mathbf{c}}\left(1\right)$.
It must be emphasized that, if
Derivative Level = 0${\mathbf{Derivative\; Level}}=0$ or
1$1$, unassigned elements of
cjac are not treated as constant; they are estimated by finite differences, at nontrivial expense. If you do not supply a value for the optional parameter
Difference Interval, an interval for each element of
x$x$ is computed automatically at the start of the optimization. The automatic procedure can usually identify constant elements of
cjac, which are then computed once only by finite differences.
See also the description of the optional parameter
Verify.
 9:
clamda(n + nclin + ncnln${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnln}}$) – double array
On initial entry: need not be set if the (default) optional parameter
Cold Start is used.
If the optional parameter
Warm Start has been chosen,
clamda(j)${\mathbf{clamda}}\left(\mathit{j}\right)$ must contain a multiplier estimate for each nonlinear constraint with a sign that matches the status of the constraint specified by the
istate array, for
j = n + nclin + 1, … ,n + nclin + ncnln$\mathit{j}={\mathbf{n}}+{\mathbf{nclin}}+1,\dots ,{\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnln}}$. The remaining elements need not be set. Note that if the
j$j$th constraint is defined as ‘inactive’ by the initial value of the
istate array (i.e.
istate(j) = 0${\mathbf{istate}}\left(j\right)=0$),
clamda(j)${\mathbf{clamda}}\left(j\right)$ should be zero; if the
j$j$th constraint is an inequality active at its lower bound (i.e.
istate(j) = 1${\mathbf{istate}}\left(j\right)=1$),
clamda(j)${\mathbf{clamda}}\left(j\right)$ should be nonnegative; if the
j$j$th constraint is an inequality active at its upper bound (i.e.
istate(j) = 2${\mathbf{istate}}\left(j\right)=2$),
clamda(j)${\mathbf{clamda}}\left(j\right)$ should be nonpositive. If necessary, the function will modify
clamda to match these rules.
 10:
objf – double scalar
On initial entry: need not be set.
On intermediate reentry: if
irevcm = 1${\mathbf{irevcm}}=1$ or
3$3$,
objf must be set to the value of the objective function at
x$x$.
 11:
objgrd(n) – double array
n, the dimension of the array, must satisfy the constraint
n > 0${\mathbf{n}}>0$.
On initial entry: need not be set.
n, the dimension of the array, must satisfy the constraint
n > 0${\mathbf{n}}>0$.
On intermediate reentry: if
irevcm = 2${\mathbf{irevcm}}=2$ or
3$3$,
objgrd must contain the available elements of the gradient evaluated at
x$x$.
See also the description of the optional parameter
Verify.
 12:
r(ldr,n) – double array
ldr, the first dimension of the array, must satisfy the constraint
ldr ≥ n$\mathit{ldr}\ge {\mathbf{n}}$.
On initial entry: need not be initialized if the (default) optional parameter
Cold Start is used.
If the optional parameter
Warm Start has been chosen,
r must contain the upper triangular Cholesky factor
R$R$ of the initial approximation of the Hessian of the Lagrangian function, with the variables in the natural order. Elements not in the upper triangular part of
r are assumed to be zero and need not be assigned.
 13:
x(n) – double array
n, the dimension of the array, must satisfy the constraint
n > 0${\mathbf{n}}>0$.
On initial entry: an initial estimate of the solution.
 14:
iwork(liwork) – int64int32nag_int array
liwork, the dimension of the array, must satisfy the constraint
liwork ≥ 3 × n + nclin + 2 × ncnln$\mathit{liwork}\ge 3\times {\mathbf{n}}+{\mathbf{nclin}}+2\times {\mathbf{ncnln}}$.
On initial entry: the dimension of the array
iwork as declared in the (sub)program from which
nag_opt_nlp1_rcomm (e04uf) is called.
Constraint:
liwork ≥ 3 × n + nclin + 2 × ncnln$\mathit{liwork}\ge 3\times {\mathbf{n}}+{\mathbf{nclin}}+2\times {\mathbf{ncnln}}$.
 15:
work(lwork) – double array
lwork, the dimension of the array, must satisfy the constraint
 if ncnln = 0${\mathbf{ncnln}}=0$ and nclin = 0${\mathbf{nclin}}=0$, lwork ≥ 21 × n + 2$\mathit{lwork}\ge 21\times {\mathbf{n}}+2$;
 if ncnln = 0${\mathbf{ncnln}}=0$ and nclin > 0${\mathbf{nclin}}>0$, lwork ≥ 2 × n^{2} + 21 × n + 11 × nclin + 2$\mathit{lwork}\ge 2\times {{\mathbf{n}}}^{2}+21\times {\mathbf{n}}+11\times {\mathbf{nclin}}+2$;
 if ncnln > 0${\mathbf{ncnln}}>0$ and nclin ≥ 0${\mathbf{nclin}}\ge 0$,
lwork ≥ 2 × n^{2} + n × nclin + 2 × n × ncnln + 21 × n + 11 × nclin + 22 × ncnln + 1$\mathit{lwork}\ge 2\times {{\mathbf{n}}}^{2}+{\mathbf{n}}\times {\mathbf{nclin}}+2\times {\mathbf{n}}\times {\mathbf{ncnln}}+21\times {\mathbf{n}}+11\times {\mathbf{nclin}}+\phantom{\rule{0ex}{0ex}}22\times {\mathbf{ncnln}}+1$.
On initial entry: the dimension of the array
work as declared in the (sub)program from which
nag_opt_nlp1_rcomm (e04uf) is called.
Constraints:
 if ncnln = 0${\mathbf{ncnln}}=0$ and nclin = 0${\mathbf{nclin}}=0$, lwork ≥ 21 × n + 2$\mathit{lwork}\ge 21\times {\mathbf{n}}+2$;
 if ncnln = 0${\mathbf{ncnln}}=0$ and nclin > 0${\mathbf{nclin}}>0$, lwork ≥ 2 × n^{2} + 21 × n + 11 × nclin + 2$\mathit{lwork}\ge 2\times {{\mathbf{n}}}^{2}+21\times {\mathbf{n}}+11\times {\mathbf{nclin}}+2$;
 if ncnln > 0${\mathbf{ncnln}}>0$ and nclin ≥ 0${\mathbf{nclin}}\ge 0$,
lwork ≥ 2 × n^{2} + n × nclin + 2 × n × ncnln + 21 × n + 11 × nclin + 22 × ncnln + 1$\mathit{lwork}\ge 2\times {{\mathbf{n}}}^{2}+{\mathbf{n}}\times {\mathbf{nclin}}+2\times {\mathbf{n}}\times {\mathbf{ncnln}}+21\times {\mathbf{n}}+11\times {\mathbf{nclin}}+\phantom{\rule{0ex}{0ex}}22\times {\mathbf{ncnln}}+1$.
The amounts of workspace provided and required may be (by default for
nag_opt_nlp1_rcomm (e04uf)) output on the current advisory message unit (as defined by
nag_file_set_unit_advisory (x04ab)). As an alternative to computing
liwork and
lwork from the formulae given above, you may prefer to obtain appropriate values from the output of a preliminary run with
liwork and
lwork set to
1$1$. (
nag_opt_nlp1_rcomm (e04uf) will then terminate with
ifail = 9${\mathbf{ifail}}={\mathbf{9}}$.)
 16:
cwsav(5$5$) – cell array of strings
 17:
lwsav(120$120$) – logical array
 18:
iwsav(610$610$) – int64int32nag_int array
 19:
rwsav(475$475$) – double array
The arrays
lwsav,
iwsav,
rwsav and
cwsav must not be altered between calls to any of the functions
nag_opt_init (e04wb),
nag_opt_nlp1_rcomm (e04uf),
(e04ue).
Optional Input Parameters
 1:
n – int64int32nag_int scalar
Default:
The dimension of the arrays
objgrd,
x and the first dimension of the array
r and the second dimension of the array
r. (An error is raised if these dimensions are not equal.)
On initial entry: n$n$, the number of variables.
Constraint:
n > 0${\mathbf{n}}>0$.
 2:
nclin – int64int32nag_int scalar
Default:
The first dimension of the array
a.
On initial entry: n_{L}${n}_{L}$, the number of general linear constraints.
Constraint:
nclin ≥ 0${\mathbf{nclin}}\ge 0$.
 3:
ncnln – int64int32nag_int scalar
Default:
The first dimension of the array
cjac The dimension of the array
c.
On initial entry: n_{N}${n}_{N}$, the number of nonlinear constraints.
Constraint:
ncnln ≥ 0${\mathbf{ncnln}}\ge 0$.
Input Parameters Omitted from the MATLAB Interface
 lda ldcj ldr liwork lwork
Output Parameters
 1:
irevcm – int64int32nag_int scalar
On intermediate exit:
specifies what values the calling program must assign to parameters of
nag_opt_nlp1_rcomm (e04uf) before reentering the function.
 irevcm = 1${\mathbf{irevcm}}=1$
 Set objf to the value of the objective function F(x)$F\left(x\right)$.
 irevcm = 2${\mathbf{irevcm}}=2$
 Set
objgrd(j)${\mathbf{objgrd}}\left(\mathit{j}\right)$ to the value ( ∂ F)/( ∂ x_{j}) $\frac{\partial F}{\partial {x}_{\mathit{j}}}$ if available, for j = 1,2, … ,n$\mathit{j}=1,2,\dots ,n$.
 irevcm = 3${\mathbf{irevcm}}=3$
 Set objf and objgrd(j)${\mathbf{objgrd}}\left(j\right)$ as for irevcm = 1${\mathbf{irevcm}}=1$ and irevcm = 2${\mathbf{irevcm}}=2$.
 irevcm = 4${\mathbf{irevcm}}=4$
 Set c(i)${\mathbf{c}}\left(i\right)$ to the value of the constraint function c_{i}(x)${c}_{i}\left(x\right)$, for each i$i$ such that needc(i) > 0${\mathbf{needc}}\left(i\right)>0$.
 irevcm = 5${\mathbf{irevcm}}=5$
 Set cjac(i,j)${\mathbf{cjac}}\left(i,j\right)$ to the value ( ∂ c_{i})/( ∂ x_{j}) $\frac{\partial {c}_{i}}{\partial {x}_{j}}$ if available, for each i$i$ such that needc(i) > 0${\mathbf{needc}}\left(i\right)>0$ and j = 1,2, … ,n$j=1,2,\dots ,n$.
 irevcm = 6${\mathbf{irevcm}}=6$
 Set c(i)${\mathbf{c}}\left(i\right)$ and cjac(i,j)${\mathbf{cjac}}\left(i,j\right)$ as for irevcm = 4${\mathbf{irevcm}}=4$ and irevcm = 5${\mathbf{irevcm}}=5$.
On final exit:
irevcm = 0${\mathbf{irevcm}}=0$.
 2:
iter – int64int32nag_int scalar
On final exit: the number of major iterations performed.
 3:
istate(n + nclin + ncnln${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnln}}$) – int64int32nag_int array
On final exit: the status of the constraints in the QP working set at the point returned in
x. The significance of each possible value of
istate(j)${\mathbf{istate}}\left(j\right)$ is as follows:
istate(j)${\mathbf{istate}}\left(j\right)$  Meaning 
− 2$2$  This constraint violates its lower bound by more than the appropriate feasibility tolerance (see the optional parameters Linear Feasibility Tolerance and Nonlinear Feasibility Tolerance). This value can occur only when no feasible point can be found for a QP subproblem. 
− 1$1$  This constraint violates its upper bound by more than the appropriate feasibility tolerance (see the optional parameters Linear Feasibility Tolerance and Nonlinear Feasibility Tolerance). This value can occur only when no feasible point can be found for a QP subproblem. 
− 0$\phantom{}0$  The constraint is satisfied to within the feasibility tolerance, but is not in the QP working set. 
− 1$\phantom{}1$  This inequality constraint is included in the QP working set at its lower bound. 
− 2$\phantom{}2$  This inequality constraint is included in the QP working set at its upper bound. 
− 3$\phantom{}3$  This constraint is included in the QP working set as an equality. This value of istate can occur only when bl(j) = bu(j)${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)$. 
 4:
c( : $:$) – double array

Note: the dimension of the array
c
must be at least
max (1,ncnln)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{ncnln}})$.
On final exit: if
ncnln > 0${\mathbf{ncnln}}>0$,
c(i)${\mathbf{c}}\left(\mathit{i}\right)$ contains the value of the
i$\mathit{i}$th nonlinear constraint function
c_{i}${c}_{\mathit{i}}$ at the final iterate, for
i = 1,2, … ,ncnln$\mathit{i}=1,2,\dots ,{\mathbf{ncnln}}$.
If
ncnln = 0${\mathbf{ncnln}}=0$, the array
c is not referenced.
 5:
cjac(ldcj, : $:$) – double array

The first dimension of the array
cjac will be
max (1,ncnln)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{ncnln}})$The second dimension of the array will be
n${\mathbf{n}}$ if
ncnln > 0${\mathbf{ncnln}}>0$, and at least
1$1$ otherwise
ldcj ≥ max (1,ncnln)$\mathit{ldcj}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{ncnln}})$.
On final exit: if
ncnln > 0${\mathbf{ncnln}}>0$,
cjac contains the Jacobian matrix of the nonlinear constraint functions at the final iterate, i.e.,
cjac(i,j)${\mathbf{cjac}}\left(\mathit{i},\mathit{j}\right)$ contains the partial derivative of the
i$\mathit{i}$th constraint function with respect to the
j$\mathit{j}$th variable, for
i = 1,2, … ,ncnln$\mathit{i}=1,2,\dots ,{\mathbf{ncnln}}$ and
j = 1,2, … ,n$\mathit{j}=1,2,\dots ,{\mathbf{n}}$.
If
ncnln = 0${\mathbf{ncnln}}=0$, the array
cjac is not referenced.
 6:
clamda(n + nclin + ncnln${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnln}}$) – double array
On final exit: the values of the QP multipliers from the last QP subproblem.
clamda(j)${\mathbf{clamda}}\left(j\right)$ should be nonnegative if
istate(j) = 1${\mathbf{istate}}\left(j\right)=1$ and nonpositive if
istate(j) = 2${\mathbf{istate}}\left(j\right)=2$.
 7:
objf – double scalar
On final exit: the value of the objective function at the final iterate.
 8:
objgrd(n) – double array
On final exit: the gradient of the objective function at the final iterate (or its finite difference approximation).
 9:
r(ldr,n) – double array
ldr ≥ n$\mathit{ldr}\ge {\mathbf{n}}$.
On final exit: if
Hessian = NO${\mathbf{Hessian}}=\mathrm{NO}$,
r contains the upper triangular Cholesky factor
R$R$ of
Q^{T}H̃Q${Q}^{\mathrm{T}}\stackrel{~}{H}Q$, an estimate of the transformed and reordered Hessian of the Lagrangian at
x$x$ (see
(6) in
Section [Overview]).
If
Hessian = YES${\mathbf{Hessian}}=\mathrm{YES}$,
r contains the upper triangular Cholesky factor
R$R$ of
H$H$, the approximate (untransformed) Hessian of the Lagrangian, with the variables in the natural order.
 10:
x(n) – double array
On intermediate exit:
the point x$x$ at which the objective function, constraint functions or their derivatives are to be evaluated.
On final exit: the final estimate of the solution.
 11:
needc(max (1,ncnln)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{ncnln}})$) – int64int32nag_int array
On intermediate exit:
if
irevcm ≥ 4${\mathbf{irevcm}}\ge 4$,
needc specifies the indices of the elements of
c and/or
cjac that must be assigned. If
needc(i) > 0${\mathbf{needc}}\left(i\right)>0$, then the
i$i$th element of
c and/or the available elements of the
i$i$th row of
cjac must be evaluated at
x$x$.
 12:
iwork(liwork) – int64int32nag_int array
 13:
work(lwork) – double array
The amounts of workspace provided and required may be (by default for
nag_opt_nlp1_rcomm (e04uf)) output on the current advisory message unit (as defined by
nag_file_set_unit_advisory (x04ab)). As an alternative to computing
liwork and
lwork from the formulae given above, you may prefer to obtain appropriate values from the output of a preliminary run with
liwork and
lwork set to
1$1$. (
nag_opt_nlp1_rcomm (e04uf) will then terminate with
ifail = 9${\mathbf{ifail}}={\mathbf{9}}$.)
 14:
cwsav(5$5$) – cell array of strings
 15:
lwsav(120$120$) – logical array
 16:
iwsav(610$610$) – int64int32nag_int array
 17:
rwsav(475$475$) – double array
 18:
ifail – int64int32nag_int scalar
ifail = 0${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see
[Error Indicators and Warnings]).
nag_opt_nlp1_rcomm (e04uf) returns with
ifail = 0${\mathbf{ifail}}={\mathbf{0}}$ if the iterates have converged to a point
x$x$ that satisfies the firstorder Kuhn–Tucker conditions (see
Section [Overview]) to the accuracy requested by the optional parameter
Optimality Tolerance. This has
default value = ε_{r}^{0.8}$\text{default value}={\epsilon}_{r}^{0.8}$, where
ε_{r}${\epsilon}_{r}$ is the value of the optional parameter
Function Precision (
default value = ε^{0.9}$\text{default value}={\epsilon}^{0.9}$, where
ε$\epsilon $ is the
machine precision). That is
ifail = 0${\mathbf{ifail}}={\mathbf{0}}$ when the projected gradient and active constraint residuals are negligible at
x$x$.
You should check whether the following four conditions are satisfied:
If all these conditions hold,
x$x$ is almost certainly a local minimum of
(1).
Error Indicators and Warnings
Note: nag_opt_nlp1_rcomm (e04uf) 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_rcomm (e04uf) because you set
irevcm < 0${\mathbf{irevcm}}<0$ during an intermediate exit. The value of
ifail will be the same as your setting of
irevcm.
 W ifail = 1${\mathbf{ifail}}=1$
The final iterate
x$x$ satisfies the firstorder Kuhn–Tucker conditions (see
Section [Overview]) to the accuracy requested, but the sequence of iterates has not yet converged.
nag_opt_nlp1_rcomm (e04uf) was terminated because no further improvement could be made in the merit function (see
Section [Printed output]).
This value of
ifail may occur in several circumstances. The most common situation is that you ask for a solution with accuracy that is not attainable with the given precision of the problem (as specified by the optional parameter
Function Precision). This condition will also occur if, by chance, an iterate is an ‘exact’ Kuhn–Tucker point, but the change in the variables was significant at the previous iteration. (This situation often happens when minimizing very simple functions, such as quadratics.)
If the four conditions listed in
Section [Parameters] for
ifail = 0${\mathbf{ifail}}={\mathbf{0}}$ are satisfied,
x$x$ is likely to be a solution of
(1) even if
ifail = 1${\mathbf{ifail}}={\mathbf{1}}$.
 W ifail = 2${\mathbf{ifail}}=2$
nag_opt_nlp1_rcomm (e04uf) has terminated without finding a feasible point for the linear constraints and bounds, which means that either no feasible point exists for the given value of the optional parameter
Linear Feasibility Tolerance, or no feasible point could be found in the number of iterations specified by the optional parameter
Minor Iteration Limit. You should check that there are no constraint redundancies. If the data for the constraints are accurate only to an absolute precision
σ$\sigma $, you should ensure that the value of the optional parameter
Linear Feasibility Tolerance is greater than
σ$\sigma $. For example, if all elements of
A_{L}${A}_{L}$ are of order unity and are accurate to only three decimal places,
Linear Feasibility Tolerance should be at least
10^{ − 3}${10}^{3}$.
 W ifail = 3${\mathbf{ifail}}=3$
No feasible point could be found for the nonlinear constraints. The problem may have no feasible solution. This means that there has been a sequence of QP subproblems for which no feasible point could be found (indicated by
I at the end of each line of intermediate printout produced by the major iterations; see
Section [Printed output]). This behaviour will occur if there is no feasible point for the nonlinear constraints. (However, there is no general test that can determine whether a feasible point exists for a set of nonlinear constraints.) If the infeasible subproblems occur from the very first major iteration, it is highly likely that no feasible point exists. If infeasibilities occur when earlier subproblems have been feasible, small constraint inconsistencies may be present. You should check the validity of constraints with negative values of
istate. If you are convinced that a feasible point does exist,
nag_opt_nlp1_rcomm (e04uf) should be restarted at a different starting point.
 ifail = 4${\mathbf{ifail}}=4$
The limiting number of iterations (as determined by the optional parameter
Major Iteration Limit) has been reached.
If the algorithm appears to be making satisfactory progress, then optional parameter
Major Iteration Limit may be too small. If so, either increase its value and rerun
nag_opt_nlp1_rcomm (e04uf) or, alternatively, rerun
nag_opt_nlp1_rcomm (e04uf) using the optional parameter
Warm Start. If the algorithm seems to be making little or no progress however, then you should check for incorrect gradients or illconditioning as described under
ifail = 6${\mathbf{ifail}}={\mathbf{6}}$.
Note that illconditioning in the working set is sometimes resolved automatically by the algorithm, in which case performing additional iterations may be helpful. However, illconditioning in the Hessian approximation tends to persist once it has begun, so that allowing additional iterations without altering
r${\mathbf{r}}$ is usually inadvisable. If the quasiNewton update of the Hessian approximation was reset during the latter major iterations (i.e., an
R occurs at the end of each line of intermediate printout; see
Section [Printed output]), it may be worthwhile to try a
Warm Start at the final point as suggested above.
 ifail = 5${\mathbf{ifail}}=5$
Not used by this function.
 W ifail = 6${\mathbf{ifail}}=6$
x$x$ does not satisfy the firstorder Kuhn–Tucker conditions (see
Section [Overview]), and no improved point for the merit function (see
Section [Printed output]) could be found during the final linesearch.
This sometimes occurs because an overly stringent accuracy has been requested, i.e., the value of the optional parameter
Optimality Tolerance (
default value = ε_{r}^{0.8}$\text{default value}={\epsilon}_{r}^{0.8}$, where
ε_{r}${\epsilon}_{r}$ is the value of the optional parameter
Function Precision) is too small. In this case you should apply the four tests described under
ifail = 0${\mathbf{ifail}}={\mathbf{0}}$ to determine whether or not the final solution is acceptable (see
Gill et al. (1981), for a discussion of the attainable accuracy).
If many iterations have occurred in which essentially no progress has been made and
nag_opt_nlp1_rcomm (e04uf) has failed completely to move from the initial point, then values set by the calling program for the objective or constraint functions or their derivatives during intermediate exits may be incorrect. You should refer to comments under
ifail = 7${\mathbf{ifail}}={\mathbf{7}}$ and check the gradients using the optional parameter
Verify. Unfortunately, there may be small errors in the objective and constraint gradients that cannot be detected by the verification process. Finite difference approximations to first derivatives are catastrophically affected by even small inaccuracies. An indication of this situation is a dramatic alteration in the iterates if the finite difference interval is altered. One might also suspect this type of error if a switch is made to central differences even when
Norm Gz and
Violtn (see
Section [Printed output]) are large.
Another possibility is that the search direction has become inaccurate because of illconditioning in the Hessian approximation or the matrix of constraints in the working set; either form of illconditioning tends to be reflected in large values of
Mnr (the number of iterations required to solve each QP subproblem; see
Section [Printed output]).
If the condition estimate of the projected Hessian (
Cond Hz; see
Section [Printed output]) is extremely large, it may be worthwhile rerunning
nag_opt_nlp1_rcomm (e04uf) from the final point with the optional parameter
Warm Start. In this situation,
istate and
clamda should be left unaltered and
r${\mathbf{r}}$ should be reset to the identity matrix.
If the matrix of constraints in the working set is illconditioned (i.e.,
Cond T is extremely large; see
Section [Description of Monitoring Information]), it may be helpful to run
nag_opt_nlp1_rcomm (e04uf) with a relaxed value of the optional parameter
Feasibility Tolerance. (Constraint dependencies are often indicated by wide variations in size in the diagonal elements of the matrix
T$T$, whose diagonals will be printed if
Major Print Level ≥ 30${\mathbf{Major\; Print\; Level}}\ge 30$.)
 ifail = 7${\mathbf{ifail}}=7$
The usersupplied derivatives of the objective function and/or nonlinear constraints appear to be incorrect.
Large errors were found in the derivatives of the objective function and/or nonlinear constraints. This value of
ifail will occur if the verification process indicated that at least one gradient or Jacobian element had no correct figures. You should refer to the printed output to determine which elements are suspected to be in error.
As a firststep, you should check that the code for the objective and constraint values is correct – for example, by computing the function at a point where the correct value is known. However, care should be taken that the chosen point fully tests the evaluation of the function. It is remarkable how often the values x = 0$x=0$ or x = 1$x=1$ are used in such a test, and how often the special properties of these numbers make the test meaningless.
Special care should be used in the test if computation of the objective function involves subsidiary data communicated in global storage. Although the first evaluation of the function may be correct, subsequent calculations may be in error because some of the subsidiary data has accidentally been overwritten.
Gradient checking will be ineffective if the objective function uses information computed by the constraints, since they are not necessarily computed before each function evaluation.
Errors in programming the function may be quite subtle in that the function value is ‘almost’ correct. For example, the function may not be accurate to full precision because of the inaccurate calculation of a subsidiary quantity, or the limited accuracy of data upon which the function depends. A common error on machines where numerical calculations are usually performed in double precision is to include even one single precision constant in the calculation of the function; since some compilers do not convert such constants to double precision, half the correct figures may be lost by such a seemingly trivial error.
 ifail = 8${\mathbf{ifail}}=8$
Not used by this function.
 ifail = 9${\mathbf{ifail}}=9$

An input parameter is invalid.
 Overflow$\mathbf{\text{Overflow}}$

If overflow occurs then either an element of C$C$ is very large, or the singular values or singular vectors have been incorrectly supplied.
Accuracy
If
ifail = 0${\mathbf{ifail}}={\mathbf{0}}$ on final exit then the vector returned in the array
x is an estimate of the solution to an accuracy of approximately
Optimality Tolerance (
default value = ε^{0.8}$\text{default value}={\epsilon}^{0.8}$, where
ε$\epsilon $ is the
machine precision).
Further Comments
Description of the Printed Output
This section describes the intermediate printout and final printout produced by
nag_opt_nlp1_rcomm (e04uf). The intermediate printout is a subset of the monitoring information produced by
nag_opt_nlp1_rcomm (e04uf) 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_rcomm (e04uf), by default no output is produced by
nag_opt_nlp1_rcomm (e04uf)).
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 (12) at the current iterate. This function will decrease at each iteration unless it was necessary to increase the penalty parameters
(see Section [The Merit Function]).
As the solution is approached, Merit Function will converge to the value of the objective function at the solution.
If the QP subproblem does not have a feasible point (signified by I at the end of the current output line) then the merit function is a large multiple of the constraint violations, weighted by the penalty parameters. During a sequence of major iterations with infeasible subproblems, the sequence of Merit Function values will decrease monotonically until either a feasible subproblem is obtained or nag_opt_nlp1_rcomm (e04uf) terminates with ifail = 3${\mathbf{ifail}}={\mathbf{3}}$ (no feasible point could be found for the nonlinear constraints).
If there are no nonlinear constraints present (i.e., ncnln = 0${\mathbf{ncnln}}=0$) then this entry contains Objective, the value of the objective function F(x)$F\left(x\right)$. The objective function will decrease monotonically to its optimal value when there are no nonlinear constraints.

Norm Gz 
is ‖Z^{T}g_{FR}‖$\Vert {Z}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, the Euclidean norm of the projected gradient
(see Section [Solution of the Quadratic Programming Subproblem]).
Norm Gz will be approximately zero in the neighbourhood of a solution.

Violtn 
is the Euclidean norm of the residuals of constraints that are violated or in the predicted active set (not printed if ncnln is zero). Violtn will be approximately zero in the neighbourhood of a solution.

Cond Hz 
is a lower bound on the condition number of the projected Hessian approximation
H_{Z}
${H}_{Z}$ (
H_{Z}
=
Z^{T}
H_{FR}
Z
=
R_{Z}^{T}
R_{Z}
${H}_{Z}={Z}^{\mathrm{T}}{H}_{\mathrm{FR}}Z={R}_{Z}^{\mathrm{T}}{R}_{Z}$; see (6)). The larger this number, the more difficult the problem.

M 
is printed if the quasiNewton update has been modified to ensure that the Hessian approximation is positive definite
(see Section [The QuasiNewton Update]).

I 
is printed if the QP subproblem has no feasible point.

C 
is printed if central differences have been used to compute the unspecified objective and constraint gradients. If the value of Step is zero then the switch to central differences was made because no lower point could be found in the linesearch. (In this case, the QP subproblem is resolved with the central difference gradient and Jacobian.) If the value of Step is nonzero then central differences were computed because Norm Gz and Violtn imply that x$x$ is close to a Kuhn–Tucker point (see Section [Overview]).

L 
is printed if the linesearch has produced a relative change in x$x$ greater than the value defined by the optional parameter Step Limit. If this output occurs frequently during later iterations of the run, optional parameter Step Limit should be set to a larger value.

R 
is printed if the approximate Hessian has been refactorized. If the diagonal condition estimator of R$R$ indicates that the approximate Hessian is badly conditioned then the approximate Hessian is refactorized using column interchanges. If necessary, R$R$ is modified so that its diagonal condition estimator is bounded.

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.
Varbl 
gives the name (V) and index j$\mathit{j}$, for j = 1,2, … ,n$\mathit{j}=1,2,\dots ,n$, of the variable.

State 
gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the Feasibility Tolerance, State will be ++ or  respectively.
A key is sometimes printed before State.
A 
Alternative optimum possible. The variable is active at one of its bounds, but its Lagrange multiplier is essentially zero. This means that if the variable were allowed to start moving away from its bound then there would be no change to the objective function. The values of the other free variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case the values of the Lagrange multipliers might also change.

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

I 
Infeasible. The variable is currently violating one of its bounds by more than the 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 unless bl(j) ≤ − bigbnd${\mathbf{bl}}\left(j\right)\le \mathit{bigbnd}$ and bu(j) ≥ bigbnd${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$, in which case the entry will be blank. If x$x$ is optimal, the multiplier should be nonnegative if State is LL and nonpositive if State is UL.

Slack 
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 linear and nonlinear constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’,
bl(j)${\mathbf{bl}}\left(j\right)$ and
bu(j)${\mathbf{bu}}\left(j\right)$ replaced by
bl(n + j)${\mathbf{bl}}\left(n+j\right)$ and
bu(n + j)${\mathbf{bu}}\left(n+j\right)$ respectively and with the following changes in the heading:
L Con 
gives the name (L) and index j$\mathit{j}$, for j = 1,2, … ,n_{L}$\mathit{j}=1,2,\dots ,{n}_{L}$, of the linear constraint.

N Con 
gives the name (N) and index (j − n_{L}$\mathit{j}{n}_{L}$), for j = n_{L} + 1, … ,n_{L} + n_{N}$\mathit{j}={n}_{L}+1,\dots ,{n}_{L}+{n}_{N}$, of the nonlinear constraint.

Note that movement off a constraint (as opposed to a variable moving away from its bound) can be interpreted as allowing the entry in the Slack column to become positive.
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_rcomm_example
function nag_opt_nlp1_rcomm_example
irevcm = int64(0);
nclin = 1;
a = [1, 1, 1, 1];
bl = [1;
1;
1;
1;
1e25;
1e25;
25];
bu = [5;
5;
5;
5;
20;
40;
1e25];
iter = int64(0);
istate = zeros(7, 1, 'int64');
c = [0;
0];
cjac = [0, 0, 0, 0;
0, 0, 0, 0];
clamda = zeros(7, 1);
objf = 0;
objgrd = zeros(4, 1);
r = zeros(4, 4);
x = [1; 5; 5; 1];
iwork = zeros(100, 1, 'int64');
work = zeros(1000, 1);
[cwsav,lwsav,iwsav,rwsav,ifail] = nag_opt_init('nag_opt_nlp1_rcomm');
if (ifail == 0)
[irevcm, iter, istate, c, cjac, clamda, objf, objgrd, r, x, needc, ...
iwork, work, cwsav, lwsav, iwsav, rwsav, ifail] = ...
nag_opt_nlp1_rcomm(irevcm, a, bl, bu, iter, istate, c, cjac, clamda, ...
objf, objgrd, r, x, iwork, work, cwsav, lwsav, iwsav, rwsav);
while (irevcm > 0)
if (irevcm == 1  irevcm == 3)
objf = x(1)*x(4)*(x(1)+x(2)+x(3)) + x(3);
end
if (irevcm == 2  irevcm == 3)
objgrd(1) = x(4)*(2*x(1)+x(2)+x(3));
objgrd(2) = x(1)*x(4);
objgrd(3) = x(1)*x(4) + 1;
objgrd(4) = x(1)*(x(1)+x(2)+x(3));
end
if (irevcm == 4  irevcm == 6)
if (needc(1) > 0)
c(1) = x(1)^2 + x(2)^2 + x(3)^2 + x(4)^2;
end
if (needc(2) > 0)
c(2) = x(1)*x(2)*x(3)*x(4);
end
end
if (irevcm == 5  irevcm == 6)
if (needc(1) > 0)
cjac(1,1) = 2*x(1);
cjac(1,2) = 2*x(2);
cjac(1,3) = 2*x(3);
cjac(1,4) = 2*x(4);
end
if (needc(2) > 0)
cjac(2,1) = x(2)*x(3)*x(4);
cjac(2,2) = x(1)*x(3)*x(4);
cjac(2,3) = x(1)*x(2)*x(4);
cjac(2,4) = x(1)*x(2)*x(3);
end
end
[irevcm, iter, istate, c, cjac, clamda, objf, objgrd, r, x, needc, ...
iwork, work, cwsav, lwsav, iwsav, rwsav, ifail] = ...
nag_opt_nlp1_rcomm(irevcm, a, bl, bu, iter, istate, c, cjac, clamda, ...
objf, objgrd, r, x, iwork, work, cwsav, lwsav, iwsav, rwsav);
end
if (ifail == 0)
fprintf('\n Varbl Istate Value Lagr Mult\n');
for i=1:4
fprintf(' V %3d %3d %14.4f %12.4f \n', i, istate(i), x(i), clamda(i));
end
ax = a*x;
fprintf('\n L Con Istate Value Lagr Mult\n')
for i=5:4+nclin
j=i4;
fprintf(' L %3d %3d %14.4f%12.4f\n', j, istate(i), ax(j), clamda(i));
end
fprintf('\n L Con Istate Value Lagr Mult\n')
for i=5+nclin:6+nclin
j=i4nclin;
fprintf(' N %3d %3d %14.4f%12.4f\n', j, istate(i), c(j), clamda(i));
end
fprintf('\n Final objective value = %15.7f\n', objf)
end
end
Varbl Istate Value Lagr Mult
V 1 1 1.0000 1.0879
V 2 0 4.7430 0.0000
V 3 0 3.8211 0.0000
V 4 0 1.3794 0.0000
L Con Istate Value Lagr Mult
L 1 0 10.9436 0.0000
L Con Istate Value Lagr Mult
N 1 2 40.0000 0.1615
N 2 1 25.0000 0.5523
Final objective value = 17.0140173
Open in the MATLAB editor:
e04uf_example
function e04uf_example
nclin = 1;
irevcm = int64(0);
a = [1, 1, 1, 1];
bl = [1;
1;
1;
1;
1e25;
1e25;
25];
bu = [5;
5;
5;
5;
20;
40;
1e25];
iter = int64(0);
istate = zeros(7, 1, 'int64');
c = [0;
0];
cjac = [0, 0, 0, 0;
0, 0, 0, 0];
clamda = zeros(7, 1);
objf = 0;
objgrd = zeros(4, 1);
r = zeros(4, 4);
x = [1; 5; 5; 1];
iwork = zeros(100, 1, 'int64');
work = zeros(1000, 1);
[cwsav,lwsav,iwsav,rwsav,ifail] = e04wb('e04uf');
if (ifail == 0)
[irevcm, iter, istate, c, cjac, clamda, objf, objgrd, r, x, needc, ...
iwork, work, cwsav, lwsav, iwsav, rwsav, ifail] = ...
e04uf(irevcm, a, bl, bu, iter, istate, c, cjac, clamda, ...
objf, objgrd, r, x, iwork, work, cwsav, lwsav, iwsav, rwsav);
while (irevcm > 0)
if (irevcm == 1  irevcm == 3)
objf = x(1)*x(4)*(x(1)+x(2)+x(3)) + x(3);
end
if (irevcm == 2  irevcm == 3)
objgrd(1) = x(4)*(2*x(1)+x(2)+x(3));
objgrd(2) = x(1)*x(4);
objgrd(3) = x(1)*x(4) + 1;
objgrd(4) = x(1)*(x(1)+x(2)+x(3));
end
if (irevcm == 4  irevcm == 6)
if (needc(1) > 0)
c(1) = x(1)^2 + x(2)^2 + x(3)^2 + x(4)^2;
end
if (needc(2) > 0)
c(2) = x(1)*x(2)*x(3)*x(4);
end
end
if (irevcm == 5  irevcm == 6)
if (needc(1) > 0)
cjac(1,1) = 2*x(1);
cjac(1,2) = 2*x(2);
cjac(1,3) = 2*x(3);
cjac(1,4) = 2*x(4);
end
if (needc(2) > 0)
cjac(2,1) = x(2)*x(3)*x(4);
cjac(2,2) = x(1)*x(3)*x(4);
cjac(2,3) = x(1)*x(2)*x(4);
cjac(2,4) = x(1)*x(2)*x(3);
end
end
[irevcm, iter, istate, c, cjac, clamda, objf, objgrd, r, x, needc, ...
iwork, work, cwsav, lwsav, iwsav, rwsav, ifail] = ...
e04uf(irevcm, a, bl, bu, iter, istate, c, cjac, clamda, ...
objf, objgrd, r, x, iwork, work, cwsav, lwsav, iwsav, rwsav);
end
if (ifail == 0)
fprintf('\n Varbl Istate Value Lagr Mult\n');
for i=1:4
fprintf(' V %3d %3d %14.4f %12.4f \n', i, istate(i), x(i), clamda(i));
end
ax = a*x;
fprintf('\n L Con Istate Value Lagr Mult\n')
for i=5:4+nclin
j=i4;
fprintf(' L %3d %3d %14.4f%12.4f\n', j, istate(i), ax(j), clamda(i));
end
fprintf('\n L Con Istate Value Lagr Mult\n')
for i=5+nclin:6+nclin
j=i4nclin;
fprintf(' N %3d %3d %14.4f%12.4f\n', j, istate(i), c(j), clamda(i));
end
fprintf('\n Final objective value = %15.7f\n', objf)
end
end
Varbl Istate Value Lagr Mult
V 1 1 1.0000 1.0879
V 2 0 4.7430 0.0000
V 3 0 3.8211 0.0000
V 4 0 1.3794 0.0000
L Con Istate Value Lagr Mult
L 1 0 10.9436 0.0000
L Con Istate Value Lagr Mult
N 1 2 40.0000 0.1615
N 2 1 25.0000 0.5523
Final objective value = 17.0140173
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_option_string (e04ue). 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_rcomm (e04uf).
Overview
nag_opt_nlp1_rcomm (e04uf) is essentially identical to the function NPSOL described in
Gill et al. (1986b).
At a solution of
(1), some of the constraints will be
active, i.e., satisfied exactly. An active simple bound constraint implies that the corresponding variable is
fixed at its bound, and hence the variables are partitioned into
fixed and
free variables. Let
C$C$ denote the
m$m$ by
n$n$ matrix of gradients of the active general linear and nonlinear constraints. The number of fixed variables will be denoted by
n_{FX}
${n}_{\mathrm{FX}}$, with
n_{FR}
${n}_{\mathrm{FR}}$ (
n_{FR}
=
n − n_{FX}
${n}_{\mathrm{FR}}=n{n}_{\mathrm{FX}}$) the number of free variables. The subscripts ‘FX’ and ‘FR’ on a vector or matrix will denote the vector or matrix composed of the elements corresponding to fixed or free variables.
A point
x$x$ is a
firstorder Kuhn–Tucker point for
(1) (see
Powell (1974)) if the following conditions hold:
(i) 
x$x$ is feasible; 
(ii) 
there exist vectors ξ$\xi $ and λ$\lambda $ (the Lagrange multiplier vectors for the bound and general constraints) such that
where g$g$ is the gradient of F$F$ evaluated at x$x$ and ξ_{j} = 0${\xi}_{j}=0$ if the j$j$th variable is free. 
(iii) 
the Lagrange multiplier corresponding to an inequality constraint active at its lower bound must be nonnegative. It is nonpositive for an inequality constraint active at its upper bound. 
Let
Z$Z$ denote a matrix whose columns form a basis for the set of vectors orthogonal to the rows of
C_{FR}${C}_{\mathrm{FR}}$; i.e.,
C_{FR}Z = 0${C}_{\mathrm{FR}}Z=0$. An equivalent statement of the condition
(2) in terms of
Z$Z$ is
The vector
Z^{T}g_{FR}${Z}^{\mathrm{T}}{g}_{\mathrm{FR}}$ is termed the
projected gradient of
F$F$ at
x$x$. Certain additional conditions must be satisfied in order for a firstorder Kuhn–Tucker point to be a solution of
(1) (see
Powell (1974)).
nag_opt_nlp1_rcomm (e04uf) implements a sequential quadratic programming (SQP) method. For an overview of SQP methods, see
Fletcher (1987),
Gill et al. (1981) and
Powell (1983).
The basic structure of
nag_opt_nlp1_rcomm (e04uf) involves
major and
minor iterations. The major iterations generate a sequence of iterates
{x_{k}}$\left\{{x}_{k}\right\}$ that converge to
x^{*}${x}^{*}$, a firstorder Kuhn–Tucker point of
(1). At a typical major iteration, the new iterate
x$\stackrel{}{x}$ is defined by
where
x$x$ is the current iterate, the nonnegative scalar
α$\alpha $ is the
step length, and
p$p$ is the
search direction. (For simplicity, we shall always consider a typical iteration and avoid reference to the index of the iteration.) Also associated with each major iteration are estimates of the Lagrange multipliers and a prediction of the active set.
The search direction
p$p$ in
(3) is the solution of a quadratic programming subproblem of the form
where
g$g$ is the gradient of
F$F$ at
x$x$, the matrix
H$H$ is a positive definite quasiNewton approximation to the Hessian of the Lagrangian function (see
Section [The QuasiNewton Update]), and
A_{N}${A}_{N}$ is the Jacobian matrix of
c$c$ evaluated at
x$x$. (Finite difference estimates may be used for
g$g$ and
A_{N}${A}_{N}$; see the optional parameter
Derivative Level.) Let
l$l$ in
(1) be partitioned into three sections:
l_{B}${l}_{B}$,
l_{L}${l}_{L}$ and
l_{N}${l}_{N}$, corresponding to the bound, linear and nonlinear constraints. The vector
l$\stackrel{}{l}$ in
(4) is similarly partitioned and is defined as
where
c$c$ is the vector of nonlinear constraints evaluated at
x$x$. The vector
u$\stackrel{}{u}$ is defined in an analogous fashion.
The estimated Lagrange multipliers at each major iteration are the Lagrange multipliers from the subproblem
(4) (and similarly for the predicted active set). (The numbers of bounds, general linear and nonlinear constraints in the QP active set are the quantities
Bnd,
Lin and
Nln in the monitoring file output of
nag_opt_nlp1_rcomm (e04uf); see
Section [Description of Monitoring Information].) In
nag_opt_nlp1_rcomm (e04uf),
(4) is solved using
nag_opt_lsq_lincon_solve (e04nc). Since solving a quadratic program is itself an iterative procedure, the
minor iterations of
nag_opt_nlp1_rcomm (e04uf) are the iterations of
nag_opt_lsq_lincon_solve (e04nc). (More details about solving the subproblem are given in
Section [Solution of the Quadratic Programming Subproblem].)
Certain matrices associated with the QP subproblem are relevant in the major iterations. Let the subscripts ‘FX’ and ‘FR’ refer to the
predicted fixed and free variables, and let
C$C$ denote the
m$m$ by
n$n$ matrix of gradients of the general linear and nonlinear constraints in the predicted active set. Firstly, we have available the
TQ$TQ$ factorization of
C_{FR}${C}_{\mathrm{FR}}$:
where
T$T$ is a nonsingular
m$m$ by
m$m$ reversetriangular matrix (i.e.,
t_{ij} = 0${t}_{ij}=0$ if
i + j < m$i+j<m$), and the nonsingular
n_{FR}${n}_{\mathrm{FR}}$ by
n_{FR}${n}_{\mathrm{FR}}$ matrix
Q_{FR}${Q}_{\mathrm{FR}}$ is the product of orthogonal transformations (see
Gill et al. (1984b)). Secondly, we have the upper triangular Cholesky factor
R$R$ of the
transformed and reordered Hessian matrix
where
H̃$\stackrel{~}{H}$ is the Hessian
H$H$ with rows and columns permuted so that the free variables are first and
Q$Q$ is the
n$n$ by
n$n$ matrix
with
I_{FX}${I}_{\mathrm{FX}}$ the identity matrix of order
n_{FX}${n}_{\mathrm{FX}}$. If the columns of
Q_{FR}${Q}_{\mathrm{FR}}$ are partitioned so that
then the
n_{Z}${n}_{Z}$ (
n_{Z} ≡ n_{FR} − m${n}_{Z}\equiv {n}_{\mathrm{FR}}m$) columns of
Z$Z$ form a basis for the null space of
C_{FR}${C}_{\mathrm{FR}}$. The matrix
Z$Z$ is used to compute the projected gradient
Z^{T}g_{FR}${Z}^{\mathrm{T}}{g}_{\mathrm{FR}}$ at the current iterate. (The values
Nz and
Norm Gz printed by
nag_opt_nlp1_rcomm (e04uf) give
n_{Z}${n}_{Z}$ and
‖Z^{T}g_{FR}‖$\Vert {Z}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, see
Section [Description of Monitoring Information].)
A theoretical characteristic of SQP methods is that the predicted active set from the QP subproblem
(4) is identical to the correct active set in a neighbourhood of
x^{*}${x}^{*}$. In
nag_opt_nlp1_rcomm (e04uf), this feature is exploited by using the QP active set from the previous iteration as a prediction of the active set for the next QP subproblem, which leads in practice to optimality of the subproblems in only one iteration as the solution is approached. Separate treatment of bound and linear constraints in
nag_opt_nlp1_rcomm (e04uf) also saves computation in factorizing
C_{FR}${C}_{\mathrm{FR}}$ and
H_{Q}${H}_{Q}$.
Once
p$p$ has been computed, the major iteration proceeds by determining a step length
α$\alpha $ that produces a ‘sufficient decrease’ in an augmented Lagrangian
merit function (see
Section [The Merit Function]). Finally, the approximation to the transformed Hessian matrix
H_{Q}${H}_{Q}$ is updated using a modified BFGS quasiNewton update (see
Section [The QuasiNewton Update]) to incorporate new curvature information obtained in the move from
x$x$ to
x$\stackrel{}{x}$.
On entry to
nag_opt_nlp1_rcomm (e04uf), an iterative procedure from
nag_opt_lsq_lincon_solve (e04nc) is executed, starting with the usersupplied initial point, to find a point that is feasible with respect to the bounds and linear constraints (using the tolerance specified by the optional parameter
Linear Feasibility Tolerance). If no feasible point exists for the bound and linear constraints,
(1) has no solution and
nag_opt_nlp1_rcomm (e04uf) terminates. Otherwise, the problem functions will thereafter be evaluated only at points that are feasible with respect to the bounds and linear constraints. The only exception involves variables whose bounds differ by an amount comparable to the finite difference interval (see the discussion of the optional parameter
Difference Interval). In contrast to the bounds and linear constraints, it must be emphasized that
the nonlinear constraints will not generally be satisfied until an optimal point is reached.
Facilities are provided to check whether the usersupplied gradients appear to be correct (see the description of the optional parameter
Verify). In general, the check is provided at the first point that is feasible with respect to the linear constraints and bounds. However, you may request that the check be performed at the initial point.
In summary, the method of
nag_opt_nlp1_rcomm (e04uf) first determines a point that satisfies the bound and linear constraints. Thereafter, each iteration includes:
(a) 
the solution of a quadratic programming subproblem; 
(b) 
a linesearch with an augmented Lagrangian merit function; and 
(c) 
a quasiNewton update of the approximate Hessian of the Lagrangian function. 
Solution of the Quadratic Programming Subproblem
The search direction
p$p$ is obtained by solving
(4) using
nag_opt_lsq_lincon_solve (e04nc) (see
Gill et al. (1986)), which was specifically designed to be used within an SQP algorithm for nonlinear programming.
nag_opt_lsq_lincon_solve (e04nc) is based on a twophase (primal) quadratic programming method. The two phases of the method are: finding an initial feasible point by minimizing the sum of infeasibilities (the
feasibility phase) and minimizing the quadratic objective function within the feasible region (the
optimality phase). The computations in both phases are performed by the same functions. The twophase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities to the quadratic objective function.
In general, a quadratic program must be solved by iteration. Let
p$p$ denote the current estimate of the solution of
(4); the new iterate
p$\stackrel{}{p}$ is defined by
where, as in
(3),
σ$\sigma $ is a nonnegative step length and
d$d$ is a search direction.
At the beginning of each iteration of
nag_opt_lsq_lincon_solve (e04nc), a
working set is defined of constraints (general and bound) that are satisfied exactly. The vector
d$d$ is then constructed so that the values of constraints in the working set remain
unaltered for any move along
d$d$. For a bound constraint in the working set, this property is achieved by setting the corresponding element of
d$d$ to zero, i.e., by fixing the variable at its bound. As before, the subscripts ‘FX’ and ‘FR’ denote selection of the elements associated with the fixed and free variables.
Let
C$C$ denote the submatrix of rows of
corresponding to general constraints in the working set. The general constraints in the working set will remain unaltered if
which is equivalent to defining
d_{FR}${d}_{\mathrm{FR}}$ as
for some vector
d_{Z}${d}_{Z}$, where
Z$Z$ is the matrix associated with the
TQ$TQ$ factorization
(5) of
C_{FR}${C}_{\mathrm{FR}}$.
The definition of
d_{Z}${d}_{Z}$ in
(10) depends on whether the current
p$p$ is feasible. If not,
d_{Z}${d}_{Z}$ is zero except for an element
γ$\gamma $ in the
j$j$th position, where
j$j$ and
γ$\gamma $ are chosen so that the sum of infeasibilities is decreasing along
d$d$. (For further details, see
Gill et al. (1986).) In the feasible case,
d_{Z}${d}_{Z}$ satisfies the equations
where
R_{Z}${R}_{Z}$ is the Cholesky factor of
Z^{T}H_{FR}Z${Z}^{\mathrm{T}}{H}_{\mathrm{FR}}Z$ and
q$q$ is the gradient of the quadratic objective function
(q = g + Hp)$(q=g+Hp)$. (The vector
Z^{T}q_{FR}${Z}^{\mathrm{T}}{q}_{\mathrm{FR}}$ is the projected gradient of the QP.) With
(11),
p + d$p+d$ is the minimizer of the quadratic objective function subject to treating the constraints in the working set as equalities.
If the QP projected gradient is zero, the current point is a constrained stationary point in the subspace defined by the working set. During the feasibility phase, the projected gradient will usually be zero only at a vertex (although it may vanish at nonvertices in the presence of constraint dependencies). During the optimality phase, a zero projected gradient implies that
p$p$ minimizes the quadratic objective function when the constraints in the working set are treated as equalities. In either case, Lagrange multipliers are computed. Given a positive constant
δ$\delta $ of the order of the
machine precision, the Lagrange multiplier
μ_{j}${\mu}_{j}$ corresponding to an inequality constraint in the working set is said to be
optimal if
μ_{j} ≤ δ${\mu}_{j}\le \delta $ when the
j$j$th constraint is at its
upper bound, or if
μ_{j} ≥ − δ${\mu}_{j}\ge \delta $ when the associated constraint is at its
lower bound. If any multiplier is nonoptimal, the current objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint from the working set.
If optimal multipliers occur during the feasibility phase and the sum of infeasibilities is nonzero, no feasible point exists. The QP algorithm will then continue iterating to determine the minimum sum of infeasibilities. At this point, the Lagrange multiplier μ_{j}${\mu}_{j}$ will satisfy − (1 + δ) ≤ μ_{j} ≤ δ$(1+\delta )\le {\mu}_{j}\le \delta $ for an inequality constraint at its upper bound, and − δ ≤ μ_{j} ≤ (1 + δ)$\delta \le {\mu}_{j}\le (1+\delta )$ for an inequality at its lower bound. The Lagrange multiplier for an equality constraint will satisfy μ_{j} ≤ 1 + δ$\left{\mu}_{j}\right\le 1+\delta $.
The choice of step length
σ$\sigma $ in the QP iteration
(8) is based on remaining feasible with respect to the satisfied constraints. During the optimality phase, if
p + d$p+d$ is feasible,
σ$\sigma $ will be taken as unity. (In this case, the projected gradient at
p$\stackrel{}{p}$ will be zero.) Otherwise,
σ$\sigma $ is set to
σ_{M}${\sigma}_{\mathrm{M}}$, the step to the ‘nearest’ constraint, which is added to the working set at the next iteration.
Each change in the working set leads to a simple change to C_{FR}${C}_{\mathrm{FR}}$: if the status of a general constraint changes, a row of C_{FR}${C}_{\mathrm{FR}}$ is altered; if a bound constraint enters or leaves the working set, a column of C_{FR}${C}_{\mathrm{FR}}$ changes. Explicit representations are recurred of the matrices T$T$, Q_{FR}${Q}_{\mathrm{FR}}$ and R$R$, and of the vectors Q^{T}q${Q}^{\mathrm{T}}q$ and Q^{T}g${Q}^{\mathrm{T}}g$.
The Merit Function
After computing the search direction as described in
Section [Solution of the Quadratic Programming Subproblem], each major iteration proceeds by determining a step length
α$\alpha $ in
(3) that produces a ‘sufficient decrease’ in the augmented Lagrangian merit function
where
x$x$,
λ$\lambda $ and
s$s$ vary during the linesearch. The summation terms in
(12) involve only the
nonlinear constraints. The vector
λ$\lambda $ is an estimate of the Lagrange multipliers for the nonlinear constraints of
(1). The nonnegative
slack variables
{s_{i}}$\left\{{s}_{i}\right\}$ allow nonlinear inequality constraints to be treated without introducing discontinuities. The solution of the QP subproblem
(4) provides a vector triple that serves as a direction of search for the three sets of variables. The nonnegative vector
ρ$\rho $ of
penalty parameters is initialized to zero at the beginning of the first major iteration. Thereafter, selected elements are increased whenever necessary to ensure descent for the merit function. Thus, the sequence of norms of
ρ$\rho $ (the printed quantity
Penalty; see
Section [Description of Monitoring Information]) is generally nondecreasing, although each
ρ_{i}${\rho}_{i}$ may be reduced a limited number of times.
The merit function
(12) and its global convergence properties are described in
Gill et al. (1986a).
The QuasiNewton Update
The matrix
H$H$ in
(4) is a
positive definite quasiNewton approximation to the Hessian of the Lagrangian function. (For a review of quasiNewton methods, see
Dennis and Schnabel (1983).) At the end of each major iteration, a new Hessian approximation
H$\stackrel{}{H}$ is defined as a ranktwo modification of
H$H$. In
nag_opt_nlp1_rcomm (e04uf), the BFGS (Broyden–Fletcher–Goldfarb–Shanno) quasiNewton update is used:
where
s = x − x$s=\stackrel{}{x}x$ (the change in
x$x$).
In
nag_opt_nlp1_rcomm (e04uf),
H$H$ is required to be positive definite. If
H$H$ is positive definite,
H$\stackrel{}{H}$ defined by
(13) will be positive definite if and only if
y^{T}s${y}^{\mathrm{T}}s$ is positive (see
Dennis and Moré (1977)). Ideally,
y$y$ in
(13) would be taken as
y_{L}${y}_{L}$, the change in gradient of the Lagrangian function
where
μ_{N}${\mu}_{N}$ denotes the QP multipliers associated with the nonlinear constraints of the original problem. If
y_{L}^{T}
s
${y}_{L}^{\mathrm{T}}s$ is not sufficiently positive, an attempt is made to perform the update with a vector
y$y$ of the form
where
ω_{i} ≥ 0${\omega}_{i}\ge 0$. If no such vector can be found, the update is performed with a scaled
y_{L}${y}_{L}$. In this case,
M is printed to indicate that the update was modified.
Rather than modifying
H$H$ itself, the Cholesky factor of the
transformed Hessian
H_{Q}${H}_{Q}$ (6) is updated, where
Q$Q$ is the matrix from
(5) associated with the active set of the QP subproblem. The update
(13) is equivalent to the following update to
H_{Q}${H}_{Q}$:
where
y_{Q} = Q^{T}y${y}_{Q}={Q}^{\mathrm{T}}y$, and
s_{Q} = Q^{T}s${s}_{Q}={Q}^{\mathrm{T}}s$. This update may be expressed as a
rankone update to
R$R$ (see
Dennis and Schnabel (1981)).
Optional Parameters
Several optional parameters in nag_opt_nlp1_rcomm (e04uf) define choices in the problem specification or the algorithm logic. In order to reduce the complexity of using nag_opt_nlp1_rcomm (e04uf) 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_option_string (e04ue) before a call to
nag_opt_nlp1_rcomm (e04uf).
nag_opt_nlp1_option_string (e04ue) can be called to supply options directly, one call being necessary for each optional parameter. For example,
[lwsav, iwsav, rwsav, inform] = e04ue('Print Level = 1', lwsav, iwsav, rwsav);
nag_opt_nlp1_option_string (e04ue) should be consulted for a full description of this method of supplying optional parameters.
All optional parameters not specified are set to their default values. Optional parameters specified are unaltered by nag_opt_nlp1_rcomm (e04uf) (unless they define invalid values) and so remain in effect for subsequent calls to nag_opt_nlp1_rcomm (e04uf).
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, where the symbol ε$\epsilon $ is a generic notation for machine precision (see nag_machine_precision (x02aj)), and ε_{r}${\epsilon}_{r}$ denotes the relative precision of the objective function Function Precision.
Keywords and character values are case and white space insensitive.
Central Difference Interval r$r$Default values are computedIf the algorithm switches to central differences because the forwarddifference approximation is not sufficiently accurate then the value of
r$r$ is used as the difference interval for every element of
x$x$. 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 [Printed output]). The use of finite differences is discussed further under the optional parameter
Difference Interval.
If you supply a value for this optional parameter, a small value between 0.0$0.0$ and 1.0$1.0$ is appropriate.
Cold Start DefaultWarm Start This option controls the specification of the initial working set in both the procedure for finding a feasible point for the linear constraints and bounds and in the first QP subproblem thereafter. With a
Cold Start, the first working set is chosen by
nag_opt_nlp1_rcomm (e04uf) based on the values of the variables and constraints at the initial point. Broadly speaking, the initial working set will include equality constraints and bounds or inequality constraints that violate or ‘nearly’ satisfy their bounds (to within
Crash Tolerance).
With a
Warm Start, you must set the
istate array and define
clamda and
r as discussed in
Section [Parameters].
istate values associated with bounds and linear constraints determine the initial working set of the procedure to find a feasible point with respect to the bounds and linear constraints.
istate values associated with nonlinear constraints determine the initial working set of the first QP subproblem after such a feasible point has been found.
nag_opt_nlp1_rcomm (e04uf) will override your specification of
istate if necessary, so that a poor choice of the working set will not cause a fatal error. For instance, any elements of
istate which are set to
− 2$2$,
− 1 or 4$1\text{ or}4$ will be reset to zero, as will any elements which are set to
3$3$ when the corresponding elements of
bl and
bu are not equal. A warm start will be advantageous if a good estimate of the initial working set is available – for example, when
nag_opt_nlp1_rcomm (e04uf) is called repeatedly to solve related problems.
Crash Tolerance r$r$Default = 0.01$\text{}=0.01$This value is used in conjunction with the optional parameter
Cold Start (the default value) when
nag_opt_nlp1_rcomm (e04uf) selects an initial working set. If
0 ≤ r ≤ 1$0\le r\le 1$, the initial working set will include (if possible) bounds or general inequality constraints that lie within
r$r$ of their bounds. In particular, a constraint of the form
a_{j}^{T}
x ≥ l
${a}_{j}^{\mathrm{T}}x\ge l$ will be included in the initial working set if
a_{j}^{T}x − l
≤
r
(1 + l)
${a}_{j}^{\mathrm{T}}xl\le r(1+\leftl\right)$. If
r < 0$r<0$ or
r > 1$r>1$, 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 derivatives are provided during intermediate exits. 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 elements of the objective gradient are not specified. 
1 
All elements of the objective gradient are provided, but some elements of the constraint Jacobian are not specified. 
0 
Some elements of both the objective gradient and the constraint Jacobian are not specified. 
The value i = 3$i=3$ should be used whenever possible, since nag_opt_nlp1_rcomm (e04uf) is more reliable (and will usually be more efficient) when all derivatives are exact.
If
i = 0 or 2$i=0\text{ or}2$,
nag_opt_nlp1_rcomm (e04uf) will estimate the unspecified elements of the objective gradient, using finite differences. The computation of finite difference approximations usually increases the total runtime, since an intermediate exit to the calling program is required for each unspecified element. Furthermore, less accuracy can be attained in the solution (see Chapter 8 of
Gill et al. (1981), for a discussion of limiting accuracy).
If
i = 0 or 1$i=0\text{ or}1$,
nag_opt_nlp1_rcomm (e04uf) will approximate unspecified elements of the constraint Jacobian. One intermediate exit is needed for each variable for which partial derivatives are not available. For example, if the Jacobian has the form
where ‘
*$*$’ indicates an element provided and ‘?’ indicates an unspecified element,
nag_opt_nlp1_rcomm (e04uf) will make an intermediate exit to the calling program 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 intermediate exits for information.)
At times, central differences are used rather than forward differences, in which case twice as many intermediate exits 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.
Difference Interval r$r$Default values are computedThis option defines an interval used to estimate derivatives by finite differences in the following circumstances:
(a) 
For verifying the objective and/or constraint gradients (see the description of the optional parameter Verify). 
(b) 
For estimating unspecified elements of the objective gradient or the constraint Jacobian. 
In general, a derivative with respect to the
j$j$th variable is approximated using the interval
δ_{j}${\delta}_{j}$, where
δ_{j} = r(1 + x̂_{j})${\delta}_{j}=r(1+\left{\hat{x}}_{j}\right)$, with
x̂$\hat{x}$ the first point feasible with respect to the bounds and linear constraints. If the functions are well scaled then the resulting derivative approximation should be accurate to
O(r)$\mathit{O}\left(r\right)$. See
Gill et al. (1981) for a discussion of the accuracy in finite difference approximations.
If a difference interval is not specified then a finite difference interval will be computed automatically for each variable by a procedure that requires up to six intermediate exits for each element. This option is recommended if the function is badly scaled or you wish to have nag_opt_nlp1_rcomm (e04uf) determine constant elements in the objective and constraint gradients.
If you supply a value for this optional parameter, a small value between 0.0$0.0$ and 1.0$1.0$ is appropriate.
Feasibility Tolerance r$r$Default = sqrt(ε)$\text{}=\sqrt{\epsilon}$The scalar
r$r$ defines the maximum acceptable
absolute violations in linear and nonlinear constraints at a ‘feasible’ point; i.e., a constraint is considered satisfied if its violation does not exceed
r$r$. If
r < ε$r<\epsilon $ or
r ≥ 1$r\ge 1$, the default value is used. Using this keyword sets both optional parameters
Linear Feasibility Tolerance and
Nonlinear Feasibility Tolerance to
r$r$, if
ε ≤ r < 1$\epsilon \le r<1$. (Additional details are given under the descriptions of these optional parameters.)
Function Precision r$r$Default = ε^{0.9}$\text{}={\epsilon}^{0.9}$This parameter defines ε_{r}${\epsilon}_{r}$, which is intended to be a measure of the accuracy with which the problem functions F(x)$F\left(x\right)$ and c(x)$c\left(x\right)$ can be computed. If r < ε$r<\epsilon $ or r ≥ 1$r\ge 1$, the default value is used.
The value of
ε_{r}${\epsilon}_{r}$ should reflect the relative precision of
1 + F(x)$1+\leftF\left(x\right)\right$; i.e.,
ε_{r}${\epsilon}_{r}$ acts as a relative precision when
F$\leftF\right$ is large and as an absolute precision when
F$\leftF\right$ is small. For example, if
F(x)$F\left(x\right)$ is typically of order
1000$1000$ and the first six significant digits are known to be correct, an appropriate value for
ε_{r}${\epsilon}_{r}$ would be
10^{ − 6}${10}^{6}$. In contrast, if
F(x)$F\left(x\right)$ is typically of order
10^{ − 4}${10}^{4}$ and the first six significant digits are known to be correct, an 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. However, when the accuracy of the computed function values is known to be significantly worse than full precision, the value of
ε_{r}${\epsilon}_{r}$ should be large enough so that
nag_opt_nlp1_rcomm (e04uf) will not attempt to distinguish between function values that differ by less than the error inherent in the calculation.
Hessian Default = NO$=\mathrm{NO}$This option controls the contents of the upper triangular matrix
R$R$ (see
Section [Parameters]).
nag_opt_nlp1_rcomm (e04uf) works exclusively with the
transformed and reordered Hessian
H_{Q}${H}_{Q}$ (6), and hence extra computation is required to form the Hessian itself. If
Hessian = NO${\mathbf{Hessian}}=\mathrm{NO}$,
r contains the Cholesky factor of the transformed and reordered Hessian. If
Hessian = YES${\mathbf{Hessian}}=\mathrm{YES}$, the Cholesky factor of the approximate Hessian itself is formed and stored in
r. You should select
Hessian = YES${\mathbf{Hessian}}=\mathrm{YES}$ if a
Warm Start will be used for the next call to
nag_opt_nlp1_rcomm (e04uf).
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<0$, the default value is used.
Infinite Step Size r$r$Default = max (bigbnd,10^{20})$\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}(\mathit{bigbnd},{10}^{20})$If r > 0$r>0$, r$r$ specifies the magnitude of the change in variables that is treated as a step to an unbounded solution. If the change in x$x$ during an iteration would exceed the value of r$r$, the objective function is considered to be unbounded below in the feasible region. If r ≤ 0$r\le 0$, the default value is used.
Line Search Tolerance r$r$Default = 0.9$\text{}=0.9$The value r$r$ (0 ≤ r < 1$0\le r<1$) controls the accuracy with which the step α$\alpha $ taken during each iteration approximates a minimum of the merit function along the search direction (the smaller the value of r$r$, the more accurate the linesearch). 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 there are no nonlinear constraints, a more accurate search may be appropriate when it is desirable to reduce the number of major iterations – for example, if the objective function is cheap to evaluate, or if a substantial number of derivatives are unspecified. If r < 0$r<0$ or r ≥ 1$r\ge 1$, the default value is used.
Linear Feasibility Tolerance r_{1}${r}_{1}$Default = sqrt(ε)$\text{}=\sqrt{\epsilon}$Nonlinear Feasibility Tolerance r_{2}${r}_{2}$Default = ε^{0.33}$\text{}={\epsilon}^{0.33}$ or sqrt(ε)$\sqrt{\epsilon}$The default value of
r_{2}${r}_{2}$ is
ε^{0.33}${\epsilon}^{0.33}$ if
Derivative Level = 0${\mathbf{Derivative\; Level}}=0$ or
1$1$, and
sqrt(ε)$\sqrt{\epsilon}$ otherwise.
The scalars r_{1}${r}_{1}$ and r_{2}${r}_{2}$ define the maximum acceptable absolute violations in linear and nonlinear constraints at a ‘feasible’ point; i.e., a linear constraint is considered satisfied if its violation does not exceed r_{1}${r}_{1}$. Similarly a nonlinear constraint is considered satisfied if its violation does not exceed r_{2}${r}_{2}$. If r_{m} < ε${r}_{\mathit{m}}<\epsilon $ or r_{m} ≥ 1${r}_{\mathit{m}}\ge 1$, the default value is used, for m = 1,2$\mathit{m}=1,2$.
On entry to nag_opt_nlp1_rcomm (e04uf), an iterative procedure is executed in order to find a point that satisfies the linear constraints and bounds on the variables to within the tolerance r_{1}${r}_{1}$. All subsequent iterates will satisfy the linear constraints to within the same tolerance (unless r_{1}${r}_{1}$ is comparable to the finite difference interval).
For nonlinear constraints, the feasibility tolerance
r_{2}${r}_{2}$ defines the largest constraint violation that is acceptable at an optimal point. Since nonlinear constraints are generally not satisfied until the final iterate, the value of optional parameter
Nonlinear Feasibility Tolerance acts as a partial termination criterion for the iterative sequence generated by
nag_opt_nlp1_rcomm (e04uf) (see the discussion of optional parameter
Optimality Tolerance).
These tolerances should reflect the precision of the corresponding constraints. For example, if the variables and the coefficients in the linear constraints are of order unity, and the latter are correct to about 6$6$ decimal digits, it would be appropriate to specify r_{1}${r}_{1}$ as 10^{ − 6}${10}^{6}$.
List Nolist Default for nag_opt_nlp1_rcomm (e04uf)Optional parameter
List may be used to turn on printing of each optional parameter specification as it is supplied.
Nolist may then be used to suppress this printing again.
Major Iteration Limit i$i$Default = max (50,3(n + n_{L}) + 10n_{N})$\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}(50,3(n+{n}_{L})+10{n}_{N})$Iteration Limit Iters Itns The value of
i$i$ specifies the maximum number of major iterations allowed before termination. Setting
i = 0$i=0$ and
Major Print Level > 0${\mathbf{Major\; Print\; Level}}>0$ means that the workspace needed will be computed and printed, but no iterations will be performed. If
i < 0$i<0$, the default value is used.
Major Print Level i$i$Print Level Default for nag_opt_nlp1_rcomm (e04uf)
= 0$\text{}=0$The value of
i$i$ controls the amount of printout produced by the major iterations of
nag_opt_nlp1_rcomm (e04uf), as indicated below. A detailed description of the printed output is given in
Section [Printed output] (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 the optional parameter
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 [Printed output]) 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
by the optional parameter
Monitoring File:
i$i$ 
Output 
< 5$\text{}<5$ 
No output. 
≥ 5$\text{}\ge 5$ 
One long line of output ( > 80$\text{}>80$ characters; see Section [Description of Monitoring Information]) for each major iteration (no printout of the final solution). 
≥ 20$\text{}\ge 20$ 
At each major iteration, the objective function, the Euclidean norm of the nonlinear constraint violations, the values of the nonlinear constraints (the vector c$c$), the values of the linear constraints (the vector A_{L}x${A}_{L}x$) and the current values of the variables (the vector x$x$). 
≥ 30$\text{}\ge 30$ 
At each major iteration, the diagonal elements of the matrix T$T$ associated with the TQ$TQ$ factorization (5) (see Section [Overview]) of the QP working set and the diagonal elements of R$R$, the triangular factor of the transformed and reordered Hessian (6) (see Section [Overview]). 
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.
Minor Iteration Limit i$i$Default = max (50,3(n + n_{L} + n_{N}))$\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}(50,3(n+{n}_{L}+{n}_{N}))$The value of i$i$ specifies the maximum number of iterations for finding a feasible point with respect to the bounds and linear constraints (if any). The value of i$i$ also specifies the maximum number of minor iterations for the optimality phase of each QP subproblem. If i ≤ 0$i\le 0$, 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_rcomm (e04uf) (i.e., the iterations of the quadratic programming algorithm), as indicated below. A detailed description of the printed output is given in
Section [Printed output] in
(e04nc) (summary output at each minor iteration and the final QP solution) and
Section [Description of Monitoring Information] in
(e04nc)) (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 
≥ 00$\phantom{\ge 0}0$ 
No output. 
≥ 01$\phantom{\ge 0}1$ 
The final QP solution only. 
≥ 05$\phantom{\ge 0}5$ 
One line of summary output ( < 80$\text{}<80$ characters; see Section [Printed output] in (e04nc))
for each minor iteration (no printout of the final QP solution). 
≥ 10$\text{}\ge 10$ 
The final QP solution and one line of summary output for each minor iteration. 
The following printout is sent to the
logical unit number defined
by the optional parameter
Monitoring File:
i$i$ 
Output 
< 5$\text{}<5$ 
No output. 
≥ 5$\text{}\ge 5$ 
One long line of output ( > 80$\text{}>80$ characters; see
Section [Printed output] in (e04nc))
for each minor iteration (no printout of the final QP solution). 
≥ 20$\text{}\ge 20$ 
At each minor iteration, the current estimates of the QP multipliers, the current estimate of the QP search direction, the QP constraint values and the status of each QP constraint. 
≥ 30$\text{}\ge 30$ 
At each minor iteration, the diagonal elements of the matrix T$T$ associated with the TQ$TQ$ factorization (5) (see Section [Overview]) of the QP working set and the diagonal elements of the Cholesky factor R$R$ of the transformed Hessian (6) (see Section [Overview]). 
If
Minor Print Level ≥ 5${\mathbf{Minor\; Print\; Level}}\ge 5$ 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 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 ≥ 5${\mathbf{Minor\; Print\; Level}}\ge 5$, monitoring information produced by
nag_opt_nlp1_rcomm (e04uf) 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 < 5${\mathbf{Minor\; Print\; Level}}<5$, no monitoring information is produced.
Optimality Tolerance r$r$Default = ε_{r}^{0.8}$\text{}={\epsilon}_{r}^{0.8}$The parameter r$r$ (ε_{r} ≤ r < 1${\epsilon}_{r}\le r<1$) specifies the accuracy to which you wish the final iterate to approximate a solution of the problem. Broadly speaking, r$r$ indicates the number of correct figures desired in the objective function at the solution. For example, if r$r$ is 10^{ − 6}${10}^{6}$ and nag_opt_nlp1_rcomm (e04uf) terminates successfully, the final value of F$F$ should have approximately six correct figures. If r < ε_{r}$r<{\epsilon}_{r}$ or r ≥ 1$r\ge 1$, the default value is used.
nag_opt_nlp1_rcomm (e04uf) will terminate successfully if the iterative sequence of
x$x$ values is judged to have converged and the final point satisfies the firstorder Kuhn–Tucker conditions (see
Section [Overview]). The sequence of iterates is considered to have converged at
x$x$ if
where
p$p$ is the search direction and
α$\alpha $ the step length from
(3). An iterate is considered to satisfy the firstorder conditions for a minimum if
and
where
Z^{T}g_{FR}${Z}^{\mathrm{T}}{g}_{\mathrm{FR}}$ is the projected gradient (see
Section [Overview]),
g_{FR}${g}_{\mathrm{FR}}$ is the gradient of
F(x)$F\left(x\right)$ with respect to the free variables,
res_{j}${\mathit{res}}_{j}$ is the violation of the
j$j$th active nonlinear constraint, and
ftol$\mathit{ftol}$ is the
Nonlinear Feasibility Tolerance.
Start Objective Check At Variable i_{1}${i}_{1}$Default = 1$\text{}=1$Stop Objective Check At Variable i_{2}${i}_{2}$Default = n$\text{}=n$Start Constraint Check At Variable i_{3}${i}_{3}$Default = 1$\text{}=1$Stop Constraint Check At Variable i_{4}${i}_{4}$Default = n$\text{}=n$These keywords take effect only if
Verify Level > 0${\mathbf{Verify\; Level}}>0$. They may be used to control the verification of gradient elements and/or Jacobian elements computed by the calling program during intermediate exits. 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, it is reasonable to specify
Start Objective Check At Variable = 31${\mathbf{Start\; Objective\; Check\; At\; Variable}}=31$. If the first
30$30$ variables appear linearly in the objective, so that the corresponding gradient elements are constant, the above choice would also be appropriate.
If i_{2m − 1} ≤ 0${i}_{2\mathit{m}1}\le 0$ or i_{2m − 1} > min (n,i_{2m})${i}_{2\mathit{m}1}>\mathrm{min}\phantom{\rule{0.125em}{0ex}}(n,{i}_{2\mathit{m}})$, the default value is used, for m = 1,2$\mathit{m}=1,2$. If i_{2m} ≤ 0${i}_{2\mathit{m}}\le 0$ or i_{2m} > n${i}_{2\mathit{m}}>n$, the default value is used, for m = 1,2$\mathit{m}=1,2$.
Step Limit r$r$Default = 2.0$\text{}=2.0$If
r > 0,r$r>0,r$ specifies the maximum change in variables at the first step of the linesearch. 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 encourage evaluation of the problem functions at meaningful points. Given any major iterate
x$x$, the first point
x̃$\stackrel{~}{x}$ at which
F$F$ and
c$c$ are evaluated during the linesearch is restricted so that
The linesearch may go on and evaluate
F$F$ and
c$c$ at points further from
x$x$ if this will result in a lower value of the merit function (indicated by
L at the end of each line of output produced by the major iterations; see
Section [Printed output]). If
L is printed for most of the iterations,
r$r$ should be set to a larger value.
Wherever possible, upper and lower bounds on
x$x$ should be used to prevent evaluation of nonlinear functions at wild values. The default value
Step Limit = 2.0${\mathbf{Step\; Limit}}=2.0$ should not affect progress on wellbehaved functions, but values such as
0.1 or 0.01$0.1\text{ or}0.01$ may be helpful when rapidly varying functions are present. If a small value of
Step Limit is selected then 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.
Verify Level i$i$Default = 0$\text{}=0$Verify Verify Constraint Gradients Verify Gradients Verify Objective Gradients These keywords refer to finite difference checks on the gradient elements computed by the calling program during intermediate exits. (Unspecified gradient elements are not checked.) The possible choices for
i$i$ are as follows:
i$i$ 
Meaning 
− 1$1$ 
No checks are performed. 
− 0$\phantom{}0$ 
Only a ‘cheap’ test will be performed. 
≥ 1$\ge 1$ 
In addition to the ‘cheap’ test, individual gradient elements will also be checked using a reliable (but more expensive) test. 
It is possible to specify
Verify Level = 0${\mathbf{Verify\; Level}}=0$ to
3$3$ in several ways. For example, the objective gradient will be verified if
Verify,
Verify = YES${\mathbf{Verify}}=\mathrm{YES}$,
Verify Gradients,
Verify Objective Gradients or
Verify Level = 1${\mathbf{Verify\; Level}}=1$ is specified. The constraint gradients will be verified if
Verify = YES${\mathbf{Verify}}=\mathrm{YES}$ or
Verify Level = 2${\mathbf{Verify\; Level}}=2$ or
Verify is specified. Similarly, the objective and the constraint gradients will be verified if
Verify = YES${\mathbf{Verify}}=\mathrm{YES}$ or
Verify Level = 3${\mathbf{Verify\; Level}}=3$ or
Verify is specified.
If 0 ≤ i ≤ 3$0\le i\le 3$, gradients will be verified at the first point that satisfies the linear constraints and bounds.
If i = 0$i=0$, only a ‘cheap’ test will be performed, requiring one intermediate exit for the objective function gradients and (if appropriate) one intermediate exit for the partial derivatives of the constraints.
If
1 ≤ i ≤ 3$1\le i\le 3$, a more reliable (but more expensive) check will be made on individual gradient elements, within the ranges specified by the
Start Objective Check At Variable and
Stop Objective Check At Variable keywords. A result of the form
OK or
BAD? is printed by
nag_opt_nlp1_rcomm (e04uf) to indicate whether or not each element appears to be correct.
If 10 ≤ i ≤ 13$10\le i\le 13$, the action is the same as for i < 10$i<10$, except that it will take place at the userspecified initial value of x$x$.
If i < − 1$i<1$ or 4 ≤ i ≤ 9$4\le i\le 9$ or i > 13$i>13$, the default value is used.
We suggest that
Verify Level = 3${\mathbf{Verify\; Level}}=3$ be used whenever a new calling program is being developed.
Description of Monitoring Information
This section describes the long line of output (
> 80$\text{}>80$ characters) which forms part of the monitoring information produced by
nag_opt_nlp1_rcomm (e04uf). (See also the description of the optional parameters
Major Print Level,
Minor Print Level and
Monitoring File.) You can control the level of printed output (see the description of the optional parameter
Major Print Level).
When
Major Print Level ≥ 5${\mathbf{Major\; Print\; Level}}\ge 5$ and
Monitoring File ≥ 0${\mathbf{Monitoring\; File}}\ge 0$, the following line of output is produced at every major iteration of
nag_opt_nlp1_rcomm (e04uf)
on the unit number
specified by
Monitoring File. 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.

Nfun 
is the cumulative number of evaluations of the objective function needed for the linesearch. Evaluations needed for the estimation of the gradients by finite differences are not included. Nfun is printed as a guide to the amount of work required for the linesearch.

Merit Function 
is the value of the augmented Lagrangian merit function (12) at the current iterate. This function will decrease at each iteration unless it was necessary to increase the penalty parameters
(see Section [The Merit Function]).
As the solution is approached, Merit Function will converge to the value of the objective function at the solution.
If the QP subproblem does not have a feasible point (signified by I at the end of the current output line) then the merit function is a large multiple of the constraint violations, weighted by the penalty parameters. During a sequence of major iterations with infeasible subproblems, the sequence of Merit Function values will decrease monotonically until either a feasible subproblem is obtained or nag_opt_nlp1_rcomm (e04uf) terminates with ifail = 3${\mathbf{ifail}}={\mathbf{3}}$ (no feasible point could be found for the nonlinear constraints).
If there are no nonlinear constraints present (i.e., ncnln = 0${\mathbf{ncnln}}=0$) then this entry contains Objective, the value of the objective function F(x)$F\left(x\right)$. The objective function will decrease monotonically to its optimal value when there are no nonlinear constraints.

Norm Gz 
is ‖Z^{T}g_{FR}‖$\Vert {Z}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, the Euclidean norm of the projected gradient
(see Section [Solution of the Quadratic Programming Subproblem]).
Norm Gz will be approximately zero in the neighbourhood of a solution.

Violtn 
is the Euclidean norm of the residuals of constraints that are violated or in the predicted active set (not printed if ncnln is zero). Violtn will be approximately zero in the neighbourhood of a solution.

Nz 
is the number of columns of Z$Z$ (see Section [Solution of the Quadratic Programming Subproblem]). The value of Nz is the number of variables minus the number of constraints in the predicted active set; i.e., Nz = n − (Bnd + Lin + Nln)$\mathtt{Nz}=n(\mathtt{Bnd}+\mathtt{Lin}+\mathtt{Nln})$.

Bnd 
is the number of simple bound constraints in the predicted active set.

Lin 
is the number of general linear constraints in the predicted working set.

Nln 
is the number of nonlinear constraints in the predicted active set (not printed if ncnln is zero).

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

Cond H 
is a lower bound on the condition number of the Hessian approximation H$H$.

Cond Hz 
is a lower bound on the condition number of the projected Hessian approximation
H_{Z}
${H}_{Z}$ (
H_{Z}
=
Z^{T}
H_{FR}
Z
=
R_{Z}^{T}
R_{Z}
${H}_{Z}={Z}^{\mathrm{T}}{H}_{\mathrm{FR}}Z={R}_{Z}^{\mathrm{T}}{R}_{Z}$; see (6)). The larger this number, the more difficult the problem.

Cond T 
is a lower bound on the condition number of the matrix of predicted active constraints.

Conv 
is a threeletter indication of the status of the three convergence tests (16)–(18) defined in the description of the optional parameter Optimality Tolerance. Each letter is T if the test is satisfied and F otherwise. The three tests indicate whether:
(i) 
the sequence of iterates has converged; 
(ii) 
the projected gradient (Norm Gz) is sufficiently small; and 
(iii) 
the norm of the residuals of constraints in the predicted active set (Violtn) is small enough. 
If any of these indicators is F when nag_opt_nlp1_rcomm (e04uf) terminates with ifail = 0${\mathbf{ifail}}={\mathbf{0}}$, you should check the solution carefully.

M 
is printed if the quasiNewton update has been modified to ensure that the Hessian approximation is positive definite
(see Section [The QuasiNewton Update]).

I 
is printed if the QP subproblem has no feasible point.

C 
is printed if central differences have been used to compute the unspecified objective and constraint gradients. If the value of Step is zero then the switch to central differences was made because no lower point could be found in the linesearch. (In this case, the QP subproblem is resolved with the central difference gradient and Jacobian.) If the value of Step is nonzero then central differences were computed because Norm Gz and Violtn imply that x$x$ is close to a Kuhn–Tucker point (see Section [Overview]).

L 
is printed if the linesearch has produced a relative change in x$x$ greater than the value defined by the optional parameter Step Limit. If this output occurs frequently during later iterations of the run, optional parameter Step Limit should be set to a larger value.


Need not be initialized if the (default) optional parameter Cold Start is used.
If the optional parameter Warm Start has been chosen, r must contain the upper triangular Cholesky factor R$R$ of the initial approximation of the Hessian of the Lagrangian function, with the variables in the natural order. Elements not in the upper triangular part of r are assumed to be zero and need not be assigned.

PDF version (NAG web site
, 64bit version, 64bit version)
© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013