NAG Library Function Document
nag_opt_nlp (e04ucc)
1 Purpose
nag_opt_nlp (e04ucc) 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_nlp (e04ucc) may also be used for unconstrained, boundconstrained and linearly constrained optimization.
2 Specification
#include <nag.h> 
#include <nage04.h> 
void 
nag_opt_nlp (Integer n,
Integer nclin,
Integer ncnlin,
const double a[],
Integer tda,
const double bl[],
const double bu[],
void 
(*objfun)(Integer n,
const double x[],
double *objf,
double g[],
Nag_Comm *comm),


double x[],
double *objf,
double g[],
Nag_E04_Opt *options,
Nag_Comm *comm,
NagError *fail) 

3 Description
nag_opt_nlp (e04ucc) 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
$F\left(x\right)$ (the
objective function) is a nonlinear function,
${A}_{L}$ is an
${n}_{L}$ by
$n$ constant matrix, and
$c\left(x\right)$ is an
${n}_{N}$ element vector of nonlinear constraint functions. (The matrix
${A}_{L}$ and the vector
$c\left(x\right)$ 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_nlp (e04ucc) 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
${l}_{i}={u}_{i}$. If certain bounds are not present, the associated elements of
$l$ or
$u$ can be set to special values that will be treated as
$\infty $ or
$+\infty $. (See the description of the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ in
Section 11.2.)
If there are no nonlinear constraints in
(1) and
$F$ is linear or quadratic, then one of
nag_opt_lp (e04mfc),
nag_opt_lin_lsq (e04ncc) or
nag_opt_qp (e04nfc) will generally be more efficient.
You must supply an initial estimate of the solution to
(1), together with functions that define
$F\left(x\right),c\left(x\right)$ and as many first partial derivatives as possible; unspecified derivatives are approximated by finite differences.
The objective function is defined by function
objfun, and the nonlinear constraints are defined by function
confun. On every call, these functions must return appropriate values of the objective and nonlinear constraints. You should also provide the available partial derivatives. Any unspecified derivatives are approximated by finite differences; see
Section 11.2 for a discussion of the optional arguments
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_deriv}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_deriv}}$. Just before either
objfun or
confun is called, each element of the current gradient array
g or
conjac is initialized to a special value. On exit, any element that retains the value is estimated by finite differences. Note that if there
are any nonlinear constraints, then the
first call to
confun will precede the
first call to
objfun.
For maximum reliability, it is preferable if you provide all partial derivatives (see Chapter 8 of
Gill et al. (1981), for a detailed discussion). If all gradients cannot be provided, it is similarly advisable to provide as many as possible. While developing the functions
objfun and
confun, the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}$ (see
Section 11.2) should be used to check the calculation of any known gradients.
The method used by nag_opt_nlp (e04ucc) is described in detail in
Section 10.
4 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 (1983) Documentation for FDCALC and FDCORE Technical Report SOL 83–6 Stanford University
Gill P E, Murray W, Saunders M A and Wright M H (1984a) 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 (1984b) 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 (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
5 Arguments
 1:
n – IntegerInput

On entry: $n$, the number of variables.
Constraint:
${\mathbf{n}}>0$.
 2:
nclin – IntegerInput

On entry: ${n}_{L}$, the number of general linear constraints.
Constraint:
${\mathbf{nclin}}\ge 0$.
 3:
ncnlin – IntegerInput

On entry: ${n}_{N}$, the number of nonlinear constraints.
Constraint:
${\mathbf{ncnlin}}\ge 0$.
 4:
a[${\mathbf{nclin}}\times {\mathbf{tda}}$] – const doubleInput

On entry: the
$\mathit{i}$th row of
a must contain the coefficients of the
$\mathit{i}$th general linear constraint (the
$\mathit{i}$th row of the matrix
${A}_{L}$ in
(1)). The
$\mathit{i}j$th element of
${A}_{L}$ must be stored in
${\mathbf{a}}\left[\mathit{i}1\times {\mathbf{tda}}+j1\right]$, for
$\mathit{i}=1,2,\dots ,{n}_{L}$.
If
${\mathbf{nclin}}=0$ then the array
a is not referenced.
 5:
tda – IntegerInput

On entry: the stride separating matrix column elements in the array
a.
Constraint:
if ${\mathbf{nclin}}>0$, ${\mathbf{tda}}\ge {\mathbf{n}}$
 6:
bl[${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$] – const doubleInput
 7:
bu[${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$] – const doubleInput

On entry:
bl must contain the lower bounds and
bu the upper bounds, for all the constraints in the following order. The first
$n$ elements of each array must contain the bounds on the variables, the next
${n}_{L}$ elements the bounds for the general linear constraints (if any), and the next
${n}_{N}$ elements the bounds for the nonlinear constraints (if any). To specify a nonexistent lower bound (i.e.,
${l}_{j}=\infty $), set
${\mathbf{bl}}\left[j1\right]\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$, and to specify a nonexistent upper bound (i.e.,
${u}_{j}=+\infty $), set
${\mathbf{bu}}\left[j1\right]\ge {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$, where
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ is one of the optional arguments (default value
${10}^{20}$, see
Section 11.2). To specify the
$j$th constraint as an equality, set
${\mathbf{bl}}\left[j1\right]={\mathbf{bu}}\left[j1\right]=\beta $, say, where
$\left\beta \right<{\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$.
Constraints:
 ${\mathbf{bl}}\left[\mathit{j}1\right]\le {\mathbf{bu}}\left[\mathit{j}1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$;
 if ${\mathbf{bl}}\left[j1\right]={\mathbf{bu}}\left[j1\right]=\beta $, $\left\beta \right<{\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$.
 8:
objfun – function, supplied by the userExternal Function

objfun must calculate the objective function
$F\left(x\right)$ and (optionally) its gradient
$g\left(x\right)=\frac{\partial F}{\partial {x}_{j}}$ for a specified
$n$ element vector
$x$.
The specification of
objfun is:
void 
objfun (Integer n,
const double x[],
double *objf,
double g[],
Nag_Comm *comm)


 1:
n – IntegerInput

On entry: $n$, the number of variables.
 2:
x[n] – const doubleInput

On entry: $x$, the vector of variables at which the value of $F$ and/or all available elements of its gradient are to be evaluated.
 3:
objf – double *Output

On exit: if
$\mathbf{comm}\mathbf{\to}\mathbf{flag}=0$ or 2,
objfun must set
objf to the value of the objective function
$F$ at the current point
$x$. If it is not possible to evaluate
$F$ then
objfun should assign a negative value to
$\mathbf{comm}\mathbf{\to}\mathbf{flag}$; nag_opt_nlp (e04ucc) will then terminate.
 4:
g[n] – doubleOutput

On exit: if
$\mathbf{comm}\mathbf{\to}\mathbf{flag}=2$,
g must contain the elements of the vector
$g\left(x\right)$ given by
where
$\frac{\partial F}{\partial {x}_{\mathit{i}}}$ is the partial derivative of the objective function with respect to the
$\mathit{i}$th variable evaluated at the point
$x$ , for
$\mathit{i}=1,2,\dots ,n$.
If the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_deriv}}=\mathrm{Nag\_TRUE}$ (the default), all elements of
g must be set; if
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_deriv}}=\mathrm{Nag\_FALSE}$, any available elements of the vector
$g\left(x\right)$ must be assigned to the elements of
g; the remaining elements
must remain unchanged.
 5:
comm – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to
objfun.
 flag – IntegerInput/Output

On entry:
objfun is called with
$\mathbf{comm}\mathbf{\to}\mathbf{flag}$ set to 0 or 2.
If
$\mathbf{comm}\mathbf{\to}\mathbf{flag}=0$ then only
objf is referenced.
If
$\mathbf{comm}\mathbf{\to}\mathbf{flag}=2$ then both
objf and
g are referenced.
On exit: if
objfun resets
$\mathbf{comm}\mathbf{\to}\mathbf{flag}$ to some negative number then nag_opt_nlp (e04ucc) will terminate immediately with the error indicator
NE_USER_STOP. If
fail is supplied to nag_opt_nlp (e04ucc),
${\mathbf{fail}}\mathbf{.}\mathbf{errnum}$ will be set to your setting of
$\mathbf{comm}\mathbf{\to}\mathbf{flag}$.
 first – Nag_BooleanInput

On entry: will be set to Nag_TRUE on the first call to
objfun and Nag_FALSE for all subsequent calls.
 nf – IntegerInput

On entry: the number of evaluations of the objective function; this value will be equal to the number of calls made to
objfun including the current one.
 user – double *
 iuser – Integer *
 p – Pointer

The type Pointer will be void * with a C compiler that defines void * and char * otherwise.
Before calling nag_opt_nlp (e04ucc) these pointers may be allocated memory and initialized with various quantities for use by
objfun when called from nag_opt_nlp (e04ucc).
Note: objfun should be tested separately before being used in conjunction with nag_opt_nlp (e04ucc). The optional arguments
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{max\_iter}}$ can be used to assist this process. The array
x must
not be changed by
objfun.
If the function
objfun does not calculate all of the gradient elements then the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_deriv}}$ should be set to Nag_FALSE.
 9:
confun – function, supplied by the userExternal Function

confun must calculate the vector
$c\left(x\right)$ of nonlinear constraint functions and (optionally) its Jacobian (
$\text{}=\frac{\partial c}{\partial x}$) for a specified
$n$ element vector
$x$. If there are no nonlinear constraints (i.e.,
${\mathbf{ncnlin}}=0$),
confun will never be called and the NAG defined null void function pointer,
NULLFN, can be supplied in the call to nag_opt_nlp (e04ucc). If there are nonlinear constraints the first call to
confun will occur before the first call to
objfun.
The specification of
confun is:
 1:
n – IntegerInput

On entry: $n$, the number of variables.
 2:
ncnlin – IntegerInput

On entry: ${n}_{N}$, the number of nonlinear constraints.
 3:
needc[ncnlin] – const IntegerInput

On entry: the indices of the elements of
conf and/or
conjac that must be evaluated by
confun. If
${\mathbf{needc}}\left[i1\right]>0$ then the
$i$th element of
conf and/or the available elements of the
$i$th row of
conjac (see argument
$\mathbf{comm}\mathbf{\to}\mathbf{flag}$ below) must be evaluated at
$x$.
 4:
x[n] – const doubleInput

On entry: the vector of variables $x$ at which the constraint functions and/or all available elements of the constraint Jacobian are to be evaluated.
 5:
conf[ncnlin] – doubleOutput

On exit: if
${\mathbf{needc}}\left[i1\right]>0$ and
$\mathbf{comm}\mathbf{\to}\mathbf{flag}=0$ or 2,
${\mathbf{conf}}\left[i1\right]$ must contain the value of the
$i$th constraint at
$x$. The remaining elements of
conf, corresponding to the nonpositive elements of
needc, are ignored.
 6:
conjac[${\mathbf{ncnlin}}\times {\mathbf{n}}$] – doubleOutput

On exit: if
${\mathbf{needc}}\left[i1\right]>0$ and
$\mathbf{comm}\mathbf{\to}\mathbf{flag}=2$, the
$\mathit{i}$th row of
conjac (i.e., the elements
${\mathbf{conjac}}\left[\left(\mathit{i}1\right)\times {\mathbf{n}}+\mathit{j}1\right]$, for
$\mathit{j}=1,2,\dots ,n$) must contain the available elements of the vector
$\nabla {c}_{i}$ given by
where
$\frac{\partial {c}_{i}}{\partial {x}_{j}}$ is the partial derivative of the
$i$th constraint with respect to the
$j$th variable, evaluated at the point
$x$. The remaining rows of
conjac, corresponding to nonpositive elements of
needc, are ignored.
If the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_deriv}}=\mathrm{Nag\_TRUE}$ (the default), all elements of
conjac must be set; if
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_deriv}}=\mathrm{Nag\_FALSE}$, then any available partial derivatives of
${c}_{i}\left(x\right)$ must be assigned to the elements of
conjac; the remaining elements
must remain unchanged.
If all elements of the constraint Jacobian are known (i.e.,
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_deriv}}=\mathrm{Nag\_TRUE}$; see
Section 11.2), any constant elements may be assigned to
conjac one time only at the start of the optimization. An element of
conjac that is not subsequently assigned in
confun will retain its initial value throughout.
Constant elements may be loaded into
conjac during the first call to
confun. The ability to preload constants is useful when many Jacobian elements are identically zero, in which case
conjac may be initialized to zero at the first call when
$\mathbf{comm}\mathbf{\to}\mathbf{first}=\mathrm{Nag\_TRUE}$.
It must be emphasized that, if
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_deriv}}=\mathrm{Nag\_FALSE}$, unassigned elements of
conjac are not treated as constant; they are estimated by finite differences, at nontrivial expense. If you do not supply a value for the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_diff\_int}}$ (the default; see
Section 11.2), an interval for each element of
$x$ is computed automatically at the start of the optimization. The automatic procedure can usually identify constant elements of
conjac, which are then computed once only by finite differences.
 7:
comm – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to
confun.
 flag – IntegerInput/Output

On entry:
confun is called with
$\mathbf{comm}\mathbf{\to}\mathbf{flag}$ set to 0 or 2.
If
$\mathbf{comm}\mathbf{\to}\mathbf{flag}=0$ then only
conf is referenced.
If
$\mathbf{comm}\mathbf{\to}\mathbf{flag}=2$ then both
conf and
conjac are referenced.
On exit: if
confun resets
$\mathbf{comm}\mathbf{\to}\mathbf{flag}$ to some negative number then nag_opt_nlp (e04ucc) will terminate immediately with the error indicator
NE_USER_STOP. If
fail is supplied to nag_opt_nlp (e04ucc)
${\mathbf{fail}}\mathbf{.}\mathbf{errnum}$ will be set to your setting of
$\mathbf{comm}\mathbf{\to}\mathbf{flag}$.
 first – Nag_BooleanInput

On entry: will be set to Nag_TRUE on the first call to
confun and Nag_FALSE for all subsequent calls.
 user – double *
 iuser – Integer *
 p – Pointer

The type Pointer will be void * with a C compiler that defines void * and char * otherwise.
Before calling nag_opt_nlp (e04ucc) these pointers may be allocated memory and initialized with various quantities for use by
confun when called from nag_opt_nlp (e04ucc).
Note: confun should be tested separately before being used in conjunction with nag_opt_nlp (e04ucc). The optional arguments
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{max\_iter}}$ can be used to assist this process. The array
x must
not be changed by
confun.
If
confun does not calculate all of the elements of the constraint gradients then the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_deriv}}$ should be set to Nag_FALSE.
 10:
x[n] – doubleInput/Output

On entry: an initial estimate of the solution.
On exit: the final estimate of the solution.
 11:
objf – double *Output

On exit: the value of the objective function at the final iterate.
 12:
g[n] – doubleOutput

On exit: the gradient of the objective function at the final iterate (or its finite difference approximation).
 13:
options – Nag_E04_Opt *Input/Output

On entry/exit: a pointer to a structure of type Nag_E04_Opt whose members are optional arguments for nag_opt_nlp (e04ucc). These structure members offer the means of adjusting some of the argument values of the algorithm and on output will supply further details of the results. A description of the members of
options is given below in
Section 11. Some of the results returned in
options can be used by nag_opt_nlp (e04ucc) to perform a ‘warm start’ (see the member
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}$ in
Section 11.2).
If any of these optional arguments are required, then the structure
options should be declared and initialized by a call to
nag_opt_init (e04xxc) immediately before being supplied as an argument to nag_opt_nlp (e04ucc).
 14:
comm – Nag_Comm *Input/Output

Note: comm is a NAG defined type (see
Section 3.2.1.1 in the Essential Introduction).
On entry/exit: structure containing pointers for communication to the usersupplied functions
objfun and
confun, and the optional userdefined printing function; see the description of
objfun and
confun and
Section 11.3.1 for details. If you do not need to make use of this communication feature the null pointer
NAGCOMM_NULL may be used in the call to nag_opt_nlp (e04ucc);
comm will then be declared internally for use in calls to usersupplied functions.
 15:
fail – NagError *Input/Output

The NAG error argument (see
Section 3.6 in the Essential Introduction).
5.1 Description of the Printed Output
Intermediate and final results are printed out by default. The level of printed output can be controlled with the structure members
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_print\_level}}$ (see
Section 11.2). The default setting of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_Soln\_Iter}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_print\_level}}=\mathrm{Nag\_NoPrint}$ provides a single line of output at each iteration and the final result. This section describes the default printout produced by nag_opt_nlp (e04ucc).
The following line of summary output (
$<80$ 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 in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution (see Section 10).
Note that Mnr may be greater than the optional argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_max\_iter}}$ (default value $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,3\left(n+{n}_{L}+{n}_{N}\right)\right)$; see Section 11.2) if some iterations are required for the feasibility phase. 
Step 
is the step taken along the computed search direction. On reasonably wellbehaved problems, the unit step 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 arguments (see Section 10.3). 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), the merit function is a large multiple of the constraint violations, weighted by the penalty arguments. 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_nlp (e04ucc) terminates with ${\mathbf{fail}}\mathbf{.}\mathbf{code}={\mathbf{NW\_NONLIN\_NOT\_FEASIBLE}}$ (no feasible point could be found for the nonlinear constraints).
If no nonlinear constraints are present (i.e., ${\mathbf{ncnlin}}=0$), this entry contains Objective, the value of the objective function $F\left(x\right)$. The objective function will decrease monotonically to its optimal value when there are no nonlinear constraints. 
Violtn 
is the Euclidean norm of the residuals of constraints that are violated or in the predicted active set (not printed if ncnlin is zero). Violtn will be approximately zero in the neighbourhood of a solution. 
Norm Gz 
is $\Vert {Z}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, the Euclidean norm of the projected gradient (see Section 10.1). Norm Gz 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 ${H}_{Z}$
$\left({H}_{Z}={Z}^{\mathrm{T}}{H}_{\mathrm{FR}}Z={R}_{Z}^{\mathrm{T}}{R}_{Z}\right)$; see (6) and (11). The larger this number, the more difficult the problem. 
The line of output may be terminated by one of the following characters:
M 
is printed if the quasiNewton update was modified to ensure that the Hessian approximation is positive definite (see Section 10.4). 
I 
is printed if the QP subproblem has no feasible point. 
C 
is printed if central differences were used to compute the unspecified objective and constraint gradients. If the value of Step is zero, the switch to central differences was made because no lower point could be found in the line search. (In this case, the QP subproblem is resolved with the central difference gradient and Jacobian.) If the value of Step is nonzero, central differences were computed because Norm Gz and Violtn imply that $x$ is close to a Kuhn–Tucker point (see Section 10.1). 
L 
is printed if the line search has produced a relative change in $x$ greater than the value defined by the optional argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}$ (default value $\text{}=2.0$; see Section 11.2). If this output occurs frequently during later iterations of the run, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}$ should be set to a larger value. 
R 
is printed if the approximate Hessian has been refactorized. If the diagonal condition estimator of $R$ indicates that the approximate Hessian is badly conditioned, the approximate Hessian is refactorized using column interchanges. If necessary, $R$ 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.
Varbl 
gives the name (V) and index $\mathit{j}$, for $\mathit{j}=1,2,\dots ,n$ of the variable. 
State 
gives the state of the variable (FR if neither bound is in the active set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound). If Value lies outside the upper or lower bounds by more than the feasibility tolerances specified by the optional arguments ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ and ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}$ (see Section 11.2), State will be ++ or  respectively.
A key is sometimes printed before State to give some additional information about the state of a variable.
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, 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 ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$. 

Value 
is the value of the variable at the final iteration. 
Lower bound 
is the lower bound specified for the variable $j$. (None indicates that ${\mathbf{bl}}\left[j1\right]\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$, where ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ is the optional argument.) 
Upper bound 
is the upper bound specified for the variable $j$. (None indicates that ${\mathbf{bu}}\left[j1\right]\ge {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$, where ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ is the optional argument.) 
Lagr Mult 
is the value of the Lagrange multiplier for the associated bound constraint. This will be zero if State is FR unless ${\mathbf{bl}}\left[j1\right]\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ and ${\mathbf{bu}}\left[j1\right]\ge {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$, in which case the entry will be blank. If $x$ is optimal, the multiplier should be nonnegative if State is LL, and nonpositive if State is UL. 
Residual 
is the difference between the variable Value and the nearer of its (finite) bounds ${\mathbf{bl}}\left[j1\right]$ and ${\mathbf{bu}}\left[j1\right]$. A blank entry indicates that the associated variable is not bounded (i.e., ${\mathbf{bl}}\left[j1\right]\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ and ${\mathbf{bu}}\left[j1\right]\ge {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$). 
The meaning of the printout for linear and nonlinear constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’,
${\mathbf{bl}}\left[j1\right]$ and
${\mathbf{bu}}\left[j1\right]$ are replaced by
${\mathbf{bl}}\left[n+j1\right]$ and
${\mathbf{bu}}\left[n+j1\right]$ respectively, and with the following changes in the heading:
L Con 
gives the name (L) and index $\mathit{j}$, for $\mathit{j}=1,2,\dots ,{n}_{L}$, of the linear constraint. 
N Con 
gives the name (N) and index $\left({jn}_{L}\right)$, for $j={n}_{L}+1,{n}_{L}+2,\dots ,{n}_{L}+{n}_{N}$ of the nonlinear constraint. 
The I key in the State column is printed for general linear constraints which currently violate one of their bounds by more than ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ and for nonlinear constraints which violate one of their bounds by more than ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}$.
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 Residual column to become positive.
Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.
6 Error Indicators and Warnings
 NE_2_INT_ARG_LT
On entry, ${\mathbf{tda}}=\u2329\mathit{\text{value}}\u232a$ while ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$. These arguments must satisfy ${\mathbf{tda}}\ge {\mathbf{n}}$.
This error message is output only if ${\mathbf{nclin}}>0$.
 NE_2_INT_OPT_ARG_CONS
On entry, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_start}}=\u2329\mathit{\text{value}}\u232a$ while ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_stop}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_start}}\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_stop}}$.
On entry, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_start}}=\u2329\mathit{\text{value}}\u232a$ while ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_stop}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_start}}\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_stop}}$.
 NE_ALLOC_FAIL
Dynamic memory allocation failed.
 NE_BAD_PARAM
On entry, argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_print\_level}}$ had an illegal value.
On entry, argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_deriv}}$ had an illegal value.
On entry, argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}$ had an illegal value.
On entry, argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}$ had an illegal value.
On entry, argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}$ had an illegal value.
 NE_BOUND
The lower bound for variable $\u2329\mathit{\text{value}}\u232a$ (array element ${\mathbf{bl}}\left[\u2329\mathit{\text{value}}\u232a\right]$) is greater than the upper bound.
 NE_BOUND_EQ
The lower bound and upper bound for variable $\u2329\mathit{\text{value}}\u232a$ (array elements ${\mathbf{bl}}\left[\u2329\mathit{\text{value}}\u232a\right]$ and ${\mathbf{bu}}\left[\u2329\mathit{\text{value}}\u232a\right]$) are equal but they are greater than or equal to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$.
 NE_BOUND_EQ_LCON
The lower bound and upper bound for linear constraint $\u2329\mathit{\text{value}}\u232a$ (array elements ${\mathbf{bl}}\left[\u2329\mathit{\text{value}}\u232a\right]$ and ${\mathbf{bu}}\left[\u2329\mathit{\text{value}}\u232a\right]$) are equal but they are greater than or equal to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$.
 NE_BOUND_EQ_NLCON
The lower bound and upper bound for nonlinear constraint $\u2329\mathit{\text{value}}\u232a$ (array elements ${\mathbf{bl}}\left[\u2329\mathit{\text{value}}\u232a\right]$ and ${\mathbf{bu}}\left[\u2329\mathit{\text{value}}\u232a\right]$) are equal but they are greater than or equal to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$.
 NE_BOUND_LCON
The lower bound for linear constraint $\u2329\mathit{\text{value}}\u232a$ (array element ${\mathbf{bl}}\left[\u2329\mathit{\text{value}}\u232a\right]$) is greater than the upper bound.
 NE_BOUND_NLCON
The lower bound for nonlinear constraint $\u2329\mathit{\text{value}}\u232a$ (array element ${\mathbf{bl}}\left[\u2329\mathit{\text{value}}\u232a\right]$) is greater than the upper bound.
 NE_DERIV_ERRORS
Large errors were found in the derivatives of the objective function and/or nonlinear constraints.
This failure 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 $x=0$ or $x=1$ are used to test function evaluation procedures, and how often the special properties of these numbers make the test meaningless.
Gradient checking will be ineffective if the objective function uses information computed by the constraints, since they are not necessarily computed prior to 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.
 NE_INT_ARG_LT
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{nclin}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nclin}}\ge 0$.
On entry, ${\mathbf{ncnlin}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{ncnlin}}\ge 0$.
 NE_INT_OPT_ARG_GT
On entry, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_start}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_start}}\le {\mathbf{n}}$.
On entry, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_stop}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_stop}}\le {\mathbf{n}}$.
On entry, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_start}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_start}}\le {\mathbf{n}}$.
On entry, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_stop}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_stop}}\le {\mathbf{n}}$.
 NE_INT_OPT_ARG_LT
On entry, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_start}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_start}}\ge 1$.
On entry, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_stop}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_stop}}\ge 1$.
On entry, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_start}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_start}}\ge 1$.
On entry, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_stop}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_stop}}\ge 1$.
 NE_INVALID_INT_RANGE_1
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{max\_iter}}$ not valid. Correct range is ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{max\_iter}}\ge 0$.
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_max\_iter}}$ not valid. Correct range is
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_max\_iter}}\ge 0$.
 NE_INVALID_REAL_RANGE_EF
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{c\_diff\_int}}$ not valid. Correct range is $\epsilon \le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{c\_diff\_int}}<1.0$.
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_diff\_int}}$ not valid. Correct range is $\epsilon \le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_diff\_int}}<1.0$.
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_prec}}$ not valid. Correct range is $\epsilon \le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_prec}}<1.0$.
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ not valid. Correct range is $\epsilon \le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}<1.0$.
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}$ not valid. Correct range is $\epsilon \le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}<1.0$.
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}$ not valid. Correct range is ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_prec}}\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}<1.0$.
 NE_INVALID_REAL_RANGE_F
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ not valid. Correct range is ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}>0.0$.
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_step}}$ not valid. Correct range is ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_step}}>0.0$.
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}$ not valid. Correct range is ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}>0.0$.
 NE_INVALID_REAL_RANGE_FF
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{crash\_tol}}$ not valid. Correct range is $0.0\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{crash\_tol}}\le 1.0$.
Value $\u2329\mathit{\text{value}}\u232a$ given to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{linesearch\_tol}}$ not valid. Correct range is $0.0\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{linesearch\_tol}}<1.0$.
 NE_NOT_APPEND_FILE
Cannot open file $\u2329\mathit{string}\u232a$ for appending.
 NE_NOT_CLOSE_FILE
Cannot close file $\u2329\mathit{string}\u232a$.
 NE_OPT_NOT_INIT
Options structure not initialized.
 NE_STATE_VAL
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}\left[\u2329\mathit{\text{value}}\u232a\right]$ is out of range. ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}\left[\u2329\mathit{\text{value}}\u232a\right]=\u2329\mathit{\text{value}}\u232a$.
 NE_USER_STOP
User requested termination, user flag value
$\text{}=\u2329\mathit{\text{value}}\u232a$.
This exit occurs if you set
$\mathbf{comm}\mathbf{\to}\mathbf{flag}$ to a negative value in
objfun or
confun. If
fail is supplied the value of
${\mathbf{fail}}\mathbf{.}\mathbf{errnum}$ will be the same as your setting of
$\mathbf{comm}\mathbf{\to}\mathbf{flag}$.
 NE_WRITE_ERROR
Error occurred when writing to file $\u2329\mathit{string}\u232a$.
 NW_KT_CONDITIONS
The current point cannot be improved upon. The final point does not satisfy the firstorder Kuhn–Tucker conditions and no improved point for the merit function could be found during the final line search.
The Kuhn–Tucker conditions are specified in
Section 10.1, and the merit function is described in
Section 10.3 and
Section 11.3.
This sometimes occurs because an overly stringent accuracy has been requested, i.e., the value of the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}$ (default value
$\text{}={\epsilon}_{r}^{0.8}$, where
${\epsilon}_{r}$ is the relative precision of
$F\left(x\right)$; see
Section 11.2) is too small. In this case you should apply the four tests described in
Section 8.1 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_nlp (e04ucc) has failed completely to move from the initial point then functions
objfun and/or
confun may be incorrect. You should refer to comments under
${\mathbf{fail}}\mathbf{.}\mathbf{code}={\mathbf{NE\_DERIV\_ERRORS}}$ and check the gradients using the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}$ (default value
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}=\mathrm{Nag\_SimpleCheck}$; see
Section 11.2). 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 5.1) are large.
Another possibility is that the search direction has become inaccurate because of ill conditioning in the Hessian approximation or the matrix of constraints in the working set; either form of ill conditioning tends to be reflected in large values of
Mnr (the number of iterations required to solve each QP subproblem; see
Section 5.1).
If the condition estimate of the projected Hessian (
Cond Hz; see
Section 5.1) is extremely large, it may be worthwhile rerunning nag_opt_nlp (e04ucc) from the final point with the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Warm}$ (see
Section 11.2). In this situation, the optional arguments
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lambda}}$ should be left unaltered and
$R$ (in optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{h}}$) should be reset to the identity matrix.
If the matrix of constraints in the working set is ill conditioned (i.e.,
Cond T is extremely large; see
Section 11.3), it may be helpful to run nag_opt_nlp (e04ucc) with a relaxed value of the optional arguments
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}$ (default values
$\sqrt{\epsilon}$,
${\epsilon}^{0.33}$ or
$\sqrt{\epsilon}$, respectively, where
$\epsilon $ is the
machine precision; see
Section 11.2). (Constraint dependencies are often indicated by wide variations in size in the diagonal elements of the matrix
$T$, whose diagonals will be printed if
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_Soln\_Iter\_Full}$ (default value
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_Soln\_Iter}$; see
Section 11.2).)
 NW_LIN_NOT_FEASIBLE
No feasible point was found for the linear constraints and bounds.
nag_opt_nlp (e04ucc) 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 argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ (default value
$\text{}=\sqrt{\epsilon}$, where
$\epsilon $ is the
machine precision; see
Section 11.2), or no feasible point could be found in the number of iterations specified by the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_max\_iter}}$ (default value
$\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,3\left(n+{n}_{L}+{n}_{N}\right)\right)$; see
Section 11.2). You should check that there are no constraint redundancies. If the data for the constraints are accurate only to an absolute precision
$\sigma $, you should ensure that the value of the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ is greater than
$\sigma $. For example, if all elements of
${A}_{L}$ are of order unity and are accurate to only three decimal places,
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ should be at least
${10}^{3}$.
 NW_NONLIN_NOT_FEASIBLE
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 terse line of output; see
Section 5.1). 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 the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$. If you are convinced that a feasible point does exist, nag_opt_nlp (e04ucc) should be restarted at a different starting point.
 NW_NOT_CONVERGED
Optimal solution found, but the sequence of iterates has not converged with the requested accuracy.
The final iterate
$x$ satisfies the firstorder Kuhn–Tucker conditions (see
Section 10.1) to the accuracy requested, but the sequence of iterates has not yet converged. nag_opt_nlp (e04ucc) was terminated because no further improvement could be made in the merit function (see
Section 11.3).
This value of
${\mathbf{fail}}\mathbf{.}\mathbf{code}$ 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 argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_prec}}$ (default value
$\text{}={\epsilon}^{0.9}$, where
$\epsilon $ is the
machine precision; see
Section 11.2)). 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 8.1 are satisfied then
$x$ is likely to be a solution of
(1) even if
${\mathbf{fail}}\mathbf{.}\mathbf{code}={\mathbf{NW\_NOT\_CONVERGED}}$.
 NW_OVERFLOW_WARN
Serious ill conditioning in the working set after adding constraint
$\u2329\mathit{\text{value}}\u232a$. Overflow may occur in subsequent iterations.
If overflow occurs preceded by this warning then serious ill conditioning has probably occurred in the working set when adding a constraint. It may be possible to avoid the difficulty by increasing the magnitude of the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ (default value
$\text{}=\sqrt{\epsilon}$, where
$\epsilon $ is the
machine precision; see
Section 11.2) and/or the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}$ (default value
${\epsilon}^{0.33}$ or
$\sqrt{\epsilon}$; see
Section 11.2), and rerunning the program. If the message recurs even after this change, the offending linearly dependent constraint
$j$ must be removed from the problem. If overflow occurs in one of the usersupplied functions (e.g., if the nonlinear functions involve exponentials or singularities), it may help to specify tighter bounds for some of the variables (i.e., reduce the gap between the appropriate
${l}_{j}$ and
${u}_{j}$).
 NW_TOO_MANY_ITER
The maximum number of iterations,
$\u2329\mathit{\text{value}}\u232a$, have been performed.
The value of the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{max\_iter}}$ may be too small. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), increase the value of the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{max\_iter}}$ and rerun nag_opt_nlp (e04ucc); alternatively, rerun nag_opt_nlp (e04ucc), setting the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Warm}$ to specify the initial working set. If the algorithm seems to be making little or no progress, however, then you should check for incorrect gradients or ill conditioning as described under
${\mathbf{fail}}\mathbf{.}\mathbf{code}={\mathbf{NW\_KT\_CONDITIONS}}$.
Note that ill conditioning in the working set is sometimes resolved automatically by the algorithm, in which case performing additional iterations may be helpful. However, ill conditioning in the Hessian approximation tends to persist once it has begun, so that allowing additional iterations without altering
$R$ is usually inadvisable. If the quasiNewton update of the Hessian approximation was reset during the latter iterations (i.e., an
R occurs at the end of each line of output; see
Section 5.1), it may be worthwhile setting
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Warm}$ and calling nag_opt_nlp (e04ucc) from the final point.
7 Accuracy
If
${\mathbf{fail}}\mathbf{.}\mathbf{code}=\mathrm{NE\_NOERROR}$ on exit, then the vector returned in the array
x is an estimate of the solution to an accuracy of approximately
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}$ (default value
$\text{}={\epsilon}_{r}^{0.8}$, where
${\epsilon}_{r}$ is the relative precision of
$F\left(x\right)$; see
Section 11.2).
8.1 Termination Criteria
The function exits with
${\mathbf{fail}}\mathbf{.}\mathbf{code}=\mathrm{NE\_NOERROR}$ if iterates have converged to a point
$x$ that satisfies the Kuhn–Tucker conditions (see
Section 10.1) to the accuracy requested by the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}$ (default value
$\text{}={\epsilon}_{r}^{0.8}$, see
Section 11.2).
You should also examine the printout from nag_opt_nlp (e04ucc) (see
Section 5.1 or
Section 11.3) to check whether the following four conditions are satisfied:
(i) 
the final value of Norm Gz is significantly less than at the starting point; 
(ii) 
during the final major iterations, the values of Step and Mnr are both one; 
(iii) 
the last few values of both Violtn and Norm Gz become small at a fast linear rate; and 
(iv) 
Cond Hz is small. 
If all these conditions hold, $x$ is almost certainly a local minimum.
9 Example
This example is based on Problem 71 in
Hock and Schittkowski (1981) and involves the minimization of the nonlinear function
subject to the bounds
to the general linear constraint
and to the nonlinear constraints
The initial point, which is infeasible, is
and
$F\left({x}_{0}\right)=16$.
The optimal solution (to five figures) is
and
$F\left({x}^{*}\right)=17.014$. One bound constraint and both nonlinear constraints are active at the solution.
The
options structure is declared and initialized by
nag_opt_init (e04xxc). Two options are read from the data file by use of
nag_opt_read (e04xyc). nag_opt_nlp (e04ucc) is then called to solve the problem using the function
objfun and
confun with elements of the objective gradient not being set at all and only some of the elements of the constraint Jacobian being provided. The memory freeing function
nag_opt_free (e04xzc) is used to free the memory assigned to the pointers in the options structure. You must
not use the standard C function
free() for this purpose.
9.1 Program Text
Program Text (e04ucce.c)
9.2 Program Data
Program Data (e04ucce.d)
Program Options (e04ucce.opt)
9.3 Program Results
Program Results (e04ucce.r)
10 Further Description
This section gives a detailed description of the algorithm used in nag_opt_nlp (e04ucc). This, and possibly the next section,
Section 11, may be omitted if the more sophisticated features of the algorithm and software are not currently of interest.
10.1 Overview
nag_opt_nlp (e04ucc) is based on the same algorithm as used in subroutine 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
$C$ denote the
$m$ by
$n$ matrix of gradients of the active general linear and nonlinear constraints. The number of fixed variables will be denoted by
${n}_{\mathrm{FX}}$, with
${n}_{\mathrm{FR}}$ $\left({n}_{\mathrm{FR}}=n{n}_{\mathrm{FX}}\right)$ 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
$x$ is a
firstorder Kuhn–Tucker point for
(1) (see, e.g.,
Powell (1974)) if the following conditions hold:
(i) 
$x$ is feasible; 
(ii) 
there exist vectors $\xi $ and $\lambda $ (the Lagrange multiplier vectors for the bound and general constraints) such that
where $g$ is the gradient of $F$ evaluated at $x$, and ${\xi}_{j}=0$ if the $j$th variable is free. 
(iii) 
The Lagrange multiplier corresponding to an inequality constraint active at its lower bound must be nonnegative, and it must be nonpositive for an inequality constraint active at its upper bound. 
Let
$Z$ denote a matrix whose columns form a basis for the set of vectors orthogonal to the rows of
${C}_{\mathrm{FR}}$; i.e.,
${C}_{\mathrm{FR}}Z=0$. An equivalent statement of the condition
(2) in terms of
$Z$ is
The vector
${Z}^{\mathrm{T}}{g}_{\mathrm{FR}}$ is termed the
projected gradient of
$F$ at
$x$. Certain additional conditions must be satisfied in order for a firstorder Kuhn–Tucker point to be a solution of
(1) (see, e.g.,
Powell (1974)). nag_opt_nlp (e04ucc) implements a sequential quadratic programming (SQP) method. For an overview of SQP methods, see, for example,
Fletcher (1987),
Gill et al. (1981) and
Powell (1983).
The basic structure of nag_opt_nlp (e04ucc) involves
major and
minor iterations. The major iterations generate a sequence of iterates
$\left\{{x}_{k}\right\}$ that converge to
${x}^{*}$, a firstorder Kuhn–Tucker point of
(1). At a typical major iteration, the new iterate
$\stackrel{}{x}$ is defined by
where
$x$ is the current iterate, the nonnegative scalar
$\alpha $ is the
step length, and
$p$ 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
$p$ in
(3) is the solution of a quadratic programming subproblem of the form
where
$g$ is the gradient of
$F$ at
$x$, the matrix
$H$ is a positive definite quasiNewton approximation to the Hessian of the Lagrangian function (see
Section 10.4), and
${A}_{N}$ is the Jacobian matrix of
$c$ evaluated at
$x$. (Finite difference estimates may be used for
$g$ and
${A}_{N}$; see the optional arguments
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_deriv}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_deriv}}$ in
Section 11.2.) Let
$l$ in
(1) be partitioned into three sections:
${l}_{B}$,
${l}_{L}$ and
${l}_{N}$, corresponding to the bound, linear and nonlinear constraints. The vector
$\stackrel{}{l}$ in
(4) is similarly partitioned, and is defined as
where
$c$ is the vector of nonlinear constraints evaluated at
$x$. The vector
$\stackrel{}{u}$ 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 output of nag_opt_nlp (e04ucc); see
Section 11.3.) In nag_opt_nlp (e04ucc),
(4) is solved using the same algorithm as used in function
nag_opt_lin_lsq (e04ncc). Since solving a quadratic program is an iterative procedure, the minor iterations of nag_opt_nlp (e04ucc) are the iterations of
nag_opt_lin_lsq (e04ncc). (More details about solving the subproblem are given in
Section 10.2.)
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
$C$ denote the
$m$ by
$n$ matrix of gradients of the general linear and nonlinear constraints in the predicted active set. First, we have available the
$TQ$ factorization of
${C}_{\mathrm{FR}}$:
where
$T$ is a nonsingular
$m$ by
$m$ reversetriangular matrix (i.e.,
${t}_{ij}=0$ if
$i+j<m$, and the nonsingular
${n}_{\mathrm{FR}}$ by
${n}_{\mathrm{FR}}$ matrix
${Q}_{\mathrm{FR}}$ is the product of orthogonal transformations (see
Gill et al. (1984a)). Second, we have the upper triangular Cholesky factor
$R$ of the
transformed and reordered Hessian matrix
where
$\stackrel{~}{H}$ is the Hessian
$H$ with rows and columns permuted so that the free variables are first, and
$Q$ is the
$n$ by
$n$ matrix
with
${I}_{\mathrm{FX}}$ the identity matrix of order
${n}_{\mathrm{FX}}$. If the columns of
${Q}_{\mathrm{FR}}$ are partitioned so that
the
${n}_{Z}$
$\left({n}_{Z}\equiv {n}_{\mathrm{FR}}m\right)$ columns of
$Z$ form a basis for the null space of
${C}_{\mathrm{FR}}$. The matrix
$Z$ is used to compute the projected gradient
${Z}^{\mathrm{T}}{g}_{\mathrm{FR}}$ at the current iterate. (The values
Nz,
Norm Gf and
Norm Gz printed by nag_opt_nlp (e04ucc) give
${n}_{Z}$ and the norms of
${g}_{\mathrm{FR}}$ and
${Z}^{\mathrm{T}}{g}_{\mathrm{FR}}$; see
Section 11.3.)
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
${x}^{*}$. In nag_opt_nlp (e04ucc), 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_nlp (e04ucc) also saves computation in factorizing
${C}_{\mathrm{FR}}$ and
${H}_{Q}$.
Once
$p$ has been computed, the major iteration proceeds by determining a step length
$\alpha $ that produces a ‘sufficient decrease’ in an augmented Lagrangian
merit function (see
Section 10.3). Finally, the approximation to the transformed Hessian matrix
${H}_{Q}$ is updated using a modified BFGS quasiNewton update (see
Section 10.4) to incorporate new curvature information obtained in the move from
$x$ to
$\stackrel{}{x}$.
On entry to nag_opt_nlp (e04ucc), an iterative procedure from
nag_opt_lin_lsq (e04ncc) is executed, starting with the initial point you provided, to find a point that is feasible with respect to the bounds and linear constraints (using the tolerance specified by
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$; see
Section 11.2). If no feasible point exists for the bound and linear constraints,
(1) has no solution and nag_opt_nlp (e04ucc) 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
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_diff\_int}}$ in
Section 11.2). 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 gradients you provided appear to be correct (see the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}$ in
Section 11.2). 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_nlp (e04ucc) first determines a point that satisfies the bound and linear constraints. Thereafter, each iteration includes:
(a) 
the solution of a quadratic programming subproblem (see Section 10.2); 
(b) 
a linesearch with an augmented Lagrangian merit function (see Section 10.3); and 
(c) 
a quasiNewton update of the approximate Hessian of the Lagrangian function (Section 10.4). 
10.2 Solution of the Quadratic Programming Subproblem
The search direction
$p$ is obtained by solving
(4) using the algorithm of
nag_opt_lin_lsq (e04ncc) (see
Gill et al. (1986)), which was specifically designed to be used within an SQP algorithm for nonlinear programming.
The method of
nag_opt_lin_lsq (e04ncc) is 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 segments of code. 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
$p$ denote the current estimate of the solution of
(4); the new iterate
$\stackrel{}{p}$ is defined by
where, as in
(3),
$\sigma $ is a nonnegative step length and
$d$ is a search direction.
At the beginning of each iteration of
nag_opt_lin_lsq (e04ncc), a
working set is defined of constraints (general and bound) that are satisfied exactly. The vector
$d$ is then constructed so that the values of constraints in the working set remain
unaltered for any move along
$d$. For a bound constraint in the working set, this property is achieved by setting the corresponding element of
$d$ 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
$C$ 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
${d}_{\mathrm{FR}}$ as
for some vector
${d}_{Z}$, where
$Z$ is the matrix associated with the
$TQ$ factorization
(5) of
${C}_{\mathrm{FR}}$.
The definition of
${d}_{Z}$ in
(10) depends on whether the current
$p$ is feasible. If not,
${d}_{Z}$ is zero except for an element
$\gamma $ in the
$j$th position, where
$j$ and
$\gamma $ are chosen so that the sum of infeasibilities is decreasing along
$d$. (For further details, see
Gill et al. (1986).) In the feasible case,
${d}_{Z}$ satisfies the equations
where
${R}_{Z}$ is the Cholesky factor of
${Z}^{\mathrm{T}}{H}_{\mathrm{FR}}Z$ and
$q$ is the gradient of the quadratic objective function
$\left(q=g+Hp\right)$. (The vector
${Z}^{\mathrm{T}}{q}_{\mathrm{FR}}$ is the projected gradient of the QP.) With
(11),
$p+d$ 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 $p$ 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 $\delta $ of the order of the machine precision, the Lagrange multiplier ${\mu}_{j}$ corresponding to an inequality constraint in the working set at its upper bound is said to be optimal if ${\mu}_{j}\le \delta $ when the $j$th constraint is at its upper bound, or if ${\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 ${\mu}_{j}$ will satisfy $\left(1+\delta \right)\le {\mu}_{j}\le \delta $ for an inequality constraint at its upper bound, and $\delta \le {\mu}_{j}\le \left(1+\delta \right)$ for an inequality at its lower bound. The Lagrange multiplier for an equality constraint will satisfy $\left{\mu}_{j}\right\le 1+\delta $.
The choice of step length
$\sigma $ in the QP iteration
(8) is based on remaining feasible with respect to the satisfied constraints. During the optimality phase, if
$p+d$ is feasible,
$\sigma $ will be taken as unity. (In this case, the projected gradient at
$\stackrel{}{p}$ will be zero.) Otherwise,
$\sigma $ is set to
${\sigma}_{M}$, 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 ${C}_{\mathrm{FR}}$: if the status of a general constraint changes, a row of ${C}_{\mathrm{FR}}$ is altered; if a bound constraint enters or leaves the working set, a column of ${C}_{\mathrm{FR}}$ changes. Explicit representations are recurred of the matrices $T$, ${Q}_{\mathrm{FR}}$ and $R$, and of the vectors ${Q}^{\mathrm{T}}q$ and ${Q}^{\mathrm{T}}g$.
10.3 The Merit Function
After computing the search direction as described in
Section 10.2, each major iteration proceeds by determining a step length
$\alpha $ in
(3) that produces a ‘sufficient decrease’ in the augmented Lagrangian merit function
where
$x$,
$\lambda $ and
$s$ vary during the
linesearch. The summation terms in
(12) involve only the
nonlinear constraints. The vector
$\lambda $ is an estimate of the Lagrange multipliers for the nonlinear constraints of
(1). The nonnegative
slack variables $\left\{{s}_{i}\right\}$ 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
$\rho $ of
penalty arguments 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
$\rho $ (the printed quantity
Penalty; see
Section 11.3) is generally nondecreasing, although each
${\rho}_{i}$ may be reduced a limited number of times.
The merit function
(12) and its global convergence properties are described in
Gill et al. (1986a).
10.4 The Quasi–Newton Update
The matrix
$H$ 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
$\stackrel{}{H}$ is defined as a ranktwo modification of
$H$. In nag_opt_nlp (e04ucc), the BFGS quasiNewton update is used:
where
$s=\stackrel{}{x}x$ (the change in
$x$).
In nag_opt_nlp (e04ucc),
$H$ is required to be positive definite. If
$H$ is positive definite,
$\stackrel{}{H}$ defined by
(13) will be positive definite if and only if
${y}^{\mathrm{T}}s$ is positive (see, e.g.,
Dennis and Moré (1977)). Ideally,
$y$ in
(13) would be taken as
${y}_{L}$, the change in gradient of the Lagrangian function
where
${\mu}_{N}$ denotes the QP multipliers associated with the nonlinear constraints of the original problem. If
${y}_{L}^{\mathrm{T}}s$ is not sufficiently positive, an attempt is made to perform the update with a vector
$y$ of the form
where
${\omega}_{i}\ge 0$. If no such vector can be found, the update is performed with a scaled
${y}_{L}$; in this case,
M is printed to indicate that the update was modified.
Rather than modifying
$H$ itself, the Cholesky factor of the
transformed Hessian ${H}_{Q}$ (6) is updated, where
$Q$ is the matrix from
(5) associated with the active set of the QP subproblem. The update
(12) is equivalent to the following update to
${H}_{Q}$:
where
${y}_{Q}={Q}^{\mathrm{T}}y$, and
${s}_{Q}={Q}^{\mathrm{T}}s$. This update may be expressed as a
rankone update to
$R$ (see
Dennis and Schnabel (1981)).
11 Optional Arguments
A number of optional input and output arguments to nag_opt_nlp (e04ucc) are available through the structure argument
options, type Nag_E04_Opt. An argument may be selected by assigning an appropriate value to the relevant structure member; those arguments not selected will be assigned default values. If no use is to be made of any of the optional arguments you should use the NAG defined null pointer,
E04_DEFAULT, in place of
options when calling nag_opt_nlp (e04ucc); the default settings will then be used for all arguments.
Before assigning values to
options directly the structure
must be initialized by a call to the function
nag_opt_init (e04xxc). Values may then be assigned to the structure members in the normal C manner.
Option settings may also be read from a text file using the function
nag_opt_read (e04xyc) in which case initialization of the
options structure will be performed automatically if not already done. Any subsequent direct assignment to the
options structure must
not be preceded by initialization.
If assignment of functions and memory to pointers in the
options structure is required, this must be done directly in the calling program; they cannot be assigned using
nag_opt_read (e04xyc).
11.1 Optional Argument Checklist and Default Values
For easy reference, the following list shows the members of
options which are valid for nag_opt_nlp (e04ucc) together with their default values where relevant. The number
$\epsilon $ is a generic notation for
machine precision (see
nag_machine_precision (X02AJC)).
Nag_Start start 
Nag_Cold 
Boolean list 
Nag_TRUE 
Nag_PrintType print_level 
Nag_Soln_Iter 
Nag_PrintType minor_print_level 
Nag_NoPrint 
char outfile[80] 
stdout 
void (*print_fun)() 
NULL 
Boolean obj_deriv 
Nag_TRUE 
Boolean con_deriv 
Nag_TRUE 
Nag_GradChk verify_grad 
Nag_SimpleCheck 
Nag_DPrintType print_deriv 
Nag_D_Full 
Integer obj_check_start 
1 
Integer obj_check_stop 
n 
Integer con_check_start 
1 
Integer con_check_stop 
n 
double f_diff_int 
Computed automatically 
double c_diff_int 
Computed automatically 
Integer max_iter 
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,3\left({\mathbf{n}}+{\mathbf{nclin}}\right)+10{\mathbf{ncnlin}}\right)$ 
Integer minor_max_iter 
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,3\left({\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}\right)\right)$ 
double f_prec 
${\epsilon}^{0.9}$ 
double optim_tol 
${{\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_prec}}}^{0.8}$ 
double lin_feas_tol 
$\sqrt{\epsilon}$ 
double nonlin_feas_tol 
${\epsilon}^{0.33}$ or $\sqrt{\epsilon}$ 
double linesearch_tol 
0.9 
double step_limit 
$2.0$ 
double crash_tol 
0.01 
double inf_bound 
${10}^{20}$ 
double inf_step 
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}},{10}^{20}\right)$ 
double *conf 
size ncnlin 
double *conjac 
size ncnlin*n 
Integer *state 
size ${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$ 
double *lambda 
size ${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$ 
double *h 
size n*n 
Boolean hessian 
Nag_FALSE 
Integer iter 
Integer nf 
11.2 Description of the Optional Arguments
start – Nag_Start   Default $\text{}=\mathrm{Nag\_Cold}$ 
On entry: specifies how the initial working set is chosen in both the procedure for finding a feasible point for the linear constraints and bounds, and in the first QP subproblem thereafter. With
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Cold}$, nag_opt_nlp (e04ucc) chooses the initial working set 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 the value of optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{crash\_tol}}$; see below).
With
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Warm}$, you must provide a valid definition of every array element of the optional arguments
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$,
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lambda}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{h}}$ (see below for their definitions). The
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$ 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. The
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$ 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_nlp (e04ucc) will override your specification of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$ if necessary, so that a poor choice of the working set will not cause a fatal error. For instance, any elements of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$ which are set to
$2$,
$1$ or 4 will be reset to zero, as will any elements which are set to 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_nlp (e04ucc) is called repeatedly to solve related problems.
Constraint:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Cold}$ or $\mathrm{Nag\_Warm}$.
list – Nag_Boolean   Default $\text{}=\mathrm{Nag\_TRUE}$ 
On entry: if ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{list}}=\mathrm{Nag\_TRUE}$ the argument settings in the call to nag_opt_nlp (e04ucc) will be printed.
print_level – Nag_PrintType   Default $\text{}=\mathbf{Nag\_Soln\_Iter}$ 
On entry: the level of results printout produced by nag_opt_nlp (e04ucc) at each major iteration. The following values are available:
Nag_NoPrint  No output. 
Nag_Soln  The final solution only. 
Nag_Iter  One line of output for each iteration. 
Nag_Iter_Long  A longer line of output for each iteration with more information (line exceeds 80 characters). 
Nag_Soln_Iter  The final solution and one line of output for each iteration. 
Nag_Soln_Iter_Long  The final solution and one long line of output for each iteration (line exceeds 80 characters). 
Nag_Soln_Iter_Const  As Nag_Soln_Iter_Long with the objective function, the values of the variables, the Euclidean norm of the nonlinear constraint violations, the nonlinear constraint values, $c$, and the linear constraint values ${A}_{L}x$ also printed at each iteration. 
Nag_Soln_Iter_Full  As Nag_Soln_Iter_Const with the diagonal elements of the upper triangular matrix $T$ associated with the $TQ$ factorization (5) of the QP working set, and the diagonal elements of $R$, the triangular factor of the transformed and reordered Hessian (6). 
Details of each level of results printout are described in
Section 11.3.
Constraint:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_NoPrint}$, $\mathrm{Nag\_Soln}$, $\mathrm{Nag\_Iter}$, $\mathrm{Nag\_Soln\_Iter}$, $\mathrm{Nag\_Iter\_Long}$, $\mathrm{Nag\_Soln\_Iter\_Long}$, $\mathrm{Nag\_Soln\_Iter\_Const}$ or $\mathrm{Nag\_Soln\_Iter\_Full}$.
minor_print_level – Nag_PrintType   Default $\text{}=\mathrm{Nag\_NoPrint}$ 
On entry: the level of results printout produced by the minor iterations of nag_opt_nlp (e04ucc) (i.e., the iterations of the QP subproblem). The following values are available:
Nag_NoPrint  No output. 
Nag_Soln  The final solution only. 
Nag_Iter  One line of output for each iteration. 
Nag_Iter_Long  A longer line of output for each iteration with more information (line exceeds 80 characters). 
Nag_Soln_Iter  The final solution and one line of output for each iteration. 
Nag_Soln_Iter_Long  The final solution and one long line of output for each iteration (line exceeds 80 characters). 
Nag_Soln_Iter_Const  As Nag_Soln_Iter_Long with the Lagrange multipliers, the variables $x$, the constraint values ${A}_{L}x$ and the constraint status also printed at each iteration. 
Nag_Soln_Iter_Full  As Nag_Soln_Iter_Const with the diagonal elements of the upper triangular matrix $T$ associated with the $TQ$ factorization (4) of the working set, and the diagonal elements of the upper triangular matrix $R$ printed at each iteration. 
Details of each level of results printout are described in
Section 11 in nag_opt_lin_lsq (e04ncc). (
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_print\_level}}$ in the present function is equivalent to
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}$.)
Constraint:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_print\_level}}=\mathrm{Nag\_NoPrint}$, $\mathrm{Nag\_Soln}$, $\mathrm{Nag\_Iter}$, $\mathrm{Nag\_Soln\_Iter}$, $\mathrm{Nag\_Iter\_Long}$, $\mathrm{Nag\_Soln\_Iter\_Long}$, $\mathrm{Nag\_Soln\_Iter\_Const}$ or $\mathrm{Nag\_Soln\_Iter\_Full}$.
outfile – const char[80]   Default $\text{}=\mathtt{stdout}$ 
On entry: the name of the file to which results should be printed. If ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{outfile}}\left[0\right]=\text{' 0 '}$ then the stdout stream is used.
print_fun – pointer to function   Default $\text{}=\text{}$ NULL 
On entry: printing function defined by you; the prototype of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_fun}}$ is
void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);
See
Section 11.3.1 for further details.
obj_deriv – Nag_Boolean   Default $\text{}=\mathrm{Nag\_TRUE}$ 
On entry: this argument indicates whether you have provided all the derivatives of the objective function in
objfun. If none or only some of the derivatives are being supplied by
objfun then
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_deriv}}$ should be set to Nag_FALSE.
Whenever possible you should supply all derivatives, since nag_opt_nlp (e04ucc) is more reliable and will usually be more efficient when all derivatives are exact.
If
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_deriv}}=\mathrm{Nag\_FALSE}$, nag_opt_nlp (e04ucc) will approximate the unspecified components of the objective gradient, using finite differences. The computation of finite difference approximations usually increases the total runtime, since a call to
objfun is required for each unspecified element. Furthermore, less accuracy can be attained in the solution (see Chapter 8 of
Gill et al. (1986b), for a discussion of limiting accuracy).
At times, central differences are used rather than forward differences, in which case twice as many calls to
objfun are needed. (The switch to central differences is not under your control.)
con_deriv – Nag_Boolean   Default $\text{}=\mathrm{Nag\_TRUE}$ 
On entry: this argument indicates whether you have provided all derivatives for the constraint Jacobian in
confun. If none or only some of the derivatives are being supplied by
confun then
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_deriv}}$ should be set to Nag_FALSE.
Whenever possible you should supply all derivatives, since nag_opt_nlp (e04ucc) is more reliable and will usually be more efficient when all derivatives are exact.
If
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_deriv}}=\mathrm{Nag\_FALSE}$, nag_opt_nlp (e04ucc) will approximate unspecified elements of the Jacobian. One call to
confun is needed for each variable for which partial derivatives are not available. For example, if the constraint Jacobian has the form
where
$\text{'*'}$ indicates a provided element and ‘?’ indicates an unspecified element, nag_opt_nlp (e04ucc) will call
confun twice: once to estimate the missing element in column 2, and again to estimate the two missing elements in column 3. (Since columns 1 and 4 are known, they require no calls to
confun.)
At times, central differences are used rather than forward differences, in which case twice as many calls to
confun are needed. (The switch to central differences is not under your control.)
verify_grad – Nag_GradChk   Default $\text{}=\mathrm{Nag\_SimpleCheck}$ 
On entry: specifies the level of derivative checking to be performed by nag_opt_nlp (e04ucc) on the gradient elements computed by
objfun and
confun.
The following values are available:
Nag_NoCheck  No derivative checking is performed. 
Nag_SimpleCheck  Perform a simple check of both the objective and constraint gradients. 
Nag_CheckObj  Perform a component check of the objective gradient elements. 
Nag_CheckCon  Perform a component check of the constraint gradient elements. 
Nag_CheckObjCon  Perform a component check of both the objective and constraint gradient elements. 
Nag_XSimpleCheck  Perform a simple check of both the objective and constraint gradients at the initial value of $x$ specified in x. 
Nag_XCheckObj  Perform a component check of the objective gradient elements at the initial value of $x$ specified in x. 
Nag_XCheckCon  Perform a component check of the constraint gradient elements at the initial value of $x$ specified in x. 
Nag_XCheckObjCon  Perform a component check of both the objective and constraint gradient elements at the initial value of $x$ specified in x. 
If
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}=\mathrm{Nag\_SimpleCheck}$ or
$\mathrm{Nag\_XSimpleCheck}$ then a simple ‘cheap’ test is performed, which requires only one call to
objfun and one call to
confun. If
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}=\mathrm{Nag\_CheckObj}$,
$\mathrm{Nag\_CheckCon}$ or
$\mathrm{Nag\_CheckObjCon}$ then a more reliable (but more expensive) test will be made on individual gradient components. This component check will be made in the range specified by the optional arguments
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_start}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_stop}}$ for the objective gradient, with default values being
$1$ and
n respectively. For the constraint gradient the range is specified by
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_start}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_stop}}$, with default values being
$1$ and
n.
The procedure for the derivative check is based on finding an interval that produces an acceptable estimate of the second derivative, and then using that estimate to compute an interval that should produce a reasonable forwarddifference approximation. The gradient element is then compared with the difference approximation. (The method of finite difference interval estimation is based on
Gill et al. (1983).) The result of the test is printed out by nag_opt_nlp (e04ucc) if optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_deriv}}\ne \mathrm{Nag\_D\_NoPrint}$.
Constraint:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}=\mathrm{Nag\_NoCheck}$, $\mathrm{Nag\_SimpleCheck}$, $\mathrm{Nag\_CheckObj}$, $\mathrm{Nag\_CheckCon}$, $\mathrm{Nag\_CheckObjCon}$, $\mathrm{Nag\_XSimpleCheck}$, $\mathrm{Nag\_XCheckObj}$, $\mathrm{Nag\_XCheckCon}$ or $\mathrm{Nag\_XCheckObjCon}$.
print_deriv – Nag_DPrintType   Default $\text{}=\mathbf{Nag\_D\_Full}$ 
On entry: controls whether the results of any derivative checking are printed out (see optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}$).
If a component derivative check has been carried out, then full details will be printed if ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_deriv}}=\mathrm{Nag\_D\_Full}$. For a printout summarising the results of a component derivative check set ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_deriv}}=\mathrm{Nag\_D\_Sum}$. If only a simple derivative check is requested then Nag_D_Sum and Nag_D_Full will give the same level of output. To prevent any printout from a derivative check set ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_deriv}}=\mathrm{Nag\_D\_NoPrint}$.
Constraint:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_deriv}}=\mathrm{Nag\_D\_NoPrint}$, $\mathrm{Nag\_D\_Sum}$ or $\mathrm{Nag\_D\_Full}$.
obj_check_start – Integer   Default $\text{}=1$ 
obj_check_stop – Integer   Default $\text{}={\mathbf{n}}$ 
These options take effect only when ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}=\mathrm{Nag\_CheckObj}$, $\mathrm{Nag\_CheckObjCon}$, $\mathrm{Nag\_XCheckObj}$ or $\mathrm{Nag\_XCheckObjCon}$.
On entry: they may be used to control the verification of gradient elements computed by the function
objfun. For example, if the first 30 elements appeared to be correct in an earlier run, so that only element 31 remains questionable, it is reasonable to specify
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_start}}=31$. If the first 30 variables appear linearly in the objective, so that the corresponding gradient elements are constant, the above choice would also be appropriate.
Constraint:
$1\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_start}}\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_stop}}\le {\mathbf{n}}$.
con_check_start – Integer   Default $\text{}=1$ 
con_check_stop – Integer   Default $\text{}={\mathbf{n}}$ 
These options take effect only when ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}=\mathrm{Nag\_CheckCon}$, $\mathrm{Nag\_CheckObjCon}$, $\mathrm{Nag\_XCheckCon}$ or $\mathrm{Nag\_XCheckObjCon}$.
On entry: these arguments may be used to control the verification of the Jacobian elements computed by the function
confun. For example, if the first 30 columns of the constraint Jacobian appeared to be correct in an earlier run, so that only column 31 remains questionable, it is reasonable to specify
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_start}}=31$.
Constraint:
$1\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_start}}\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_stop}}\le {\mathbf{n}}$.
f_diff_int – double   Default $\text{}=\text{computed automatically}$ 
On entry: 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 argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}$). 
(b) 
For estimating unspecified elements of the objective and/or constraint Jacobian matrix. 
In general, using the notation
$r={\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_diff\_int}}$, a derivative with respect to the
$j$th variable is approximated using the interval
${\delta}_{j}$, where
${\delta}_{j}=r\left(1+\left{\hat{x}}_{j}\right\right)$, with
$\hat{x}$ the first point feasible with respect to the bounds and linear constraints. If the functions are well scaled, the resulting derivative approximation should be accurate to O
$\left(r\right)$. See
Gill et al. (1981) for a discussion of the accuracy in finite difference approximations.
If you do not specify a difference interval, a finite difference interval will be computed automatically for each variable by a procedure that requires up to six calls of
confun and
objfun for each element. This option is recommended if the function is badly scaled or you wish to have nag_opt_nlp (e04ucc) determine constant elements in the objective and constraint gradients (see the descriptions of
confun and
objfun in
Section 5).
Constraint:
$\epsilon \le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_diff\_int}}<1.0$.
c_diff_int – double   Default $\text{}=\text{computed automatically}$ 
On entry: if the algorithm switches to central differences because the forwarddifference approximation is not sufficiently accurate the value of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{c\_diff\_int}}$ is used as the difference interval for every element of
$x$. 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 5.1). The use of finite differences is discussed under the option
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_diff\_int}}$.
Constraint:
$\epsilon \le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{c\_diff\_int}}<1.0$.
max_iter – Integer   Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,3\left({\mathbf{n}}+{\mathbf{nclin}}\right)+10{\mathbf{ncnlin}}\right)$ 
On entry: the maximum number of major iterations allowed before termination.
Constraint:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{max\_iter}}\ge 0$.
minor_max_iter – Integer   Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,3\left({\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}\right)\right)$ 
On entry: the maximum number of iterations for finding a feasible point with respect to the bounds and linear constraints (if any). The value also specifies the maximum number of minor iterations for the optimality phase of each QP subproblem.
Constraint:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_max\_iter}}\ge 0$.
f_prec – double   Default $\text{}={\epsilon}^{0.9}$ 
On entry: this argument defines
${\epsilon}_{r}$, which is intended to be a measure of the accuracy with which the problem functions
$F\left(x\right)$ and
$c\left(x\right)$ can be computed.
The value of
${\epsilon}_{r}$ should reflect the relative precision of
$1+\leftF\left(x\right)\right$; i.e.,
${\epsilon}_{r}$ acts as a relative precision when
$\leftF\right$ is large, and as an absolute precision when
$\leftF\right$ is small. For example, if
$F\left(x\right)$ is typically of order 1000 and the first six significant digits are known to be correct, an appropriate value for
${\epsilon}_{r}$ would be
${10}^{6}$. In contrast, if
$F\left(x\right)$ is typically of order
${10}^{4}$ and the first six significant digits are known to be correct, an appropriate value for
${\epsilon}_{r}$ would be
${10}^{10}$. The choice of
${\epsilon}_{r}$ 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
${\epsilon}_{r}$ should be large enough so that nag_opt_nlp (e04ucc) will not attempt to distinguish between function values that differ by less than the error inherent in the calculation.
Constraint:
$\epsilon \le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_prec}}<1.0$.
optim_tol – double   Default $\text{}={{\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_prec}}}^{0.8}$ 
On entry: specifies the accuracy to which you wish the final iterate to approximate a solution of the problem. Broadly speaking,
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}$ indicates the number of correct figures desired in the objective function at the solution. For example, if
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}$ is
${10}^{6}$ and nag_opt_nlp (e04ucc) terminates successfully, the final value of
$F$ should have approximately six correct figures.
nag_opt_nlp (e04ucc) will terminate successfully if the iterative sequence of
$x$values is judged to have converged and the final point satisfies the firstorder Kuhn–Tucker conditions (see
Section 10.1). The sequence of iterates is considered to have converged at
$x$ if
where
$p$ is the search direction and
$\alpha $ the step length from
(3), and
$r$ is the value of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}$. An iterate is considered to satisfy the firstorder conditions for a minimum if
and
where
${{Z}^{\mathrm{T}}}_{\mathrm{FR}}{g}_{\mathrm{FR}}$ is the projected gradient (see
Section 10.1),
${g}_{\mathrm{FR}}$ is the gradient of
$F\left(x\right)$ with respect to the free variables,
${res}_{j}$ is the violation of the
$j$th active nonlinear constraint, and
$\mathit{ftol}$ the value of the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}$.
Constraint:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{f\_prec}}\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}<1.0$.
lin_feas_tol – double   Default $\text{}=\sqrt{\epsilon}$ 
On entry: defines the maximum acceptable
absolute violations in the linear constraints at a ‘feasible’ point; i.e., a linear constraint is considered satisfied if its violation does not exceed
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$.
On entry to nag_opt_nlp (e04ucc), 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 specified by ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$. All subsequent iterates will satisfy the constraints to within the same tolerance (unless ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ is comparable to the finite difference interval).
This tolerance should reflect the precision of the linear 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 decimal digits, it would be appropriate to specify ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ as ${10}^{6}$.
Constraint:
$\epsilon \le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}<1.0$.
nonlin_feas_tol – double   Default $\text{}={\epsilon}^{0.33}\text{ or}\sqrt{\epsilon}$ 
The default is ${\epsilon}^{0.33}$ if ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_deriv}}=\mathrm{Nag\_FALSE}$, and $\sqrt{\epsilon}$ otherwise.
On entry: defines the maximum acceptable violations in the nonlinear constraints at a ‘feasible’ point; i.e., a nonlinear constraint is considered satisfied if its violation does not exceed
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}$.
The tolerance 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 ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}$ acts as a partial termination criteria for the iterative sequence generated by nag_opt_nlp (e04ucc) (see the discussion of ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}$ ).
This tolerance should reflect the precision of the nonlinear constraint functions calculated by
confun.
Constraint:
$\epsilon \le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}<1.0$.
linesearch_tol – double   Default $\text{}=0.9$ 
On entry: controls the accuracy with which the step
$\alpha $ taken during each iteration approximates a minimum of the merit function along the search direction (the smaller the value of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{linesearch\_tol}}$, the more accurate the line search). The default value 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.
Constraint:
$0.0\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{linesearch\_tol}}<1.0$.
step_limit – double   Default $\text{}=2.0$ 
On entry: specifies the maximum change in the variables at the first step of the line search. In some cases, such as
$F\left(x\right)={ae}^{bx}$ or
$F\left(x\right)={ax}^{b}$, even a moderate change in the elements of
$x$ can lead to floating point overflow. The argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}$ is therefore used to encourage evaluation of the problem functions at meaningful points. Given any major iterate
$x$, the first point
$\stackrel{~}{x}$ at which
$F$ and
$c$ are evaluated during the line search is restricted so that
where
$r$ is the value of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}$.
The line search may go on and evaluate
$F$ and
$c$ at points further from
$x$ if this will result in a lower value of the merit function. In this case, the character
L is printed at the end of each line of output produced by the major iterations (see
Section 5.1). If
L is printed for most of the iterations,
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}$ should be set to a larger value.
Wherever possible, upper and lower bounds on $x$ should be used to prevent evaluation of nonlinear functions at wild values. The default value of ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}=2.0$ should not affect progress on wellbehaved functions, but values such as 0.1 or 0.01 may be helpful when rapidly varying functions are present. If a small value of ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}$ is selected, a good starting point may be required. An important application is to the class of nonlinear least squares problems.
Constraint:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}>0.0$.
crash_tol – double   Default $\text{}=0.01$ 
On entry: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{crash\_tol}}$ is used during a ‘cold start’ when nag_opt_nlp (e04ucc) selects an initial working set (${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Cold}$). The initial working set will include (if possible) bounds or general inequality constraints that lie within ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{crash\_tol}}$ of their bounds. In particular, a constraint of the form ${a}_{j}^{\mathrm{T}}x\ge l$ will be included in the initial working set if $\left{a}_{j}^{\mathrm{T}}xl\right\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{crash\_tol}}\times \left(1+\leftl\right\right)$.
Constraint:
$0.0\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{crash\_tol}}\le 1.0$.
inf_bound – double   Default $\text{}={10}^{20}$ 
On entry: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ defines the ‘infinite’ bound in the definition of the problem constraints. Any upper bound greater than or equal to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ will be regarded as $+\infty $ (and similarly any lower bound less than or equal to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ will be regarded as $\infty $).
Constraint:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}>0.0$.
inf_step – double   Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}},{10}^{20}\right)$ 
On entry: ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_step}}$ specifies the magnitude of the change in variables that will be considered a step to an unbounded solution. If the change in $x$ during an iteration would exceed the value of ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_step}}$, the objective function is considered to be unbounded below in the feasible region.
Constraint:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_step}}>0.0$.
conf – double   Default $\text{}={\mathbf{ncnlin}}$ 
On entry:
ncnlin values of memory will be automatically allocated by nag_opt_nlp (e04ucc) and this is the recommended method of use of
conf. However you may supply memory from the calling program.
On exit: if
${\mathbf{ncnlin}}>0$,
${\mathbf{conf}}\left[i1\right]$ contains the value of the
$i$th nonlinear constraint function
${c}_{i}$ at the final iterate.
If
${\mathbf{ncnlin}}=0$ then
conf will not be referenced.
conjac – double   Default $\text{}={\mathbf{ncnlin}}\times {\mathbf{n}}$ 
On entry: ${\mathbf{ncnlin}}\times {\mathbf{n}}$ values of memory will be automatically allocated by nag_opt_nlp (e04ucc) and this is the recommended method of use of ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{conjac}}$. However you may supply memory from the calling program.
On exit: if
${\mathbf{ncnlin}}>0$,
conjac contains the Jacobian matrix of the nonlinear constraint functions at the final iterate, i.e.,
${\mathbf{conjac}}\left[\left(\mathit{i}1\right)\times {\mathbf{n}}+\mathit{j}1\right]$ contains the partial derivative of the
$\mathit{i}$th constraint function with respect to the
$\mathit{j}$th variable, for
$\mathit{i}=1,2,\dots ,{\mathbf{ncnlin}}$ and
$\mathit{j}=1,2,\dots ,{\mathbf{n}}$. (See the discussion of the argument
conjac under
confun.)
If
${\mathbf{ncnlin}}=0$ then
conjac will not be referenced.
state – Integer   Default $\text{}={\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$ 
On entry:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$ need not be set if the default option of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Cold}$ is used as
${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$ values of memory will be automatically allocated by nag_opt_nlp (e04ucc).
If the option
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Warm}$ has been chosen,
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$ must point to a minimum of
${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$ elements of memory. This memory will already be available if the
options structure has been used in a previous call to nag_opt_nlp (e04ucc) from the calling program, with
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Cold}$ and the same values of
n,
nclin and
ncnlin. If a previous call has not been made, you must allocate sufficient memory.
When a ‘warm start’ is chosen
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$ should specify the status of the bounds and linear constraints at the start of the feasibility phase. More precisely, the first
n elements of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$ refer to the upper and lower bounds on the variables, the next
nclin elements refer to the general linear constraints and the following
ncnlin elements refer to the nonlinear constraints. Possible values for
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}\left[j\right]$ are as follows:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}\left[j\right]$ 
Meaning 
$0$ 
The corresponding constraint is not in the initial QP working set. 
$1$ 
This inequality constraint should be in the initial working set at its lower bound. 
$2$ 
This inequality constraint should be in the initial working set at its upper bound. 
$3$ 
This equality constraint should be in the initial working set. This value must only be specified if ${\mathbf{bl}}\left[j\right]={\mathbf{bu}}\left[j\right]$. 
The values
$2$,
$1$ and 4 are also acceptable but will be reset to zero by the function, as will any elements which are set to 3 when the corresponding elements of
bl and
bu are not equal. If nag_opt_nlp (e04ucc) has been called previously with the same values of
n,
nclin and
ncnlin, then
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$ already contains satisfactory information. (See also the description of the optional argument
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}$.) The function also adjusts (if necessary) the values supplied in
x to be consistent with the values supplied in
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$.
Constraint:
$2\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}\left[\mathit{j}1\right]\le 4$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$.
On exit: the status of the constraints in the QP working set at the point returned in
x. The significance of each possible value of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}\left[j\right]$ is as follows:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}\left[j\right]$ 
Meaning 
$2$ 
The constraint violates its lower bound by more than the appropriate feasibility tolerance (see the options ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ and ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}$). This value can occur only when no feasible point can be found for a QP subproblem. 
$1$ 
The constraint violates its upper bound by more than the appropriate feasibility tolerance (see the options ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ and ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}$). This value can occur only when no feasible point can be found for a QP subproblem. 
$\phantom{}0$ 
The constraint is satisfied to within the feasibility tolerance, but is not in the QP working set. 
$\phantom{}1$ 
This inequality constraint is included in the QP working set at its lower bound. 
$\phantom{}2$ 
This inequality constraint is included in the QP working set at its upper bound. 
$\phantom{}3$ 
This constraint is included in the working set as an equality. This value of ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$ can occur only when ${\mathbf{bl}}\left[j\right]={\mathbf{bu}}\left[j\right]$. 
lambda – double   Default $\text{}={\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$ 
On entry:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lambda}}$ need not be set if the default option of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Cold}$ is used as
${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$ values of memory will be automatically allocated by nag_opt_nlp (e04ucc).
If the option
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Warm}$ has been chosen,
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lambda}}$ must point to a minimum of
${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$ elements of memory. This memory will already be available if the
options structure has been used in a previous call to nag_opt_nlp (e04ucc) from the calling program, with
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Cold}$ and the same values of
n,
nclin and
ncnlin. If a previous call has not been made, you must allocate sufficient memory.
When a ‘warm start’ is chosen ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lambda}}\left[j1\right]$ must contain a multiplier estimate for each nonlinear constraint with a sign that matches the status of the constraint specified by ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$, for $j={\mathbf{n}}+{\mathbf{nclin}}+1$, ${\mathbf{n}}+{\mathbf{nclin}}+2$,$\dots $, ${\mathbf{n}}+{\mathbf{nclin}}+{\mathbf{ncnlin}}$. The remaining elements need not be set.
Note that if the $j$th constraint is defined as ‘inactive’ by the initial value of the ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}$ array (i.e., ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}\left[j1\right]=0$), ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lambda}}\left[j1\right]$ should be zero; if the $j$th constraint is an inequality active at its lower bound (i.e., ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}\left[j1\right]=1$), ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lambda}}\left[j1\right]$ should be nonnegative; if the $j$th constraint is an inequality active at its upper bound (i.e., ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}\left[j1\right]=2$), ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lambda}}\left[j1\right]$ should be nonpositive. If necessary, the function will modify ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lambda}}$ to match these rules.
On exit: the values of the Lagrange multipliers from the last QP subproblem. ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lambda}}\left[j1\right]$ should be nonnegative if ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}\left[j1\right]=1$ and nonpositive if ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{state}}\left[j1\right]=2$.
h – double   Default $\text{}={\mathbf{n}}\times {\mathbf{n}}$ 
On entry:
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{h}}$ need not be set if the default option of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Cold}$ is used, as
${\mathbf{n}}\times {\mathbf{n}}$ values of memory will be automatically allocated by nag_opt_nlp (e04ucc).
If the option
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Warm}$ has been chosen,
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{h}}$ must point to a minimum of
${\mathbf{n}}\times {\mathbf{n}}$ elements of memory. This memory will already be available if the calling program has used the
options structure in a previous call to nag_opt_nlp (e04ucc) with
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Cold}$ and the same value of
n. If a previous call has not been made you must allocate sufficient memory.
When ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Warm}$ is chosen, the memory pointed to by ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{h}}$ must contain the upper triangular Cholesky factor $R$ 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. If a previous call has been made, with ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{hessian}}=\mathrm{Nag\_TRUE}$, then ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{h}}$ will already have been set correctly.
On exit: if
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{hessian}}=\mathrm{Nag\_FALSE}$,
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{h}}$ contains the upper triangular Cholesky factor
$R$ of
${Q}^{\mathrm{T}}\stackrel{~}{H}Q$, an estimate of the transformed and reordered Hessian of the Lagrangian at
$x$ (see
(6)).
If ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{hessian}}=\mathrm{Nag\_TRUE}$, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{h}}$ contains the upper triangular Cholesky factor $R$ of $H$, the approximate (untransformed) Hessian of the Lagrangian, with the variables in the natural order.
hessian – Nag_Boolean   Default $\text{}=\mathrm{Nag\_FALSE}$ 
On entry: controls the contents of the optional argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{h}}$ on return from nag_opt_nlp (e04ucc). nag_opt_nlp (e04ucc) works exclusively with the transformed and reordered Hessian ${H}_{Q}$, and hence extra computation is required to form the Hessian itself. If ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{hessian}}=\mathrm{Nag\_FALSE}$, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{h}}$ contains the Cholesky factor of the transformed and reordered Hessian. If ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{hessian}}=\mathrm{Nag\_TRUE}$, the Cholesky factor of the approximate Hessian itself is formed and stored in ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{h}}$. This information is required by nag_opt_nlp (e04ucc) if the next call to nag_opt_nlp (e04ucc) will be made with optional argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{start}}=\mathrm{Nag\_Warm}$.
On exit: the number of major iterations which have been performed in nag_opt_nlp (e04ucc).
On exit: the number of times the objective function has been evaluated (i.e., number of calls of
objfun). The total excludes any calls made to
objfun for purposes of derivative checking.
11.3 Description of Printed Output
The level of printed output can be controlled with the structure members
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{list}}$,
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_deriv}}$,
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_print\_level}}$ (see
Section 11.2). If
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{list}}=\mathrm{Nag\_TRUE}$ then the argument values to nag_opt_nlp (e04ucc) are listed, followed by the result of any derivative check if
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_deriv}}=\mathrm{Nag\_D\_Sum}$ or
$\mathrm{Nag\_D\_Full}$. The printout of results is governed by the values of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_print\_level}}$. The default of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_Soln\_Iter}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_print\_level}}=\mathrm{Nag\_NoPrint}$ provides a single line of output at each iteration and the final result. This section describes all of the possible levels of results printout available from nag_opt_nlp (e04ucc).
If a simple derivative check, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}=\mathrm{Nag\_SimpleCheck}$, is requested then a statement indicating success or failure is given. The largest error found in the constraint Jacobian is output together with the directional derivative, ${g}^{\mathrm{T}}p$, of the objective gradient and its finite difference approximation, where $p$ is a random vector of unit length.
When a component derivative check (see
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}$ in
Section 11.2) is selected the element with the largest relative error is identified for the objective gradient and the constraint Jacobian.
If the value of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_deriv}}=\mathrm{Nag\_D\_Full}$ then the following results are printed for each component:
x[i]  the element of $x$. 
dx[i]  the optimal finite difference interval. 
g[i] or Jacobian value  the gradient/Jacobian element. 
Difference approxn.  the finite difference approximation. 
Itns  the number of trials performed to find a suitable difference interval. 
The indicator,
OK or
BAD? , states whether the gradient/Jacobian element and finite difference approximation are in agreement. If the derivatives are believed to be in error nag_opt_nlp (e04ucc) will exit with
${\mathbf{fail}}\mathbf{.}\mathbf{code}={\mathbf{NE\_DERIV\_ERRORS}}$.
When
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_Iter}$ or
$\mathrm{Nag\_Soln\_Iter}$ the following line of output is produced at every 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 in the later iterations, since theoretical analysis predicts that the correct active set will be identified near the solution (see Section 10).
Note that Mnr may be greater than the optional argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_max\_iter}}$ (default value $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,3\left(n+{n}_{L}+{n}_{N}\right)\right)$; see Section 11.2) if some iterations are required for the feasibility phase. 
Step 
is the step taken along the computed search direction. On reasonably wellbehaved problems, the unit step 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 arguments (see Section 10.3). 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), the merit function is a large multiple of the constraint violations, weighted by the penalty arguments. 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_nlp (e04ucc) terminates with ${\mathbf{fail}}\mathbf{.}\mathbf{code}={\mathbf{NW\_NONLIN\_NOT\_FEASIBLE}}$ (no feasible point could be found for the nonlinear constraints).
If no nonlinear constraints are present (i.e., ${\mathbf{ncnlin}}=0$), this entry contains Objective, the value of the objective function $F\left(x\right)$. The objective function will decrease monotonically to its optimal value when there are no nonlinear constraints. 
Violtn 
is the Euclidean norm of the residuals of constraints that are violated or in the predicted active set (not printed if ncnlin is zero). Violtn will be approximately zero in the neighbourhood of a solution. 
Norm Gz 
is $\Vert {Z}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, the Euclidean norm of the projected gradient (see Section 10.1). Norm Gz 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 ${H}_{Z}$
$\left({H}_{Z}={Z}^{\mathrm{T}}{H}_{\mathrm{FR}}Z={R}_{Z}^{\mathrm{T}}{R}_{Z}\right)$; see (6) and (11). The larger this number, the more difficult the problem. 
The line of output may be terminated by one of the following characters:
M 
is printed if the quasiNewton update was modified to ensure that the Hessian approximation is positive definite (see Section 10.4). 
I 
is printed if the QP subproblem has no feasible point. 
C 
is printed if central differences were used to compute the unspecified objective and constraint gradients. If the value of Step is zero, the switch to central differences was made because no lower point could be found in the line search. (In this case, the QP subproblem is resolved with the central difference gradient and Jacobian.) If the value of Step is nonzero, central differences were computed because Norm Gz and Violtn imply that $x$ is close to a Kuhn–Tucker point (see Section 10.1). 
L 
is printed if the line search has produced a relative change in $x$ greater than the value defined by the optional argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}$ (default value $\text{}=2.0$; see Section 11.2). If this output occurs frequently during later iterations of the run, ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}$ should be set to a larger value. 
R 
is printed if the approximate Hessian has been refactorized. If the diagonal condition estimator of $R$ indicates that the approximate Hessian is badly conditioned, the approximate Hessian is refactorized using column interchanges. If necessary, $R$ is modified so that its diagonal condition estimator is bounded. 
If
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_Iter\_Long}$,
$\mathrm{Nag\_Soln\_Iter\_Long}$,
$\mathrm{Nag\_Soln\_Iter\_Const}$ or
$\mathrm{Nag\_Soln\_Iter\_Full}$ the line of printout at every iteration is extended to give the following additional information. (Note this longer line extends over more than 80 characters.)
Nfun 
is the cumulative number of evaluations of the objective function needed for the line search. 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. 
Nz 
is the number of columns of $Z$ (see Section 10.1). The value of Nz is the number of variables minus the number of constraints in the predicted active set; i.e., $\mathtt{Nz}=n\left(\mathtt{Bnd}+\mathtt{Lin}+\mathtt{Nln}\right)$. 
Bnd 
is the number of simple bound constraints in the predicted active set. 
Lin 
is the number of general linear constraints in the predicted active set. 
Nln 
is the number of nonlinear constraints in the predicted active set (not printed if ncnlin is zero). 
Penalty 
is the Euclidean norm of the vector of penalty arguments used in the augmented Lagrangian merit function (not printed if ncnlin is zero). 
Norm Gf 
is the Euclidean norm of ${g}_{\mathrm{FR}}$, the gradient of the objective function with respect to the free variables. 
Cond H 
is a lower bound on the condition number of the Hessian approximation $H$. 
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 argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}$ in Section 11.2. 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_nlp (e04ucc) terminates with the error indicator ${\mathbf{fail}}\mathbf{.}\mathbf{code}=\mathrm{NE\_NOERROR}$, you should check the solution carefully. 
When
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_Soln\_Iter\_Const}$ or
$\mathrm{Nag\_Soln\_Iter\_Full}$ more detailed results are given at each iteration. If
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_Soln\_Iter\_Const}$ these additional values are: the value of
$x$ currently held in
x; the current value of the objective function; the Euclidean norm of nonlinear constraint violations; the values of the nonlinear constraints (the vector
$c$); and the values of the linear constraints, (the vector
${A}_{L}x$).
If
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_Soln\_Iter\_Full}$ then the diagonal elements of the matrix
$T$ associated with the
$TQ$ factorization
(5) of the QP working set and the diagonal elements of
$R$, the triangular factor of the transformed and reordered Hessian
(6) (see
Section 10.1) are also output at each iteration.
When
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_Soln}$,
$\mathrm{Nag\_Soln\_Iter}$,
$\mathrm{Nag\_Soln\_Iter\_Long}$,
$\mathrm{Nag\_Soln\_Iter\_Const}$ or
$\mathrm{Nag\_Soln\_Iter\_Full}$ the final printout from nag_opt_nlp (e04ucc) includes a listing of the status of every variable and constraint. The following describes the printout for each variable.
Varbl 
gives the name (V) and index $\mathit{j}$, for $\mathit{j}=1,2,\dots ,n$ of the variable. 
State 
gives the state of the variable (FR if neither bound is in the active set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound). If Value lies outside the upper or lower bounds by more than the feasibility tolerances specified by the optional arguments ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ and ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}$ (see Section 11.2), State will be ++ or  respectively.
A key is sometimes printed before State to give some additional information about the state of a variable.
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, 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 ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$. 

Value 
is the value of the variable at the final iteration. 
Lower bound 
is the lower bound specified for the variable $j$. (None indicates that ${\mathbf{bl}}\left[j1\right]\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$, where ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ is the optional argument.) 
Upper bound 
is the upper bound specified for the variable $j$. (None indicates that ${\mathbf{bu}}\left[j1\right]\ge {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$, where ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ is the optional argument.) 
Lagr Mult 
is the value of the Lagrange multiplier for the associated bound constraint. This will be zero if State is FR unless ${\mathbf{bl}}\left[j1\right]\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ and ${\mathbf{bu}}\left[j1\right]\ge {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$, in which case the entry will be blank. If $x$ is optimal, the multiplier should be nonnegative if State is LL, and nonpositive if State is UL. 
Residual 
is the difference between the variable Value and the nearer of its (finite) bounds ${\mathbf{bl}}\left[j1\right]$ and ${\mathbf{bu}}\left[j1\right]$. A blank entry indicates that the associated variable is not bounded (i.e., ${\mathbf{bl}}\left[j1\right]\le {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$ and ${\mathbf{bu}}\left[j1\right]\ge {\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{inf\_bound}}$). 
The meaning of the printout for linear and nonlinear constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, ${\mathbf{bl}}\left[j1\right]$ and ${\mathbf{bu}}\left[j1\right]$ are replaced by ${\mathbf{bl}}\left[n+j1\right]$ and ${\mathbf{bu}}\left[n+j1\right]$ respectively, and with the following changes in the heading:
L Con 
gives the name (L) and index $\mathit{j}$, for $\mathit{j}=1,2,\dots ,{n}_{L}$ of the linear constraint. 
N Con 
gives the name (N) and index $\left({jn}_{L}\right)$, for $j={n}_{L}+1,{n}_{L}+2,\dots ,{n}_{L}+{n}_{N}$ of the nonlinear constraint. 
The I key in the State column is printed for general linear constraints which currently violate one of their bounds by more than ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{lin\_feas\_tol}}$ and for nonlinear constraints which violate one of their bounds by more than ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{nonlin\_feas\_tol}}$.
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 Residual column to become positive.
Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.
For the output governed by
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_print\_level}}$, you are referred to the documentation for
nag_opt_lin_lsq (e04ncc). This option is equivalent to
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}$.
If ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_NoPrint}$ then printout will be suppressed; you can print the final solution when nag_opt_nlp (e04ucc) returns to the calling program.
11.3.1 Output of results via a userdefined printing function
You may also specify your own print function for output of iteration results and the final solution by use of the
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_fun}}$ function pointer, which has prototype
void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);
This section may be skipped if you wish to use the default printing facilities.
When a userdefined function is assigned to
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_fun}}$ this will be called in preference to the internal print function of nag_opt_nlp (e04ucc). Calls to the userdefined function are again controlled by means of the
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}$,
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{minor\_print\_level}}$ and
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_deriv}}$ members. Information is provided through
st and
comm, the two structure arguments to
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_fun}}$.
If $\mathbf{comm}\mathbf{\to}\mathbf{it\_maj\_prt}=\mathrm{Nag\_TRUE}$ then results from the last major iteration of nag_opt_nlp (e04ucc) are provided through st. Note that ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_fun}}$ will be called with $\mathbf{comm}\mathbf{\to}\mathbf{it\_maj\_prt}=\mathrm{Nag\_TRUE}$ only if ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_Iter}$, $\mathrm{Nag\_Soln\_Iter}$, $\mathrm{Nag\_Soln\_Iter\_Long}$, $\mathrm{Nag\_Soln\_Iter\_Const}$ or $\mathrm{Nag\_Soln\_Iter\_Full}$. The following members of st are set:
 n – Integer

The number of variables.
 nclin – Integer

The number of linear constraints.
 ncnlin – Integer

The number of nonlinear constraints.
 nactiv – Integer

The total number of active elements in the current set.
 iter – Integer

The major iteration count.
 minor_iter – Integer

The minor iteration count for the feasibility and the optimality phases of the QP subproblem.
 step – double

The step taken along the computed search direction.
 nfun – Integer

The cumulative number of objective function evaluations needed for the line search.
 merit – double

The value of the augmented Lagrangian merit function at the current iterate.
 objf – double

The current value of the objective function.
 norm_nlnviol – double

The Euclidean norm of nonlinear constraint violations (only available if $\mathbf{st}\mathbf{\to}\mathbf{ncnlin}>0$).
 violtn – double

The Euclidean norm of the residuals of constraints that are violated or in the predicted active set (only available if $\mathbf{st}\mathbf{\to}\mathbf{ncnlin}>0$).
 norm_gz – double

$\Vert {Z}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, the Euclidean norm of the projected gradient.
 nz – Integer

The number of columns of
$Z$ (see
Section 10.1).
 bnd – Integer

The number of simple bound constraints in the predicted active set.
 lin – Integer

The number of general linear constraints in the predicted active set.
 nln – Integer

The number of nonlinear constraints in the predicted active set (only available if $\mathbf{st}\mathbf{\to}\mathbf{ncnlin}>0$).
 penalty – double

The Euclidean norm of the vector of penalty arguments used in the augmented Lagrangian merit function (only available if $\mathbf{st}\mathbf{\to}\mathbf{ncnlin}>0$).
 norm_gf – double

The Euclidean norm of ${g}_{\mathrm{FR}}$, the gradient of the objective function with respect to the free variables.
 cond_h – double

A lower bound on the condition number of the Hessian approximation $H$.
 cond_hz – double

A lower bound on the condition number of the projected Hessian approximation ${H}_{Z}$.
 cond_t – double

A lower bound on the condition number of the matrix of predicted active constraints.
 iter_conv – Nag_Boolean

Nag_TRUE if the sequence of iterates has converged, i.e., convergence condition
(16) (see the description of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}$) is satisfied.
 norm_gz_small – Nag_Boolean

Nag_TRUE if the projected gradient is sufficiently small, i.e., convergence condition
(17) (see the description of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}$) is satisfied.
 violtn_small – Nag_Boolean

Nag_TRUE if the violations of the nonlinear constraints are sufficiently small, i.e., convergence condition
(18) (see the description of
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{optim\_tol}}$) is satisfied.
 update_modified – Nag_Boolean

Nag_TRUE if the quasiNewton update was modified to ensure that the Hessian is positive definite.
 qp_not_feasible – Nag_Boolean

Nag_TRUE if the QP subproblem has no feasible point.
 c_diff – Nag_Boolean

Nag_TRUE if central differences were used to compute the unspecified objective and constraint gradients.
 step_limit_exceeded – Nag_Boolean

Nag_TRUE if the line search produced a relative change in $x$ greater than the value defined by the optional argument ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{step\_limit}}$.
 refactor – Nag_Boolean

Nag_TRUE if the approximate Hessian has been refactorized.
 x – double *

Contains the components ${\mathbf{x}}\left[\mathit{j}1\right]$ of the current point $x$, for $\mathit{j}=1,2,\dots ,\mathbf{st}\mathbf{\to}\mathbf{n}$.
 state – Integer *

Contains the status of the
$\mathbf{st}\mathbf{\to}\mathbf{n}$ variables,
$\mathbf{st}\mathbf{\to}\mathbf{nclin}$ linear, and
$\mathbf{st}\mathbf{\to}\mathbf{ncnlin}$ nonlinear constraints (if any). See
Section 11.2 for a description of the possible status values.
 ax – double *

If $\mathbf{st}\mathbf{\to}\mathbf{nclin}>0$, $\mathbf{st}\mathbf{\to}\mathbf{ax}\left[\mathit{j}1\right]$ contains the current value of the $\mathit{j}$th linear constraint, for $\mathit{j}=1,2,\dots ,\mathbf{st}\mathbf{\to}\mathbf{nclin}$.
 cx – double *

If $\mathbf{st}\mathbf{\to}\mathbf{ncnlin}>0$, $\mathbf{st}\mathbf{\to}\mathbf{cx}\left[j1\right]$ contains the current value of nonlinear constraint ${c}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{st}\mathbf{\to}\mathbf{ncnlin}$.
 diagt – double *

If $\mathbf{st}\mathbf{\to}\mathbf{nactiv}>0$, the $\mathbf{st}\mathbf{\to}\mathbf{nactiv}$ elements of the diagonal of the matrix $T$.
 diagr – double *

Contains the $\mathbf{st}\mathbf{\to}\mathbf{n}$ elements of the diagonal of the upper triangular matrix $R$.
If $\mathbf{comm}\mathbf{\to}\mathbf{sol\_sqp\_prt}=\mathrm{Nag\_TRUE}$ then the final result from nag_opt_nlp (e04ucc) is provided through st. Note that ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_fun}}$ will be called with $\mathbf{comm}\mathbf{\to}\mathbf{sol\_sqp\_prt}=\mathrm{Nag\_TRUE}$ only if ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_level}}=\mathrm{Nag\_Soln}$, $\mathrm{Nag\_Soln\_Iter}$, $\mathrm{Nag\_Soln\_Iter\_Long}$, $\mathrm{Nag\_Soln\_Iter\_Const}$ or $\mathrm{Nag\_Soln\_Iter\_Full}$. The following members of st are set:
 iter – Integer

The number of iterations performed.
 n – Integer

The number of variables.
 nclin – Integer

The number of linear constraints.
 ncnlin – Integer

The number of nonlinear constraints.
 x – double *

Contains the components ${\mathbf{x}}\left[\mathit{j}1\right]$ of the final point $x$, for $\mathit{j}=1,2,\dots ,\mathbf{st}\mathbf{\to}\mathbf{n}$.
 state – Integer *

Contains the status of the
$\mathbf{st}\mathbf{\to}\mathbf{n}$ variables,
$\mathbf{st}\mathbf{\to}\mathbf{nclin}$ linear, and
$\mathbf{st}\mathbf{\to}\mathbf{ncnlin}$ nonlinear constraints (if any). See
Section 11.2 for a description of the possible status values.
 ax – double *

If $\mathbf{st}\mathbf{\to}\mathbf{nclin}>0$, $\mathbf{st}\mathbf{\to}\mathbf{ax}\left[\mathit{j}1\right]$ contains the final value of the $\mathit{j}$th linear constraint, for $\mathit{j}=1,2,\dots ,\mathbf{st}\mathbf{\to}\mathbf{nclin}$.
 cx – double *

If $\mathbf{st}\mathbf{\to}\mathbf{ncnlin}>0$, $\mathbf{st}\mathbf{\to}\mathbf{cx}\left[j1\right]$ contains the final value of nonlinear constraint ${c}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,\mathbf{st}\mathbf{\to}\mathbf{ncnlin}$.
 bl – double *

Contains the $\mathbf{st}\mathbf{\to}\mathbf{n}+\mathbf{st}\mathbf{\to}\mathbf{nclin}+\mathbf{st}\mathbf{\to}\mathbf{ncnlin}$ lower bounds on the variables.
 bu – double *

Contains the $\mathbf{st}\mathbf{\to}\mathbf{n}+\mathbf{st}\mathbf{\to}\mathbf{nclin}+\mathbf{st}\mathbf{\to}\mathbf{ncnlin}$ upper bounds on the variables.
 lambda – double *

Contains the $\mathbf{st}\mathbf{\to}\mathbf{n}+\mathbf{st}\mathbf{\to}\mathbf{nclin}+\mathbf{st}\mathbf{\to}\mathbf{ncnlin}$ final values of the Lagrange multipliers.
If $\mathbf{comm}\mathbf{\to}\mathbf{g\_prt}=\mathrm{Nag\_TRUE}$ then the results from derivative checking are provided through st. Note that ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_fun}}$ will be called with $\mathbf{comm}\mathbf{\to}\mathbf{g\_prt}$ only if ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_deriv}}=\mathrm{Nag\_D\_Sum}$ or $\mathrm{Nag\_D\_Full}$. The following members of st are set:
 n – Integer

The number of variables.
 ncnlin – Integer

The number of nonlinear constraints.
 x – double *

Contains the components ${\mathbf{x}}\left[\mathit{j}1\right]$ of the initial point ${x}_{0}$, for $\mathit{j}=1,2,\dots ,\mathbf{st}\mathbf{\to}\mathbf{n}$.
 g – double *

Contains the components ${\mathbf{g}}\left[\mathit{j}1\right]$ of the gradient vector $g\left(x\right)={\left(\frac{\partial F}{\partial {x}_{1}},\frac{\partial F}{\partial {x}_{2}},\dots ,\frac{\partial F}{\partial {x}_{n}}\right)}^{\mathrm{T}}$ at the initial point ${x}_{0}$, for $\mathit{j}=1,2,\dots ,\mathbf{st}\mathbf{\to}\mathbf{n}$.
 conjac – double *

Contains the elements of the Jacobian matrix of nonlinear constraints at the initial point ${x}_{0}$ ($\frac{\partial {f}_{\mathit{i}}}{\partial {x}_{\mathit{j}}}$ is held at location ${\mathbf{conjac}}\left[\left(\mathit{i}1\right)\times \mathbf{st}\mathbf{\to}\mathbf{n}+\mathit{j}1\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{st}\mathbf{\to}\mathbf{ncnlin}$ and $\mathit{j}=1,2,\dots ,\mathbf{st}\mathbf{\to}\mathbf{n}$).
In this case details of the derivative check performed by nag_opt_nlp (e04ucc) are held in the following substructure of st:
 gprint – Nag_GPrintSt *

Which in turn contains three substructures $\mathbf{st}\mathbf{\to}\mathbf{g\_chk}$, $\mathbf{st}\mathbf{\to}\mathbf{f\_sim}$, $\mathbf{st}\mathbf{\to}\mathbf{c\_sim}$ and two pointers to arrays of substructures, $\mathbf{st}\mathbf{\to}\mathbf{f\_comp}$ and $\mathbf{st}\mathbf{\to}\mathbf{c\_comp}$.
 g_chk – Nag_Grad_Chk_St *

The substructure $\mathbf{st}\mathbf{\to}\mathbf{g\_chk}$ contains the members:
 type – Nag_GradChk

The type of derivative check performed by nag_opt_nlp (e04ucc). This will be the same value as in ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{verify\_grad}}$.
 g_error – Integer

This member will be equal to one of the error codes NE_NOERROR or
NE_DERIV_ERRORS according to whether the derivatives were found to be correct or not.
 obj_start – Integer

Specifies the gradient element at which any component check started. This value will be equal to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_start}}$.
 obj_stop – Integer

Specifies the gradient element at which any component check ended. This value specifies the element at which any component check of the constraint gradient ended. This value will be equal to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{obj\_check\_stop}}$.
 con_start – Integer

Specifies the element at which any component check of the constraint gradient started. This value will be equal to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_start}}$.
 con_stop – Integer

Specifies the element at which any component check of the constraint gradient ended. This value will be equal to ${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{con\_check\_stop}}$.
 f_sim – Nag_SimSt *

The result of a simple derivative check of the objective gradient, $\mathbf{st}\mathbf{\to}\mathbf{gprint}\mathbf{\to}\mathbf{g\_chk}\mathbf{.}\mathbf{type}=\mathrm{Nag\_SimpleCheck}$, will be held in this substructure in members:
 correct – Nag_Boolean

If Nag_TRUE then the objective gradient is consistent with the finite difference approximation according to a simple check.
 dir_deriv – double

The directional derivative ${g}^{\mathrm{T}}p$ where $p$ is a random vector of unit length with elements of approximately equal magnitude.
 fd_approx – double

The finite difference approximation, $\left(F\left(x+hp\right)F\left(x\right)\right)/h$, to the directional derivative.
 c_sim – Nag_SimSt *

The result of a simple derivative check of the constraint Jacobian, $\mathbf{st}\mathbf{\to}\mathbf{gprint}\mathbf{\to}\mathbf{g\_chk}\mathbf{.}\mathbf{type}=\mathrm{Nag\_SimpleCheck}$, will be held in this substructure in members:
 n_elements – Integer

The number of columns of the constraint Jacobian for which a simple check has been carried out, i.e., those columns which do not contain unknown elements.
 correct – Nag_Boolean

If Nag_TRUE then the Jacobian is consistent with the finite difference approximation according to a simple check.
 max_error – double

The maximum error found between the norm of a constraint gradient and its finite difference approximation.
 max_constraint – Integer

The constraint gradient which has the maximum error between its norm and its finite difference approximation.
 f_comp – Nag_CompSt *

The results of a requested component derivative check of the objective gradient,
$\mathbf{st}\mathbf{\to}\mathbf{gprint}\mathbf{\to}\mathbf{g\_chk}\mathbf{.}\mathbf{type}=\mathrm{Nag\_CheckObj}\text{ or}\mathrm{Nag\_CheckObjCon}$, will be held in the array of
$\mathbf{st}\mathbf{\to}\mathbf{n}$ substructures of type Nag_CompSt pointed to by
$\mathbf{st}\mathbf{\to}\mathbf{gprint}\mathbf{\to}\mathbf{f\_comp}$. The procedure for the derivative check is based on finding an interval that produces an acceptable estimate of the second derivative, and then using that estimate to compute an interval that should produce a reasonable forwarddifference approximation. The gradient element is then compared with the difference approximation. (The method of finite difference interval estimation is based on
Gill et al. (1983).)
 correct – Nag_Boolean

If Nag_TRUE then this gradient element is consistent with its finite difference approximation.
 hopt – double

The optimal finite difference interval.
 gdiff – double

The finite difference approximation for this gradient component.
 iter – Integer

The number of trials performed to find a suitable difference interval.

A character string which describes the possible nature of the reason for which an estimation of the finite difference interval failed to produce a satisfactory relative condition error of the secondorder difference. Possible strings are: "Constant?", "Linear or odd?", "Too nonlinear?" and "Small derivative?".
 c_comp – Nag_CompSt *

The results of a requested component derivative check of the Jacobian of nonlinear constraint functions,
$\mathbf{st}\mathbf{\to}\mathbf{gprint}\mathbf{\to}\mathbf{g\_chk}\mathbf{.}\mathbf{type}=\mathrm{Nag\_CheckCon}\text{ or}\mathrm{Nag\_CheckObjCon}$, will be held in the array of
$\mathbf{st}\mathbf{\to}\mathbf{ncnlin}\times \mathbf{st}\mathbf{\to}\mathbf{n}$ substructures of type Nag_CompSt pointed to by
$\mathbf{st}\mathbf{\to}\mathbf{gprint}\mathbf{\to}\mathbf{c\_comp}$. The element
$\mathbf{st}\mathbf{\to}\mathbf{gprint}\mathbf{\to}\mathbf{f\_comp}\left[\left(\mathit{i}1\right)\times \mathbf{st}\mathbf{\to}\mathbf{n}+\mathit{j}1\right]$ will hold the details of the component derivative check for Jacobian element
$\mathit{i},\mathit{j}$, for
$\mathit{i}=1,2,\dots ,\mathbf{st}\mathbf{\to}\mathbf{ncnlin}$ and
$\mathit{j}=1,2,\dots ,\mathbf{st}\mathbf{\to}\mathbf{n}$. The procedure for the derivative check is based on finding an interval that produces an acceptable estimate of the second derivative, and then using that estimate to compute an interval that should produce a reasonable forwarddifference approximation. The Jacobian element is then compared with the difference approximation. (The method of finite difference interval estimation is based on
Gill et al. (1983).)
The members of $\mathbf{st}\mathbf{\to}\mathbf{gprint}\mathbf{\to}\mathbf{c\_comp}$ are as for $\mathbf{st}\mathbf{\to}\mathbf{gprint}\mathbf{\to}\mathbf{f\_comp}$ where $\mathbf{st}\mathbf{\to}\mathbf{gprint}\mathbf{\to}\mathbf{f\_comp}\mathbf{.}\mathbf{gdiff}$ gives the difference approximation for the Jacobian element.
The relevant members of the structure
comm are:
 g_prt – Nag_Boolean

Will be Nag_TRUE only when the print function is called with the result of the derivative check of
objfun and
confun.
 it_maj_prt – Nag_Boolean

Will be Nag_TRUE when the print function is called with information about the current major iteration.
 sol_sqp_prt – Nag_Boolean

Will be Nag_TRUE when the print function is called with the details of the final solution.
 it_prt – Nag_Boolean

Will be Nag_TRUE when the print function is called with information about the current minor iteration (i.e., an iteration of the current QP subproblem). See the documentation for
nag_opt_lin_lsq (e04ncc) for details of which members of
st are set.
 new_lm – Nag_Boolean

Will be Nag_TRUE when the Lagrange multipliers have been updated in a QP subproblem. See the documentation for
nag_opt_lin_lsq (e04ncc) for details of which members of
st are set.
 sol_prt – Nag_Boolean

Will be Nag_TRUE when the print function is called with the details of the solution of a QP subproblem, i.e., the solution at the end of a major iteration. See the documentation for
nag_opt_lin_lsq (e04ncc) for details of which members of
st are set.
 user – double
 iuser – Integer
 p – Pointer

Pointers for communication of user information. If used they must be allocated memory either before entry to nag_opt_nlp (e04ucc) or during a call to
objfun,
confun or
${\mathbf{options}}\mathbf{.}\textcolor[rgb]{}{\mathbf{print\_fun}}$. The type Pointer will be
void * with a C compiler that defines
void * and
char * otherwise.