NAG Library Routine Document

e04vhf (nlp2_sparse_solve)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{start}$ – IntegerInput

On entry: indicates how a starting point is to be obtained.

$\mathbf{start}=0$

Requests that the Crash procedure be used, unless a Basis file is provided via optional parameters Old Basis File, Insert File or Load File.

$\mathbf{start}=1$

Is the same as $\mathbf{start}=0$ but is more meaningful when a Basis file is given.

$\mathbf{start}=2$

Means that xstate and fstate define a valid starting point (probably from an earlier call, though not necessarily).

Constraint: $\mathbf{start}=0$, $1$ or $2$.

2:     $\mathbf{nf}$ – IntegerInput

On entry: $\mathit{nf}$, the number of problem functions in $F\left(x\right)$, including the objective function (if any) and the linear and nonlinear constraints. Upper and lower bounds on $x$ can be defined using the arguments xlow and xupp and should not be included in $F$.

Constraint: $\mathbf{nf}>0$.

3:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of variables.

Constraint: $\mathbf{n}>0$.

4:     $\mathbf{nxname}$ – IntegerInput

On entry: the number of names provided in the array xnames.

$\mathbf{nxname}=1$

There are no names provided and generic names will be used in the output.

$\mathbf{nxname}=\mathbf{n}$

Names for all variables must be provided and will be used in the output.

Constraint: $\mathbf{nxname}=1\text{ or }\mathbf{n}$.

5:     $\mathbf{nfname}$ – IntegerInput

On entry: the number of names provided in the array fnames.

$\mathbf{nfname}=1$

There are no names provided and generic names will be used in the output.

$\mathbf{nfname}=\mathbf{nf}$

Names for all functions must be provided and will be used in the output.

Constraint: $\mathbf{nfname}=1\text{ or }\mathbf{nf}$.

Note: if $\mathbf{nxname}=1$ then nfname must also be $1$ (and vice versa). Similarly, if $\mathbf{nxname}=\mathbf{n}$ then nfname must be nf (and vice versa).

6:     $\mathbf{objadd}$ – Real (Kind=nag_wp)Input

On entry: is a constant that will be added to the objective row $F_{\mathrm{obj}}$ for printing purposes. Typically, $\mathbf{objadd}=\text{0.0E+0}$.

7:     $\mathbf{objrow}$ – IntegerInput

On entry: says which row of $F\left(x\right)$ is to act as the objective function. If there is no such row, set $\mathbf{objrow}=0$. Then e04vhf will seek a feasible point such that $l_F\le F\left(x\right)\le u_F$ and $l_x\le x\le u_x$.

Constraint: $1\le \mathbf{objrow}\le \mathbf{nf}\text{ or }\mathbf{objrow}=0$ (or a feasible point problem).

8:     $\mathbf{prob}$ – Character(8)Input

On entry: is an $8$-character name for the problem. prob is used in the printed solution and in some routines that output Basis files. A blank name may be used.

9:     $\mathbf{usrfun}$ – Subroutine, supplied by the user.External Procedure

usrfun must define the nonlinear portion $f\left(x\right)$ of the problem functions $F\left(x\right)=f\left(x\right)+Ax$, along with its gradient elements $G_{ij}\left(x\right)=\frac{\partial f_i\left(x\right)}{\partial x_j}$. (A dummy subroutine is needed even if $f\equiv 0$ and all functions are linear.)

In general, usrfun should return all function and gradient values on every entry except perhaps the last. This provides maximum reliability and corresponds to the default option setting, $\mathbf{Derivative\; Option}=1$.

The elements of $G\left(x\right)$ are stored in the array $\mathbf{g}\left(1:\mathbf{leng}\right)$ in the order specified by the input arrays igfun and jgvar.

In practice it is often convenient not to code gradients. e04vhf is able to estimate them by finite differences, using a call to usrfun for each variable $x_j$ for which some $\frac{\partial f_i\left(x\right)}{\partial x_j}$ needs to be estimated. However, this reduces the reliability of the optimization algorithm, and it can be very expensive if there are many such variables $x_j$.

As a compromise, e04vhf allows you to code as many gradients as you like. This option is implemented as follows. Just before usrfun is called, each element of the derivative array g is initialized to a specific value. On exit, any element retaining that value must be estimated by finite differences.

Some rules of thumb follow:

(i)	for maximum reliability, compute all gradients;
(ii)	if the gradients are expensive to compute, specify optional parameter Nonderivative Linesearch and use the value of the input argument needg to avoid computing them on certain entries. (There is no need to compute gradients if $\mathbf{needg}=0$ on entry to usrfun.);
(iii)	if not all gradients are known, you must specify $\mathbf{Derivative\; Option}=0$. You should still compute as many gradients as you can. (It often happens that some of them are constant or zero.);
(iv)	again, if the known gradients are expensive, don't compute them if $\mathbf{needg}=0$ on entry to usrfun;
(v)	use the input argument status to test for special actions on the first or last entries;
(vi)	while usrfun is being developed, use the optional parameter Verify Level to check the computation of gradients that are supposedly known;
(vii)	usrfun is not called until the linear constraints and bounds on $x$ are satisfied. This helps confine $x$ to regions where the functions $f_i\left(x\right)$ are likely to be defined. However, be aware of the optional parameter Minor Feasibility Tolerance if the functions have singularities on the constraint boundaries;
(viii)	set $\mathbf{status}=-1$ if some of the functions are undefined. The linesearch will shorten the step and try again;
(ix)	set $\mathbf{status}\le -2$ if you want e04vhf to stop.

The specification of usrfun is:

Fortran Interface

Subroutine usrfun ( status, n, x, needf, nf, f, needg, leng, g, cuser, iuser, ruser) Integer, Intent (In) :: n, needf, nf, needg, leng Integer, Intent (Inout) :: status, iuser(*) Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Inout) :: f(nf), g(leng), ruser(*) Character (8), Intent (Inout) :: cuser(*)

C Header Interface

#include <nagmk26.h> void  usrfun (Integer *status, const Integer *n, const double x[], const Integer *needf, const Integer *nf, double f[], const Integer *needg, const Integer *leng, double g[], char cuser[], Integer iuser[], double ruser[], const Charlen length_cuser)

1:     $\mathbf{status}$ – IntegerInput/Output

On entry: indicates the first and last calls to usrfun.

$\mathbf{status}=0$

There is nothing special about the current call to usrfun.

$\mathbf{status}=1$

e04vhf is calling your subroutine for the first time. You may wish to do something special such as read data from a file.

$\mathbf{status}\ge 2$

e04vhf is calling your subroutine for the last time. This argument setting allows you to perform some additional computation on the final solution.

$\mathbf{status}=2$

The current x is optimal.

$\mathbf{status}=3$

The problem appears to be infeasible.

$\mathbf{status}=4$

The problem appears to be unbounded.

$\mathbf{status}=5$

An iterations limit was reached.

If the functions are expensive to evaluate, it may be desirable to do nothing on the last call. The first executable statement could be

If (status .ge. 2) Return.

On exit: may be used to indicate that you are unable to evaluate $f$ or its gradients at the current $x$. (For example, the problem functions may not be defined there.)

During the linesearch, $f\left(x\right)$ is evaluated at points $x=x_k+\alpha p_k$ for various step lengths $\alpha$, where $f\left(x_k\right)$ has already been evaluated satisfactorily. For any such $x$, if you set $\mathbf{status}=-1$, e04vhf will reduce $\alpha$ and evaluate $f$ again (closer to $x_k$, where $f\left(x_k\right)$ is more likely to be defined).

If for some reason you wish to terminate the current problem, set $\mathbf{status}\le -2$.

2:     $\mathbf{n}$ – IntegerInput

On entry: $n$, the number of variables, as defined in the call to e04vhf.

3:     $\mathbf{x}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the variables $x$ at which the problem functions are to be calculated. The array $x$ must not be altered.

4:     $\mathbf{needf}$ – IntegerInput

On entry: indicates whether f must be assigned during this call of usrfun.

$\mathbf{needf}=0$

f is not required and is ignored.

$\mathbf{needf}>0$

The components of $f\left(x\right)$ corresponding to the nonlinear part of $F\left(x\right)$ must be calculated and assigned to f.

If $F_i\left(x\right)$ is linear and completely defined by the $i$th row of $A$, $A_i^{\prime }$, the associated value $f_i\left(x\right)$ is ignored and need not be assigned. However, if $F_i\left(x\right)$ has a nonlinear portion $f_i\left(x\right)$ that happens to be zero at $x$, it is still necessary to set $f_i\left(x\right)=0$. If the linear part $A_i^{\prime }$ of a nonlinear $F_i\left(x\right)$ is provided using the arrays iafun, javar and a, it must not be computed again as part of $f_i\left(x\right)$.

To simplify the code, you may ignore the value of needf and compute $f\left(x\right)$ on every entry to usrfun.

needf may also be ignored with Derivative Linesearch and $\mathbf{Derivative\; Option}=1$. In this case, needf is always $1$, and f must always be assigned.

5:     $\mathbf{nf}$ – IntegerInput

On entry: is the length of the full vector $F\left(x\right)=f\left(x\right)+Ax$ as defined in the call to e04vhf.

6:     $\mathbf{f}\left(\mathbf{nf}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: concerns the calculation of $f\left(x\right)$.

On exit: f contains the computed functions $f\left(x\right)$ (except perhaps if $\mathbf{needf}=0$).

7:     $\mathbf{needg}$ – IntegerInput

On entry: indicates whether g must be assigned during this call of usrfun.

$\mathbf{needg}=0$

g is not required and is ignored.

$\mathbf{needg}>0$

The partial derivatives of $f\left(x\right)$ must be calculated and assigned to g. The value of $\mathbf{g}\left(k\right)$ should be $\frac{\partial f_i\left(x\right)}{\partial x_j}$, where $i=\mathbf{igfun}\left(k\right)$, $j=\mathbf{jgvar}\left(k\right)$ and $k=1,2,\dots ,\mathbf{leng}$.

8:     $\mathbf{leng}$ – IntegerInput

On entry: is the length of the coordinate arrays jgvar and igfun in the call to e04vhf.

9:     $\mathbf{g}\left(\mathbf{leng}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: concerns the calculations of the derivatives of the function $f\left(x\right)$.

On exit: contains the computed derivatives $G\left(x\right)$ (unless $\mathbf{needg}=0$).

These derivative elements must be stored in g in exactly the same positions as implied by the definitions of arrays igfun and jgvar. There is no internal check for consistency (except indirectly via the optional parameter Verify Level), so great care is essential.

10:   $\mathbf{cuser}\left(*\right)$ – Character(8) arrayUser Workspace

usrfun is called with the argument cuser as supplied to e04vhf. You should use the array cuser to supply information to usrfun.

11:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

usrfun is called with the argument iuser as supplied to e04vhf. You should use the array iuser to supply information to usrfun.

12:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

usrfun is called with the argument ruser as supplied to e04vhf. You should use the array ruser to supply information to usrfun.

usrfun must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which e04vhf is called. Arguments denoted as Input must not be changed by this procedure.

Note: usrfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by e04vhf. If your code inadvertently does return any NaNs or infinities, e04vhf is likely to produce unexpected results.

10:   $\mathbf{iafun}\left(\mathbf{lena}\right)$ – Integer arrayInput

11:   $\mathbf{javar}\left(\mathbf{lena}\right)$ – Integer arrayInput

12:   $\mathbf{a}\left(\mathbf{lena}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: define the coordinates $\left(i,j\right)$ and values $A_{ij}$ of the nonzero elements of the linear part $A$ of the function $F\left(x\right)=f\left(x\right)+Ax$.

In particular, nea triples $\left(\mathbf{iafun}\left(k\right),\mathbf{javar}\left(k\right),\mathbf{a}\left(k\right)\right)$ define the row and column indices $i=\mathbf{iafun}\left(k\right)$ and $j=\mathbf{javar}\left(k\right)$ of the element $A_{ij}=\mathbf{a}\left(k\right)$.

The coordinates may define the elements of $A$ in any order.

13:   $\mathbf{lena}$ – IntegerInput

On entry: the dimension of the arrays iafun, javar and a that hold $\left(i,j,A_{ij}\right)$ as declared in the (sub)program from which e04vhf is called.

Constraint: $\mathbf{lena}\ge 1$.

14:   $\mathbf{nea}$ – IntegerInput

On entry: is the number of nonzero entries in $A$ such that $F\left(x\right)=f\left(x\right)+Ax$.

Constraint: $0\le \mathbf{nea}\le \mathbf{lena}$.

15:   $\mathbf{igfun}\left(\mathbf{leng}\right)$ – Integer arrayInput

16:   $\mathbf{jgvar}\left(\mathbf{leng}\right)$ – Integer arrayInput

On entry: define the coordinates $\left(i,j\right)$ of the nonzero elements of $G$, the nonlinear part of the derivative $J\left(x\right)=G\left(x\right)+A$ of the function $F\left(x\right)=f\left(x\right)+Ax$. e04vjf may be used to define these two arrays.

The coordinates can define the elements of $G$ in any order. However, usrfun must define the actual elements of g in exactly the same order as defined by the coordinates $\left(\mathbf{igfun},\mathbf{jgvar}\right)$.

17:   $\mathbf{leng}$ – IntegerInput

On entry: the dimension of the arrays igfun and jgvar that define the varying Jacobian elements $\left(i,j,G_{ij}\right)$ as declared in the (sub)program from which e04vhf is called.

Constraint: $\mathbf{leng}\ge 1$.

18:   $\mathbf{neg}$ – IntegerInput

On entry: the number of nonzero entries in $G$.

Constraint: $0\le \mathbf{neg}\le \mathbf{leng}$.

19:   $\mathbf{xlow}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput

20:   $\mathbf{xupp}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: contain the lower and upper bounds $l_x$ and $u_x$ on the variables $x$.

To specify a nonexistent lower bound ${\left[l_x\right]}_j=-\infty$, set $\mathbf{xlow}\left(j\right)\le -\mathit{bigbnd}$, where $\mathit{bigbnd}$ is the optional parameter Infinite Bound Size. To specify a nonexistent upper bound ${\left[u_x\right]}_j=\infty$, set $\mathbf{xupp}\left(j\right)\ge \mathit{bigbnd}$.

To fix the $j$th variable at $x_j=\beta$, where $\left|\beta \right|<\mathit{bigbnd}$, set $\mathbf{xlow}\left(j\right)=\mathbf{xupp}\left(j\right)=\beta$.

Constraint: $\mathbf{xlow}\left(\mathit{i}\right)\le \mathbf{xupp}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

21:   $\mathbf{xnames}\left(\mathbf{nxname}\right)$ – Character(8) arrayInput

On entry: the optional names for the variables.

If $\mathbf{nxname}=1$, xnames is not referenced and default names will be used for output.

If $\mathbf{nxname}=\mathbf{n}$, $\mathbf{xnames}\left(j\right)$ should contain the $8$-character name of the $j$th variable.

22:   $\mathbf{flow}\left(\mathbf{nf}\right)$ – Real (Kind=nag_wp) arrayInput

23:   $\mathbf{fupp}\left(\mathbf{nf}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: contain the lower and upper bounds $l_F$ and $u_F$ on $F\left(x\right)$.

To specify a nonexistent lower bound ${\left[l_F\right]}_i=-\infty$, set $\mathbf{flow}\left(i\right)\le -\mathit{bigbnd}$. For a nonexistent upper bound ${\left[u_F\right]}_i=\infty$, set $\mathbf{fupp}\left(i\right)\ge \mathit{bigbnd}$.

To make the $i$th constraint an equality at $F_i=\beta$, where $\left|\beta \right|<\mathit{bigbnd}$, set $\mathbf{flow}\left(i\right)=\mathbf{fupp}\left(i\right)=\beta$.

Constraint: $\mathbf{flow}\left(\mathit{i}\right)\le \mathbf{fupp}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{n}$.

24:   $\mathbf{fnames}\left(\mathbf{nfname}\right)$ – Character(8) arrayInput

On entry: the optional names for the problem functions.

If $\mathbf{nfname}=1$, fnames is not referenced and default names will be used for output.

If $\mathbf{nfname}=\mathbf{nf}$, $\mathbf{fnames}\left(i\right)$ should contain the $8$-character name of the $i$th row of $F$.

25:   $\mathbf{x}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: an initial estimate of the variables $x$. See the following description of xstate.

On exit: the final values of the variable $x$.

26:   $\mathbf{xstate}\left(\mathbf{n}\right)$ – Integer arrayInput/Output

On entry: the initial state for each variable $x$.

If $\mathbf{start}=0$ or $1$ and no basis information is provided (the optional parameters Old Basis File, Insert File and Load File are all set to $0$; the default) x and xstate must be defined.

If nothing special is known about the problem, or if there is no wish to provide special information, you may set $\mathbf{x}\left(j\right)=0.0$, $\mathbf{xstate}\left(j\right)=0$, for all $j=1,2,\dots ,\mathbf{n}$. If you set $\mathbf{x}\left(j\right)=\mathbf{xlow}\left(j\right)$ set $\mathbf{xstate}\left(j\right)=4$; if you set $\mathbf{x}\left(j\right)=\mathbf{xupp}\left(j\right)$ then set $\mathbf{xstate}\left(j\right)=5$. In this case a Crash procedure is used to select an initial basis.

If $\mathbf{start}=0$ or $1$ and basis information is provided (at least one of the optional parameters Old Basis File, Insert File and Load File is nonzero) x and xstate need not be set.

If $\mathbf{start}=2$ (Warm Start), x and xstate must be set (probably from a previous call). In this case $\mathbf{xstate}\left(\mathit{j}\right)$ must be $0$, $1$, $2\text{ or }3$, for $\mathit{j}=1,2,\dots ,\mathbf{n}$.

On exit: the final state of the variables.

$\mathbf{xstate}\left(j\right)$	State of variable $j$	Usual value of $\mathbf{x}\left(j\right)$
0	nonbasic	$\mathbf{xlow}\left(j\right)$
1	nonbasic	$\mathbf{xupp}\left(j\right)$
2	superbasic	Between $\mathbf{xlow}\left(j\right)$ and $\mathbf{xupp}\left(j\right)$
3	basic	Between $\mathbf{xlow}\left(j\right)$ and $\mathbf{xupp}\left(j\right)$

Basic and superbasic variables may be outside their bounds by as much as the optional parameter Minor Feasibility Tolerance. Note that if scaling is specified, the feasibility tolerance applies to the variables of the scaled problem. In this case, the variables of the original problem may be as much as $0.1$ outside their bounds, but this is unlikely unless the problem is very badly scaled. Check the value of Primal infeasibility output to the unit number associated with the optional parameter Print File.

Very occasionally some nonbasic variables may be outside their bounds by as much as the optional parameter Minor Feasibility Tolerance, and there may be some nonbasics for which $\mathbf{x}\left(j\right)$ lies strictly between its bounds.

If $\mathbf{ninf}>0$, some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by sinf if scaling was not used).

Constraint: $0\le \mathbf{xstate}\left(j\right)\le 5\text{, for }j=1,2,\dots ,\mathbf{n}$.

27:   $\mathbf{xmul}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the vector of the dual variables (Lagrange multipliers) for the simple bounds $l_x\le x\le u_x$.

28:   $\mathbf{f}\left(\mathbf{nf}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: an initial value for the problem functions $F$. See the following description of fstate.

On exit: the final values for the problem functions $F$ (the values $F$ at the final point x).

29:   $\mathbf{fstate}\left(\mathbf{nf}\right)$ – Integer arrayInput/Output

On entry: the initial state for the problem functions $F$.

If $\mathbf{start}=0$ or $1$ and no basis information is provided (the optional parameters Old Basis File, Insert File and Load File are all set to $0$; the default, f and fstate must be defined.

If nothing special is known about the problem, or if there is no wish to provide special information, you may set $\mathbf{f}\left(i\right)=0.0$, $\mathbf{fstate}\left(i\right)=0$, for all $i=1,2,\dots ,\mathbf{nf}$. Less trivially, to say that the optimal value of function $\mathbf{f}\left(i\right)$ will probably be equal to one of its bounds, set $\mathbf{f}\left(i\right)=\mathbf{flow}\left(i\right)$ and $\mathbf{fstate}\left(i\right)=4$ or $\mathbf{f}\left(i\right)=\mathbf{fupp}\left(i\right)$ and $\mathbf{fstate}\left(i\right)=5$ as appropriate. In this case a Crash procedure is used to select an initial basis.

If $\mathbf{start}=0$ or $1$ and basis information is provided (at least one of the optional parameters Old Basis File, Insert File and Load File is nonzero), f and fstate need not be set.

If $\mathbf{start}=2$ (Warm Start), f and fstate must be set (probably from a previous call). In this case $\mathbf{fstate}\left(\mathit{i}\right)$ must be $0$, $1$, $2$ or $3$, for $\mathit{i}=1,2,\dots ,\mathbf{nf}$.

On exit: the final state of the variables. The elements of fstate have the following meaning:

$\mathbf{fstate}\left(i\right)$	State of the corresponding slack variable	Usual value of $\mathbf{f}\left(i\right)$
0	nonbasic	$\mathbf{flow}\left(i\right)$
1	nonbasic	$\mathbf{fupp}\left(i\right)$
2	superbasic	Between $\mathbf{flow}\left(i\right)$ and $\mathbf{fupp}\left(i\right)$
3	basic	Between $\mathbf{flow}\left(i\right)$ and $\mathbf{fupp}\left(i\right)$

Basic and superbasic slack variables may lead to the corresponding functions being outside their bounds by as much as the optional parameter Minor Feasibility Tolerance.

Very occasionally some functions may be outside their bounds by as much as the optional parameter Minor Feasibility Tolerance, and there may be some nonbasics for which $\mathbf{f}\left(i\right)$ lies strictly between its bounds.

If $\mathbf{ninf}>0$, some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by sinf if scaling was not used).

Constraint: $0\le \mathbf{fstate}\left(i\right)\le 5\text{, for }i=1,2,\dots ,\mathbf{nf}$.

30:   $\mathbf{fmul}\left(\mathbf{nf}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: an estimate of $\gamma$, the vector of Lagrange multipliers (shadow prices) for the constraints $l_F\le F\left(x\right)\le u_F$. All nf components must be defined. If nothing is known about $\gamma$, set $\mathbf{fmul}\left(\mathit{i}\right)=0.0$, for $\mathit{i}=1,2,\dots ,\mathbf{nf}$. For warm start use the values from a previous call.

On exit: the vector of the dual variables (Lagrange multipliers) for the general constraints $l_F\le F\left(x\right)\le u_F$ 

31:   $\mathbf{ns}$ – IntegerInput/Output

On entry: the number of superbasic variables. ns need not be specified for cold starts, but should retain its value from a previous call when warm start is used.

On exit: the final number of superbasic variables.

32:   $\mathbf{ninf}$ – IntegerOutput

33:   $\mathbf{sinf}$ – Real (Kind=nag_wp)Output

On exit: are the number and the sum of the infeasibilities of constraints that lie outside one of their bounds by more than the optional parameter Minor Feasibility Tolerance before the solution is unscaled.

If any linear constraints are infeasible, $x$ minimizes the sum of the infeasibilities of the linear constraints subject to the upper and lower bounds being satisfied. In this case ninf gives the number of variables and linear constraints lying outside their upper or lower bounds. The nonlinear constraints are not evaluated.

Otherwise, $x$ minimizes the sum of infeasibilities of the nonlinear constraints subject to the linear constraints and upper and lower bounds being satisfied. In this case ninf gives the number of components of $F\left(x\right)$ lying outside their bounds by more than the optional parameter Minor Feasibility Tolerance. Again this is before the solution is unscaled.

34:   $\mathbf{cw}\left(\mathbf{lencw}\right)$ – Character(8) arrayCommunication Array

35:   $\mathbf{lencw}$ – IntegerInput

On entry: the dimension of the array cw as declared in the (sub)program from which e04vhf is called.

Constraint: $\mathbf{lencw}\ge 600$.

36:   $\mathbf{iw}\left(\mathbf{leniw}\right)$ – Integer arrayCommunication Array

37:   $\mathbf{leniw}$ – IntegerInput

On entry: the dimension of the array iw as declared in the (sub)program from which e04vhf is called.

Constraint: $\mathbf{leniw}\ge 600$.

38:   $\mathbf{rw}\left(\mathbf{lenrw}\right)$ – Real (Kind=nag_wp) arrayCommunication Array

39:   $\mathbf{lenrw}$ – IntegerInput

On entry: the dimension of the array rw as declared in the (sub)program from which e04vhf is called.

Constraint: $\mathbf{lenrw}\ge 600$.

40:   $\mathbf{cuser}\left(*\right)$ – Character(8) arrayUser Workspace

41:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

42:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

cuser, iuser and ruser are not used by e04vhf, but are passed directly to usrfun and may be used to pass information to this routine.

43:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if $\mathbf{ifail}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of ifail on exit.

On exit: $\mathbf{ifail}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

e04vhf returns with $\mathbf{ifail}=\mathbf{0}$ if the iterates have converged to a point $x$ that satisfies the first-order Kuhn–Tucker (see Section 13.2) conditions to the accuracy requested by the optional parameter Major Optimality Tolerance, i.e., the projected gradient and active constraint residuals are negligible at $x$.

You should check whether the following four conditions are satisfied:

(i)	the final value of rgNorm (see Section 13.2) is significantly less than that at the starting point;
(ii)	during the final major iterations, the values of Step and Minors (see Section 13.1) are both one;
(iii)	the last few values of both rgNorm and SumInf (see Section 13.2) become small at a fast linear rate; and
(iv)	condHz (see Section 13.1) is small.

If all these conditions hold, $x$ is almost certainly a local minimum of (1).

One caution about ‘Optimal solutions’. Some of the variables or slacks may lie outside their bounds more than desired, especially if scaling was requested. Max Primal infeas in the Print file (see Section 13) refers to the largest bound infeasibility and which variable is involved. If it is too large, consider restarting with a smaller Minor Feasibility Tolerance (say $10$ times smaller) and perhaps $\mathbf{Scale\; Option}=0$.

Similarly, Max Dual infeas in the Print file indicates which variable is most likely to be at a nonoptimal value. Broadly speaking, if

$$\mathtt{Max\; Dual\; infeas}/\mathtt{Max\; pi}={10}^{-d}\text{,}$$

then the objective function would probably change in the $d$th significant digit if optimization could be continued. If $d$ seems too large, consider restarting with a smaller Major Optimality Tolerance.

Finally, Nonlinear constraint violn in the Print file shows the maximum infeasibility for nonlinear rows. If it seems too large, consider restarting with a smaller Major Feasibility Tolerance.

6

Error Indicators and Warnings

$\mathbf{ifail}=1$

On entry, $\mathbf{lencw}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lencw}\ge 600$.

On entry, $\mathbf{leniw}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{leniw}\ge 600$.

On entry, $\mathbf{lenrw}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lenrw}\ge 600$.

The initialization routine e04vgf has not been called.

$\mathbf{ifail}=2$

Array element $\mathbf{igfun}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$ is out of range $1$ to $\mathbf{nf}=〈\mathit{\text{value}}〉$, or array element $\mathbf{jgvar}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$ is out of range $1$ to $\mathbf{n}=〈\mathit{\text{value}}〉$.

Basis file dimensions do not match this problem.

On entry, bounds flow and fupp for $〈\mathit{\text{value}}〉$ are equal and infinite. $\mathbf{flow}=\mathbf{fupp}=〈\mathit{\text{value}}〉$ and $\mathit{infbnd}=〈\mathit{\text{value}}〉$.

On entry, bounds flow and fupp for variable $〈\mathit{\text{value}}〉$ are equal and infinite. $\mathbf{flow}=\mathbf{fupp}=〈\mathit{\text{value}}〉$ and $\mathit{infbnd}=〈\mathit{\text{value}}〉$.

On entry, bounds for $〈\mathit{\text{value}}〉$ are inconsistent. $\mathbf{flow}=〈\mathit{\text{value}}〉$ and $\mathbf{fupp}=〈\mathit{\text{value}}〉$.

On entry, bounds for $〈\mathit{\text{value}}〉$ are inconsistent. $\mathbf{xlow}=〈\mathit{\text{value}}〉$ and $\mathbf{xupp}=〈\mathit{\text{value}}〉$.

On entry, bounds for variable $〈\mathit{\text{value}}〉$ are inconsistent. $\mathbf{flow}=〈\mathit{\text{value}}〉$ and $\mathbf{fupp}=〈\mathit{\text{value}}〉$.

On entry, bounds for variable $〈\mathit{\text{value}}〉$ are inconsistent. $\mathbf{xlow}=〈\mathit{\text{value}}〉$ and $\mathbf{xupp}=〈\mathit{\text{value}}〉$.

On entry, bounds xlow and xupp for $〈\mathit{\text{value}}〉$ are equal and infinite. $\mathbf{xlow}=\mathbf{xupp}=〈\mathit{\text{value}}〉$ and $\mathit{infbnd}=〈\mathit{\text{value}}〉$.

On entry, bounds xlow and xupp for variable $〈\mathit{\text{value}}〉$ are equal and infinite. $\mathbf{xlow}=\mathbf{xupp}=〈\mathit{\text{value}}〉$ and $\mathit{infbnd}=〈\mathit{\text{value}}〉$.

On entry, $\mathbf{lena}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lena}\ge 1$.

On entry, $\mathbf{leng}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{leng}\ge 1$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 1$.

On entry, $\mathbf{nea}=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$ and $\mathbf{nf}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nea}\le \mathbf{n}\times \mathbf{nf}$.

On entry, $\mathbf{nea}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nea}\ge 0$.

On entry, $\mathbf{neg}=〈\mathit{\text{value}}〉$, $\mathbf{n}=〈\mathit{\text{value}}〉$ and $\mathbf{nf}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{neg}\le \mathbf{n}\times \mathbf{nf}$.

On entry, $\mathbf{neg}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{neg}\ge 0$.

On entry, $\mathbf{nf}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nf}\ge 1$.

On entry, $\mathbf{nfname}=〈\mathit{\text{value}}〉$ and $\mathbf{nf}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nfname}=1$ or nf.

On entry, $\mathbf{nxname}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nxname}=1$ or n.

On entry, $\mathbf{objrow}=〈\mathit{\text{value}}〉$ and $\mathbf{nf}=〈\mathit{\text{value}}〉$.
Constraint: $0\le \mathbf{objrow}\le \mathbf{nf}$.

On entry, one but not both of nxname and nfname is equal to $1$. $\mathbf{nxname}=〈\mathit{\text{value}}〉$ and $\mathbf{nfname}=〈\mathit{\text{value}}〉$.

On entry, $\mathbf{start}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{start}=0$, $1$ or $2$.

$\mathbf{ifail}=3$

The requested accuracy could not be achieved.

A feasible solution has been found, but the requested accuracy in the dual infeasibilities could not be achieved. An abnormal termination has occurred, but e04vhf is within ${10}^{-2}$ of satisfying the Major Optimality Tolerance. Check that the Major Optimality Tolerance is not too small.

$\mathbf{ifail}=4$

The linear constraints appear to be infeasible.

The problem appears to be infeasible. Infeasibilites have been minimized.

The problem appears to be infeasible. Nonlinear infeasibilites have been minimized.

The problem appears to be infeasible. The linear equality constraints could not be satisfied.

When the constraints are linear, this message is based on a relatively reliable indicator of infeasibility. Feasibility is measured with respect to the upper and lower bounds on the variables and slacks. Among all the points satisfying the general constraints $Ax-s=0$ (see (6) and (7) in Section 11.2), there is apparently no point that satisfies the bounds on $x$ and $s$. Violations as small as the Minor Feasibility Tolerance are ignored, but at least one component of $x$ or $s$ violates a bound by more than the tolerance.

When nonlinear constraints are present, infeasibility is much harder to recognize correctly. Even if a feasible solution exists, the current linearization of the constraints may not contain a feasible point. In an attempt to deal with this situation, when solving each QP subproblem, e04vhf is prepared to relax the bounds on the slacks associated with nonlinear rows.

If a QP subproblem proves to be infeasible or unbounded (or if the Lagrange multiplier estimates for the nonlinear constraints become large), e04vhf enters so-called ‘nonlinear elastic’ mode. The subproblem includes the original QP objective and the sum of the infeasibilities – suitably weighted using the optional parameter Elastic Weight. In elastic mode, some of the bounds on the nonlinear rows are ‘elastic’ – i.e., they are allowed to violate their specific bounds. Variables subject to elastic bounds are known as elastic variables. An elastic variable is free to violate one or both of its original upper or lower bounds. If the original problem has a feasible solution and the elastic weight is sufficiently large, a feasible point eventually will be obtained for the perturbed constraints, and optimization can continue on the subproblem. If the nonlinear problem has no feasible solution, e04vhf will tend to determine a ‘good’ infeasible point if the elastic weight is sufficiently large. (If the elastic weight were infinite, e04vhf would locally minimize the nonlinear constraint violations subject to the linear constraints and bounds.)

Unfortunately, even though e04vhf locally minimizes the nonlinear constraint violations, there may still exist other regions in which the nonlinear constraints are satisfied. Wherever possible, nonlinear constraints should be defined in such a way that feasible points are known to exist when the constraints are linearized.

$\mathbf{ifail}=5$

The problem appears to be unbounded. The constraint violation limit has been reached.

The problem appears to be unbounded. The objective function is unbounded.

The problem appears to be unbounded (or badly scaled).

For linear problems, unboundedness is detected by the simplex method when a nonbasic variable can be increased or decreased by an arbitrary amount without causing a basic variable to violate a bound. Consider adding an upper or lower bound to the variable. Also, examine the constraints that have nonzeros in the associated column, to see if they have been formulated as intended.

Very rarely, the scaling of the problem could be so poor that numerical error will give an erroneous indication of unboundedness. Consider using the optional parameter Scale Option.

For nonlinear problems, e04vhf monitors both the size of the current objective function and the size of the change in the variables at each step. If either of these is very large (as judged by the ‘Unbounded’ optional parameters (see Section 12)), the problem is terminated and declared unbounded. To avoid large function values, it may be necessary to impose bounds on some of the variables in order to keep them away from singularities in the nonlinear functions.

The message may indicate an abnormal termination while enforcing the limit on the constraint violations. This exit implies that the objective is not bounded below in the feasible region defined by expanding the bounds by the value of the Violation Limit.

$\mathbf{ifail}=6$

Iteration limit reached.

Major iteration limit reached.

The value of the optional parameter Superbasics Limit is too small.

Either the Iterations Limit or the Major Iterations Limit was exceeded before the required solution could be found. Check the iteration log to be sure that progress was being made. If so, and if you caused a basis file to be saved by using the optional parameter New Basis File, consider restarting the run using the optional parameter Old Basis File to see whether further progress can be made. If you have no basis file available, you might rerun the problem after increasing the optional parameters Minor Iterations Limit and/or Major Iterations Limit.

If none of the above limits have been reached, this error may mean that the problem appears to be more nonlinear than anticipated. The current set of basic and superbasic variables have been optimized as much as possible and a pricing operation (where a nonbasic variable is selected to become superbasic) is necessary to continue, but it can't continue as the number of superbasic variables has already reached the limit specified by the optional parameter Superbasics Limit. In general, raise the Superbasics Limit $s$ by a reasonable amount, bearing in mind the storage needed for the reduced Hessian.

$\mathbf{ifail}=7$

Numerical difficulties have been encountered and no further progress can be made.

Several circumstances could lead to this exit.

1.	usrfun could be returning accurate function values but inaccurate gradients (or vice versa). This is the most likely cause. Study the comments given for $\mathbf{ifail}=\mathbf{8}$, and do your best to ensure that the coding is correct.
2.	The function and gradient values could be consistent, but their precision could be too low. For example, accidental use of a low precision data type when a higher precision was intended would lead to a relative function precision of about ${10}^{-6}$ instead of something like ${10}^{-15}$. The default Major Optimality Tolerance of $2\times {10}^{-6}$ would need to be raised to about ${10}^{-3}$ for optimality to be declared (at a rather suboptimal point). Of course, it is better to revise the function coding to obtain as much precision as economically possible.
3.	If function values are obtained from an expensive iterative process, they may be accurate to rather few significant figures, and gradients will probably not be available. One should specify Function Precision $t$ Major Optimality Tolerance $\sqrt{t}$ but even then, if $t$ is as large as ${10}^{-5}$ or ${10}^{-6}$ (only $5$ or $6$ significant figures), the same exit condition may occur. At present the only remedy is to increase the accuracy of the function calculation.
4.	An $LU$ factorization of the basis has just been obtained and used to recompute the basic variables $x_B$, given the present values of the superbasic and nonbasic variables. A step of ‘iterative refinement’ has also been applied to increase the accuracy of $x_B$. However, a row check has revealed that the resulting solution does not satisfy the current constraints $Ax-s=0$ sufficiently well. This probably means that the current basis is very ill-conditioned. If there are some linear constraints and variables, try $\mathbf{Scale\; Option}=1$ if scaling has not yet been used. For certain highly structured basis matrices (notably those with band structure), a systematic growth may occur in the factor $U$. Consult the description of Umax and Growth in Section 13.4 and set the LU Factor Tolerance to $2.0$ (or possibly even smaller, but not less than $1.0$).
5.	The first factorization attempt will have found the basis to be structurally or numerically singular. (Some diagonals of the triangular matrix $U$ were respectively zero or smaller than a certain tolerance.) The associated variables are replaced by slacks and the modified basis is refactorized, but singularity persists. This must mean that the problem is badly scaled, or the LU Factor Tolerance is too much larger than $1.0$. This is highly unlikely to occur.

$\mathbf{ifail}=8$

User-supplied function computes incorrect constraint derivatives.

User-supplied function computes incorrect objective derivatives.

A check has been made on some elements of the Jacobian as returned in the argument g of usrfun. At least one value disagrees remarkably with its associated forward difference estimate (the relative difference between the computed and estimated values is $1.0$ or more). This exit is a safeguard, since e04vhf will usually fail to make progress when the computed gradients are seriously inaccurate. In the process it may expend considerable effort before terminating with $\mathbf{ifail}=\mathbf{7}$.

Check the function and Jacobian computation very carefully in usrfun. A simple omission could explain everything. If a component is very large, then give serious thought to scaling the function or the nonlinear variables.

If you feel certain that the computed Jacobian is correct (and that the forward-difference estimate is therefore wrong), you can specify $\mathbf{Verify\; Level}=0$ to prevent individual elements from being checked. However, the optimization procedure may have difficulty.

$\mathbf{ifail}=9$

Unable to proceed into undefined region of user-supplied function.

User-supplied function is undefined at the first feasible point.

User-supplied function is undefined at the initial point.

You have indicated that the problem functions are undefined by assigning the value $\mathbf{status}=-1$ on exit from usrfun. e04vhf attempts to evaluate the problem functions closer to a point at which the functions are already known to be defined. This exit occurs if e04vhf is unable to find a point at which the functions are defined. This will occur in the case of:

–	undefined functions with no recovery possible;
–	undefined functions at the first point;
–	undefined functions at the first feasible point; or
–	undefined functions when checking derivatives.

$\mathbf{ifail}=10$

User-supplied function requested termination.

User requested termination.

You have indicated the wish to terminate solution of the current problem by setting status to a value $<-1$ on exit from usrfun.

$\mathbf{ifail}=11$

Internal error: memory allocation failed when attempting to allocate workspace sizes $〈\mathit{\text{value}}〉$, $〈\mathit{\text{value}}〉$ and $〈\mathit{\text{value}}〉$. Please contact NAG.

$\mathbf{ifail}=12$

Internal memory allocation was insufficient. Please contact NAG.

$\mathbf{ifail}=13$

An error has occurred in the basis package. Check that arrays iafun, javar, igfun and jgvar contain values in the appropriate ranges and do not define duplicate elements of a or g. Set the optional parameter Print File and examine the output carefully for further information.

$\mathbf{ifail}=14$

An unexpected error has occurred. Set the optional parameter Print File and examine the output carefully for further information.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

9.1

The Final Output

9.1.1

The ROWS section

9.1.2

The COLUMNS section

10

Example

10.1

Program Text

Program Text (e04vhfe.f90)

10.2

Program Data

Program Data (e04vhfe.d)

10.3

Program Results

Program Results (e04vhfe.r)

11

Algorithmic Details

11.1

Constraints and Slack Variables

11.2

Major Iterations

11.3

Minor Iterations

11.4

The Merit Function

11.5

Treatment of Constraint Infeasibilities

12

Optional Parameters

• Backup Basis File
• Central Difference Interval
• Check Frequency
• Crash Option
• Crash Tolerance
• Defaults
• Derivative Linesearch
• Derivative Option
• Difference Interval
• Dump File
• Elastic Weight
• Expand Frequency
• Factorization Frequency
• Feasibility Tolerance
• Feasible Point
• Function Precision
• Hessian Frequency
• Hessian Full Memory
• Hessian Limited Memory
• Hessian Updates
• Infinite Bound Size
• Insert File
• Iterations Limit
• Linesearch Tolerance
• List
• Load File
• LU Complete Pivoting
• LU Density Tolerance
• LU Factor Tolerance
• LU Partial Pivoting
• LU Rook Pivoting
• LU Singularity Tolerance
• LU Update Tolerance
• Major Feasibility Tolerance
• Major Iterations Limit
• Major Optimality Tolerance
• Major Print Level
• Major Step Limit
• Maximize
• Minimize
• Minor Feasibility Tolerance
• Minor Iterations Limit
• Minor Print Level
• New Basis File
• New Superbasics Limit
• Nolist
• Nonderivative Linesearch
• Old Basis File
• Partial Price
• Pivot Tolerance
• Print File
• Print Frequency
• Proximal Point Method
• Punch File
• Save Frequency
• Scale Option
• Scale Print
• Scale Tolerance
• Solution File
• Summary File
• Summary Frequency
• Superbasics Limit
• Suppress Parameters
• System Information No
• System Information Yes
• Timing Level
• Unbounded Objective
• Unbounded Step Size
• Verify Level
• Violation Limit