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NAG Toolbox: nag_opt_lp_solve (e04mf)
Purpose
nag_opt_lp_solve (e04mf) solves general linear programming problems. It is not intended for large sparse problems.
Syntax
[
istate,
x,
iter,
obj,
ax,
clamda,
lwsav,
iwsav,
rwsav,
ifail] = e04mf(
a,
bl,
bu,
cvec,
istate,
x,
lwsav,
iwsav,
rwsav, 'n',
n, 'nclin',
nclin)
[
istate,
x,
iter,
obj,
ax,
clamda,
lwsav,
iwsav,
rwsav,
ifail] = nag_opt_lp_solve(
a,
bl,
bu,
cvec,
istate,
x,
lwsav,
iwsav,
rwsav, 'n',
n, 'nclin',
nclin)
Before calling
nag_opt_lp_solve (e04mf), or
the option setting function
nag_opt_lp_option_string (e04mh),
nag_opt_init (e04wb) must be called.
Description
nag_opt_lp_solve (e04mf) is designed to solve linear programming (LP) problems of the form
where
$c$ is an
$n$element vector and
$A$ is an
${m}_{L}$ by
$n$ matrix.
This is the default type of problem, referred to as type LP. The optional parameter
Problem Type may be used to specify an alternative problem type FP, in which the objective function is omitted and the function attempts to find a feasible point for the set of constraints.
The constraints involving
$A$ are called the
general constraints. Note that upper and lower bounds are specified for all the variables and for all the general 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 parameter
Infinite Bound Size.)
You must supply an initial estimate of the solution.
The method used by
nag_opt_lp_solve (e04mf) is described in detail in
Algorithmic Details.
References
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 and Murray W (1978) Numerically stable methods for quadratic programming Math. Programming 14 349–372
Gill P E, Murray W, Saunders M A and Wright M H (1984) 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 (1989) A practical anticycling procedure for linearly constrained optimization Math. Programming 45 437–474
Gill P E, Murray W, Saunders M A and Wright M H (1991) Inertiacontrolling methods for general quadratic programming SIAM Rev. 33 1–36
Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press
Parameters
Compulsory Input Parameters
 1:
$\mathrm{a}\left(\mathit{lda},:\right)$ – double array

The first dimension of the array
a must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nclin}}\right)$.
The second dimension of the array
a must be at least
${\mathbf{n}}$ if
${\mathbf{nclin}}>0$ and at least
$1$ if
${\mathbf{nclin}}=0$.
The
$\mathit{i}$th row of
a must contain the coefficients of the
$\mathit{i}$th general linear constraint, for
$\mathit{i}=1,2,\dots ,{m}_{L}$.
If
${\mathbf{nclin}}=0$,
a is not referenced.
 2:
$\mathrm{bl}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – double array
 3:
$\mathrm{bu}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – double array

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, and the next
${m}_{L}$ elements the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e.,
${l}_{j}=\infty $), set
${\mathbf{bl}}\left(j\right)\le \mathit{bigbnd}$, and to specify a nonexistent upper bound (i.e.,
${u}_{j}=+\infty $), set
${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$; the default value of
$\mathit{bigbnd}$ is
${10}^{20}$, but this may be changed by the optional parameter
Infinite Bound Size. To specify the
$j$th constraint as an
equality, set
${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)=\beta $, say, where
$\left\beta \right<\mathit{bigbnd}$.
Constraints:
 ${\mathbf{bl}}\left(\mathit{j}\right)\le {\mathbf{bu}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{nclin}}$;
 if ${\mathbf{bl}}\left(\mathit{j}\right)={\mathbf{bu}}\left(\mathit{j}\right)=\beta $, $\left\beta \right<\mathit{bigbnd}$.
 4:
$\mathrm{cvec}\left(:\right)$ – double array

The dimension of the array
cvec
must be at least
${\mathbf{n}}$ if the problem is of type LP (the default), and at least
$1$ otherwise
The coefficients of the objective function when the problem is of type LP.
If the problem is of type FP,
cvec is not referenced.
 5:
$\mathrm{istate}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – int64int32nag_int array

Need not be set if the (default) optional parameter
Cold Start is used.
If the optional parameter
Warm Start has been chosen,
istate specifies the desired status of the constraints at the start of the feasibility phase. More precisely, the first
$n$ elements of
istate refer to the upper and lower bounds on the variables, and the next
${m}_{L}$ elements refer to the general linear constraints (if any). Possible values for
${\mathbf{istate}}\left(j\right)$ are as follows:
${\mathbf{istate}}\left(j\right)$  Meaning 
0  The corresponding constraint should not be in the initial working set. 
1  The constraint should be in the initial working set at its lower bound. 
2  The constraint should be in the initial working set at its upper bound. 
3  The constraint should be in the initial working set as an equality. This value must not be specified unless ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)$. 
The values
$2$,
$1$ and
$4$ are also acceptable but will be reset to zero by the function. If
nag_opt_lp_solve (e04mf) has been called previously with the same values of
n and
nclin,
istate already contains satisfactory information. (See also the description of the optional parameter
Warm Start.) The function also adjusts (if necessary) the values supplied in
x to be consistent with
istate.
Constraint:
$2\le {\mathbf{istate}}\left(\mathit{j}\right)\le 4$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}+{\mathbf{nclin}}$.
 6:
$\mathrm{x}\left({\mathbf{n}}\right)$ – double array

An initial estimate of the solution.
 7:
$\mathrm{lwsav}\left(120\right)$ – logical array
 8:
$\mathrm{iwsav}\left(610\right)$ – int64int32nag_int array
 9:
$\mathrm{rwsav}\left(475\right)$ – double array

The arrays
lwsav,
iwsav and
rwsav must not be altered between calls to any of the functions
nag_opt_lp_solve (e04mf),
nag_opt_lp_option_string (e04mh) or
nag_opt_init (e04wb).
Optional Input Parameters
 1:
$\mathrm{n}$ – int64int32nag_int scalar

Default:
the dimension of the array
x.
$n$, the number of variables.
Constraint:
${\mathbf{n}}>0$.
 2:
$\mathrm{nclin}$ – int64int32nag_int scalar

Default:
the first dimension of the array
a.
${m}_{L}$, the number of general linear constraints.
Constraint:
${\mathbf{nclin}}\ge 0$.
Output Parameters
 1:
$\mathrm{istate}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – int64int32nag_int array

The status of the constraints in the working set at the point returned in
x. The significance of each possible value of
${\mathbf{istate}}\left(j\right)$ is as follows:
${\mathbf{istate}}\left(j\right)$  Meaning 
$2$  The constraint violates its lower bound by more than the feasibility tolerance. 
$1$  The constraint violates its upper bound by more than the feasibility tolerance. 
$\phantom{}0$  The constraint is satisfied to within the feasibility tolerance, but is not in the working set. 
$\phantom{}1$  This inequality constraint is included in the working set at its lower bound. 
$\phantom{}2$  This inequality constraint is included in the working set at its upper bound. 
$\phantom{}3$  This constraint is included in the working set as an equality. This value of istate can occur only when ${\mathbf{bl}}\left(j\right)={\mathbf{bu}}\left(j\right)$. 
$\phantom{}4$  This corresponds to optimality being declared with ${\mathbf{x}}\left(j\right)$ being temporarily fixed at its current value. This value of istate can occur only when ${\mathbf{ifail}}={\mathbf{1}}$ on exit. 
 2:
$\mathrm{x}\left({\mathbf{n}}\right)$ – double array

The point at which
nag_opt_lp_solve (e04mf) terminated. If
${\mathbf{ifail}}={\mathbf{0}}$,
${\mathbf{1}}$ or
${\mathbf{4}}$,
x contains an estimate of the solution.
 3:
$\mathrm{iter}$ – int64int32nag_int scalar

The total number of iterations performed.
 4:
$\mathrm{obj}$ – double scalar

The value of the objective function at
$x$ if
$x$ is feasible, or the sum of infeasibiliites at
$x$ otherwise. If the problem is of type FP and
$x$ is feasible,
obj is set to zero.
 5:
$\mathrm{ax}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nclin}}\right)\right)$ – double array

The final values of the linear constraints
$Ax$.
If
${\mathbf{nclin}}=0$,
ax is not referenced.
 6:
$\mathrm{clamda}\left({\mathbf{n}}+{\mathbf{nclin}}\right)$ – double array

The values of the Lagrange multipliers for each constraint with respect to the current working set. The first $n$ elements contain the multipliers for the bound constraints on the variables, and the next ${m}_{L}$ elements contain the multipliers for the general linear constraints (if any). If ${\mathbf{istate}}\left(j\right)=0$ (i.e., constraint $j$ is not in the working set), ${\mathbf{clamda}}\left(j\right)$ is zero. If $x$ is optimal, ${\mathbf{clamda}}\left(j\right)$ should be nonnegative if ${\mathbf{istate}}\left(j\right)=1$, nonpositive if ${\mathbf{istate}}\left(j\right)=2$ and zero if ${\mathbf{istate}}\left(j\right)=4$.
 7:
$\mathrm{lwsav}\left(120\right)$ – logical array
 8:
$\mathrm{iwsav}\left(610\right)$ – int64int32nag_int array
 9:
$\mathrm{rwsav}\left(475\right)$ – double array

 10:
$\mathrm{ifail}$ – int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see
Error Indicators and Warnings).
nag_opt_lp_solve (e04mf) returns with
${\mathbf{ifail}}={\mathbf{0}}$ if
$x$ is a strong local minimizer, i.e., the reduced gradient (
Norm Gz; see
Printed Output) is negligible and the Lagrange multipliers (
Lagr Mult; see
Printed Output) are optimal.
Error Indicators and Warnings
Note: nag_opt_lp_solve (e04mf) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:
Cases prefixed with W are classified as warnings and
do not generate an error of type NAG:error_n. See nag_issue_warnings.
 W ${\mathbf{ifail}}=1$

$x$ is a weak local minimum (the projected gradient is negligible and the Lagrange multipliers are optimal but there is a small multiplier). This means that the solution is not unique.
 W ${\mathbf{ifail}}=2$

The solution appears to be unbounded, i.e., the objective function is not bounded below in the feasible region. This value of
ifail occurs if a step larger than
Infinite Step Size (
$\text{default value}={10}^{20}$) would have to be taken in order to continue the algorithm, or the next step would result in an element of
$x$ having magnitude larger than optional parameter
Infinite Bound Size (
$\text{default value}={10}^{20}$).
 W ${\mathbf{ifail}}=3$

No feasible point was found, i.e., it was not possible to satisfy all the constraints to within the feasibility tolerance. In this case, the constraint violations at the final
$x$ will reveal a value of the tolerance for which a feasible point will exist – for example, when the feasibility tolerance for each violated constraint exceeds its
Slack (see
Printed Output) at the final point. The modified problem (with an altered feasibility tolerance) may then be solved using a
Warm Start. You should check that there are no constraint redundancies. If the data for the constraints are accurate only to the absolute precision
$\sigma $, you should ensure that the value of the optional parameter
Feasibility Tolerance (
$\text{default value}=\sqrt{\epsilon}$, where
$\epsilon $ is the
machine precision) is
greater than
$\sigma $. For example, if all elements of
$A$ are of order unity and are accurate only to three decimal places, the
Feasibility Tolerance should be at least
${10}^{3}$.
 ${\mathbf{ifail}}=4$

The limiting number of iterations was reached before normal termination occurred.
The value of the optional parameter
Iteration Limit (
$\text{default value}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5\left(n+{m}_{L}\right)\right)$) may be too small. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), either rerun
nag_opt_lp_solve (e04mf) with a larger value of
Iteration Limit or, alternatively, rerun
nag_opt_lp_solve (e04mf) using the
Warm Start facility to specify the initial working set.
 ${\mathbf{ifail}}=5$
Not used by this function.
 ${\mathbf{ifail}}=6$

An input argument is invalid.
 ${\mathbf{ifail}}=7$

The designated problem type was not FP or LP. Rerun
nag_opt_lp_solve (e04mf) with the optional parameter
Problem Type set to one of these values.
 $\mathbf{\text{Overflow}}$

If the printed output before the overflow error contains a warning about serious illconditioning in the working set when adding the
$j$th constraint, it may be possible to avoid the difficulty by increasing the magnitude of the
Feasibility Tolerance (
$\text{default value}=\sqrt{\epsilon}$, where
$\epsilon $ is the
machine precision) and rerunning the program. If the message recurs even after this change, the offending linearly dependent constraint (with index ‘
$j$’) must be removed from the problem.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
Accuracy
nag_opt_lp_solve (e04mf) implements a numerically stable active set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.
Further Comments
This section contains some comments on scaling and a description of the printed output.
Scaling
Sensible scaling of the problem is likely to reduce the number of iterations required and make the problem less sensitive to perturbations in the data, thus improving the condition of the problem. In the absence of better information it is usually sensible to make the Euclidean lengths of each constraint of comparable magnitude. See the
E04 Chapter Introduction and
Gill et al. (1981) for further information and advice.
Description of the Printed Output
The following line of summary output (
$\text{}<80$ characters) is produced at every iteration. In all cases, the values of the quantities printed are those in effect
on completion of the given iteration.
Itn 
is the iteration count.

Step 
is the step taken along the computed search direction. If a constraint is added during the current iteration, Step will be the step to the nearest constraint. When the problem is of type LP, the step can be greater than one during the optimality phase.

Ninf 
is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.

Sinf/Objective 
is the value of the current objective function. If $x$ is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If $x$ is feasible, Objective is the value of the objective function of (1). The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point. During the optimality phase the value of the objective function will be nonincreasing. During the feasibility phase the number of constraint infeasibilities will not increase until either a feasible point is found or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.

Norm Gz 
is $\Vert {Z}_{R}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, the Euclidean norm of the reduced gradient with respect to ${Z}_{R}$. During the optimality phase, this norm will be approximately zero after a unit step.
(See Definition of Search Direction and Choosing the Initial Working Set.) 
The final printout includes a listing of the status of every variable and constraint.
The following describes the printout for each variable. A full stop (.) is printed for any numerical value that is zero.
Varbl 
gives the name (V) and index $\mathit{j}$, for $\mathit{j}=1,2,\dots ,n$, of the variable.

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

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

I 
Infeasible. The variable is currently violating one of its bounds by more than the Feasibility Tolerance.


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

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

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

Lagr Mult 
is the Lagrange multiplier for the associated bound. This will be zero if State is FR unless ${\mathbf{bl}}\left(j\right)\le \mathit{bigbnd}$ and ${\mathbf{bu}}\left(j\right)\ge \mathit{bigbnd}$, 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.

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

The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’,
${\mathbf{bl}}\left(j\right)$ and
${\mathbf{bu}}\left(j\right)$ are replaced by
${\mathbf{bl}}\left(n+j\right)$ and
${\mathbf{bu}}\left(n+j\right)$ respectively, and with the following change 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.

Note that movement off a constraint (as opposed to a variable moving away from its bound) can be interpreted as allowing the entry in the Slack column to become positive.
Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.
Example
This example minimizes the function
subject to the bounds
and the general constraints
The initial point, which is infeasible, is
The optimal solution (to five figures) is
Four bound constraints and three general constraints are active at the solution.
Open in the MATLAB editor:
e04mf_example
function e04mf_example
fprintf('e04mf example results\n\n');
a = [1, 1, 1, 1, 1, 1, 1;
0.15, 0.04, 0.02, 0.04, 0.02, 0.01, 0.03;
0.03, 0.05, 0.08, 0.02, 0.06, 0.01, 0;
0.02, 0.04, 0.01, 0.02, 0.02, 0, 0;
0.02, 0.03, 0, 0, 0.01, 0, 0;
0.7, 0.75, 0.8, 0.75, 0.8, 0.97, 0;
0.02, 0.06, 0.08, 0.12, 0.02, 0.01, 0.97];
bl = [0.01;0.1; 0.01; 0.04; 0.1; 0.01; 0.01;
0.13;1e25; 1e25; 1e25; 1e25; 0.0992;0.003];
bu = [ 0.01; 0.15; 0.03; 0.02; 0.05; 1e25; 1e25;
0.13;0.0049;0.0064;0.0037;0.0012; 1e25; 0.002];
cvec = [0.02; 0.2;0.2; 0.2; 0.2; 0.04; 0.04];
istate = zeros(14, 1, 'int64');
x = [0.01; 0.03; 0; 0.01; 0.1; 0.02; 0.01];
[cwsav,lwsav,iwsav,rwsav,ifail] = e04wb('e04mf');
[lwsav, iwsav, rwsav, inform] = ...
e04mh('Nolist', lwsav, iwsav, rwsav);
[lwsav, iwsav, rwsav, inform] = ...
e04mh('Print Level = 1', lwsav, iwsav, rwsav);
[istate, x, iter, obj, ax, clamda, lwsav, iwsav, rwsav, ifail] = ...
e04mf(...
a, bl, bu, cvec, istate, x, lwsav, iwsav, rwsav);
fprintf('Minimum value : %9.4f\n\n',obj);
fprintf('Found after %3d iterations at x:\n ',iter);
fprintf(' %9.4f',x);
fprintf('\nLinear contrained values Ax:\n ');
fprintf(' %9.4f',ax);
fprintf('\n');
e04mf example results
Minimum value : 0.0236
Found after 7 iterations at x:
0.0100 0.1000 0.0300 0.0200 0.0675 0.0023 0.0002
Linear contrained values Ax:
0.1300 0.0055 0.0066 0.0048 0.0039 0.0992 0.0030
Note: the remainder of this document is intended for more advanced users. Algorithmic Details contains a detailed description of the algorithm which may be needed in order to understand Optional Parameters and Description of Monitoring Information. Optional Parameters describes the optional parameters which may be set by calls to nag_opt_lp_option_string (e04mh). Description of Monitoring Information describes the quantities which can be requested to monitor the course of the computation.
Algorithmic Details
This section contains a detailed description of the method used by nag_opt_lp_solve (e04mf).
Overview
nag_opt_lp_solve (e04mf) is based on an inertiacontrolling method due to
Gill and Murray (1978), and is described in detail by
Gill et al. (1991). Here we briefly summarise the main features of the method. Where possible, explicit reference is made to the names of variables that are arguments of
nag_opt_lp_solve (e04mf) or appear in the printed output.
nag_opt_lp_solve (e04mf) has two phases: finding an initial feasible point by minimizing the sum of infeasibilities (the
feasibility phase), and minimizing the linear objective function within the feasible region (the
optimality phase). The computations in both phases are performed by the same functions. The twophase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities to the linear objective function. The feasibility phase does
not perform the standard simplex method (i.e., it does not necessarily find a vertex), except in the case when
${m}_{L}\le n$. Once any iterate is feasible, all subsequent iterates remain feasible.
In general, an iterative process is required to solve a linear program. (For simplicity, we shall always consider a typical iteration and avoid reference to the index of the iteration.) Each new iterate
$\stackrel{}{x}$ is defined by
where the
step length
$\alpha $ is a nonnegative scalar, and
$p$ is called the
search direction.
At each point
$x$, a
working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied ‘exactly’ (to within the tolerance defined by the optional parameter
Feasibility Tolerance). The working set is the current prediction of the constraints that hold with equality at a solution of an LP problem. The search direction is constructed so that the constraints in the working set remain
unaltered for any value of the step length. For a bound constraint in the working set, this property is achieved by setting the corresponding element of the search direction to zero. Thus, the associated variable is
fixed, and specification of the working set induces a partition of
$x$ into
fixed and
free variables. During a given iteration, the fixed variables are effectively removed from the problem; since the relevant elements of the search direction are zero, the columns of
$A$ corresponding to fixed variables may be ignored.
Let
${m}_{\mathrm{W}}$ denote the number of general constraints in the working set and let
${n}_{\mathrm{FX}}$ denote the number of variables fixed at one of their bounds (
${m}_{\mathrm{W}}$ and
${n}_{\mathrm{FX}}$ are the quantities
Lin and
Bnd in the monitoring file output from
nag_opt_lp_solve (e04mf); see
Description of Monitoring Information). Similarly, let
${n}_{\mathrm{FR}}$ (
${n}_{\mathrm{FR}}=n{n}_{\mathrm{FX}}$) denote the number of free variables. At every iteration,
the variables are reordered so that the last
${n}_{\mathrm{FX}}$ variables are fixed, with all other relevant vectors and matrices ordered accordingly.
Definition of Search Direction
Let
${A}_{\mathrm{FR}}$ denote the
${m}_{\mathrm{W}}$ by
${n}_{\mathrm{FR}}$ submatrix of general constraints in the working set corresponding to the free variables, and let
${p}_{\mathrm{FR}}$ denote the search direction with respect to the free variables only. The general constraints in the working set will be unaltered by any move along
$p$ if
In order to compute
${p}_{\mathrm{FR}}$, the
$TQ$ factorization of
${A}_{\mathrm{FR}}$ is used:
where
$T$ is a nonsingular
${m}_{\mathrm{W}}$ by
${m}_{\mathrm{W}}$ upper triangular matrix (i.e.,
${t}_{ij}=0$ if
$i>j$), and the nonsingular
${n}_{\mathrm{FR}}$ by
${n}_{\mathrm{FR}}$ matrix
${Q}_{\mathrm{FR}}$ is the product of orthogonal transformations (see
Gill et al. (1984)). If the columns of
${Q}_{\mathrm{FR}}$ are partitioned so that
where
$Y$ is
${n}_{\mathrm{FR}}$ by
${m}_{\mathrm{W}}$, then the
${n}_{Z}$ (
${n}_{Z}={n}_{\mathrm{FR}}{m}_{\mathrm{W}}$) columns of
$Z$ form a basis for the null space of
${A}_{\mathrm{FR}}$. Let
${n}_{R}$ be an integer such that
$0\le {n}_{R}\le {n}_{Z}$, and let
${Z}_{R}$ denote a matrix whose
${n}_{R}$ columns are a subset of the columns of
$Z$. (The integer
${n}_{R}$ is the quantity
Zr in the monitoring file output from
nag_opt_lp_solve (e04mf). In many cases,
${Z}_{R}$ will include
all the columns of
$Z$.) The direction
${p}_{\mathrm{FR}}$ will satisfy
(2) if
where
${p}_{R}$ is any
${n}_{R}$vector.
Main Iteration
Let
$Q$ denote the
$n$ by
$n$ matrix
where
${I}_{\mathrm{FX}}$ is the identity matrix of order
${n}_{\mathrm{FX}}$. Let
${g}_{Q}$ denote the
transformed gradient
and let the vector of the first
${n}_{R}$ elements of
${g}_{Q}$ be denoted by
${g}_{R}$. The quantity
${g}_{R}$ is known as the
reduced gradient of
${c}^{\mathrm{T}}x$. If the reduced gradient is zero,
$x$ is a constrained stationary point in the subspace defined by
$Z$. During the feasibility phase, the reduced gradient will usually be zero only at a vertex (although it may be zero at nonvertices in the presence of constraint dependencies). During the optimality phase, a zero reduced gradient implies that
$x$ minimizes the linear objective when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers
${\lambda}_{C}$ and
${\lambda}_{B}$ for the general and bound constraints are defined from the equations
Given a positive constant
$\delta $ of the order of the
machine precision, a Lagrange multiplier
${\lambda}_{j}$ corresponding to an inequality constraint in the working set is said to be
optimal if
${\lambda}_{j}\le \delta $ when the associated constraint is at its
upper bound, or if
${\lambda}_{j}\ge \delta $ when the associated constraint is at its
lower bound. If a multiplier is nonoptimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint (with index
Jdel; see
Description of Monitoring Information) from the working set.
If optimal multipliers occur during the feasibility phase and the sum of infeasibilities is nonzero, there is no feasible point, and you can force
nag_opt_lp_solve (e04mf) to continue until the minimum value of the sum of infeasibilities has been found; see the discussion of the optional parameter
Minimum Sum of Infeasibilities. At such a point, the Lagrange multiplier
${\lambda}_{j}$ corresponding to an inequality constraint in the working set will be such that
$\left(1+\delta \right)\le {\lambda}_{j}\le \delta $ when the associated constraint is at its
upper bound, and
$\delta \le {\lambda}_{j}\le \left(1+\delta \right)$ when the associated constraint is at its
lower bound. Lagrange multipliers for equality constraints will satisfy
$\left{\lambda}_{j}\right\le 1+\delta $.
If the reduced gradient is not zero, Lagrange multipliers need not be computed and the nonzero elements of the search direction $p$ are given by ${Z}_{R}{p}_{R}$. The choice of step length is influenced by the need to maintain feasibility with respect to the satisfied constraints.
Each change in the working set leads to a simple change to ${A}_{\mathrm{FR}}$: if the status of a general constraint changes, a row of ${A}_{\mathrm{FR}}$ is altered; if a bound constraint enters or leaves the working set, a column of ${A}_{\mathrm{FR}}$ changes. Explicit representations are recurred of the matrices $T$ and ${Q}_{\mathrm{FR}}$; and of vectors ${Q}^{\mathrm{T}}g$, and ${Q}^{\mathrm{T}}c$.
One of the most important features of
nag_opt_lp_solve (e04mf) is its control of the conditioning of the working set, whose nearness to linear dependence is estimated by the ratio of the largest to smallest diagonal elements of the
$TQ$ factor
$T$ (the printed value
Cond T; see
Description of Monitoring Information). In constructing the initial working set, constraints are excluded that would result in a large value of
Cond T.
nag_opt_lp_solve (e04mf) includes a rigorous procedure that prevents the possibility of cycling at a point where the active constraints are nearly linearly dependent (see
Gill et al. (1989)). The main feature of the anticycling procedure is that the feasibility tolerance is increased slightly at the start of every iteration. This not only allows a positive step to be taken at every iteration, but also provides, whenever possible, a
choice of constraints to be added to the working set. Let
${\alpha}_{\mathrm{M}}$ denote the maximum step at which
$x+{\alpha}_{\mathrm{M}}p$ does not violate any constraint by more than its feasibility tolerance. All constraints at a distance
$\alpha $ (
$\alpha \le {\alpha}_{\mathrm{M}}$) along
$p$ from the current point are then viewed as acceptable candidates for inclusion in the working set. The constraint whose normal makes the largest angle with the search direction is added to the working set.
Choosing the Initial Working Set
Let
$Z$ be partitioned as
$Z=\left({Z}_{R}{Z}_{A}\right)$. A working set for which
${Z}_{R}$ defines the null space can be obtained by including
the rows of ${Z}_{A}^{\mathrm{T}}$ as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by
${Z}_{R}$, as described in
Definition of Search Direction.
The artificially augmented working set is given by
so that
${p}_{\mathrm{FR}}$ will satisfy
${A}_{\mathrm{FR}}{p}_{\mathrm{FR}}=0$ and ${Z}_{A}^{\mathrm{T}}{p}_{\mathrm{FR}}=0$. By definition of the
$TQ$ factorization,
${\stackrel{}{A}}_{\mathrm{FR}}$ automatically satisfies the following:
where
and hence the
$TQ$ factorization of
(6) is available trivially from
$T$ and
${Q}_{\mathrm{FR}}$ without additional expense.
The matrix
${Z}_{A}$ is not kept fixed, since its role is purely to define an appropriate null space; the
$TQ$ factorization can therefore be updated in the normal fashion as the iterations proceed. No work is required to ‘delete’ the artificial constraints associated with
${Z}_{A}$ when
${Z}_{R}^{\mathrm{T}}{g}_{\mathrm{FR}}=0$, since this simply involves repartitioning
${Q}_{\mathrm{FR}}$. The ‘artificial’ multiplier vector associated with the rows of
${Z}_{A}^{\mathrm{T}}$ is equal to
${Z}_{A}^{\mathrm{T}}{g}_{\mathrm{FR}}$, and the multipliers corresponding to the rows of the ‘true’ working set are the multipliers that would be obtained if the artificial constraints were not present. If an artificial constraint is ‘deleted’ from the working set, an
A appears alongside the entry in the
Jdel column of the monitoring file output (see
Description of Monitoring Information).
The number of columns in
${Z}_{A}$ and
${Z}_{R}$ and the Euclidean norm of
${Z}_{R}^{\mathrm{T}}{g}_{\mathrm{FR}}$, appear in the monitoring file output as
Art,
Zr and
Norm Gz respectively (see
Description of Monitoring Information).
Under some circumstances, a different type of artificial constraint is used when solving a linear program. Although the algorithm of
nag_opt_lp_solve (e04mf) does not usually perform simplex steps (in the traditional sense), there is one exception: a linear program with fewer general constraints than variables (i.e.,
${m}_{L}\le n$). Use of the simplex method in this situation leads to savings in storage. At the starting point, the ‘natural’ working set (the set of constraints exactly or nearly satisfied at the starting point) is augmented with a suitable number of ‘temporary’ bounds, each of which has the effect of temporarily fixing a variable at its current value. In subsequent iterations, a temporary bound is treated as a standard constraint until it is deleted from the working set, in which case it is never added again. If a temporary bound is ‘deleted’ from the working set, an
F (for ‘Fixed’) appears alongside the entry in the
Jdel column of the monitoring file output (see
Description of Monitoring Information).
Optional Parameters
Several optional parameters in nag_opt_lp_solve (e04mf) define choices in the problem specification or the algorithm logic. In order to reduce the number of formal arguments of nag_opt_lp_solve (e04mf) these optional parameters have associated default values that are appropriate for most problems. Therefore, you need only specify those optional parameters whose values are to be different from their default values.
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in
Description of the s.
Optional parameters may be specified by calling
nag_opt_lp_option_string (e04mh) before a call to
nag_opt_lp_solve (e04mf).
nag_opt_lp_option_string (e04mh) can be called to supply options directly, one call being necessary for each optional parameter. For example,
[lwsav, iwsav, rwsav, inform] = e04mh('Print Level = 5', lwsav, iwsav, rwsav);
nag_opt_lp_option_string (e04mh) should be consulted for a full description of this method of supplying optional parameters.
All optional parameters not specified by you are set to their default values. Optional parameters specified by you are unaltered by nag_opt_lp_solve (e04mf) (unless they define invalid values) and so remain in effect for subsequent calls unless altered by you.
Description of the Optional Parameters
For each option, we give a summary line, a description of the optional parameter and details of constraints.
The summary line contains:
 the keywords, where the minimum abbreviation of each keyword is underlined (if no characters of an optional qualifier are underlined, the qualifier may be omitted);
 a parameter value,
where the letters $a$, $i$ and $r$ denote options that take character, integer and real values respectively;
 the default value, where the symbol $\epsilon $ is a generic notation for machine precision (see nag_machine_precision (x02aj)).
Keywords and character values are case and white space insensitive.
Check Frequency $i$Default $\text{}=50$
Every $i$th iteration, a numerical test is made to see if the current solution $x$ satisfies the constraints in the working set. If the largest residual of the constraints in the working set is judged to be too large, the current working set is refactorized and the variables are recomputed to satisfy the constraints more accurately. If $i\le 0$, the default value is used.
Cold Start Default
Warm Start
This option specifies how the initial working set is chosen. With a
Cold Start,
nag_opt_lp_solve (e04mf) 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
Crash Tolerance).
With a
Warm Start, you must provide a valid definition of every element of the array
istate.
nag_opt_lp_solve (e04mf) will override your specification of
istate if necessary, so that a poor choice of the working set will not cause a fatal error. For instance, any elements of
istate which are set to
$2$,
$1\text{ or}4$ 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_lp_solve (e04mf) is called repeatedly to solve related problems.
Crash Tolerance $r$Default $\text{}=0.01$
This value is used in conjunction with the optional parameter
Cold Start (the default value) when
nag_opt_lp_solve (e04mf) selects an initial working set. If
$0\le r\le 1$, the initial working set will include (if possible) bounds or general inequality constraints that lie within
$r$ 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
$\left{a}_{j}^{\mathrm{T}}xl\right\le r\left(1+\leftl\right\right)$. If
$r<0$ or
$r>1$, the default value is used.
Defaults
This special keyword may be used to reset all optional parameters to their default values.
Expand Frequency $i$Default $\text{}=5$
This option is part of an anticycling procedure designed to guarantee progress even on highly degenerate problems.
The strategy is to force a positive step at every iteration, at the expense of violating the constraints by a small amount. Suppose that the value of the optional parameter
Feasibility Tolerance is
$\delta $. Over a period of
$i$ iterations, the feasibility tolerance actually used by
nag_opt_lp_solve (e04mf) (i.e., the
working feasibility tolerance) increases from
$0.5\delta $ to
$\delta $ (in steps of
$0.5\delta /i$).
At certain stages the following ‘resetting procedure’ is used to remove constraint infeasibilities. First, all variables whose upper or lower bounds are in the working set are moved exactly onto their bounds. A count is kept of the number of nontrivial adjustments made. If the count is positive, iterative refinement is used to give variables that satisfy the working set to (essentially)
machine precision. Finally, the working feasibility tolerance is reinitialized to
$0.5\delta $.
If a problem requires more than $i$ iterations, the resetting procedure is invoked and a new cycle of $i$ iterations is started with $i$ incremented by $10$. (The decision to resume the feasibility phase or optimality phase is based on comparing any constraint infeasibilities with $\delta $.)
The resetting procedure is also invoked when nag_opt_lp_solve (e04mf) reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any nontrivial adjustments are made, iterations are continued.
If $i\le 0$, the default value is used. If $i\ge 9999999$, no anticycling procedure is invoked.
Feasibility Tolerance $r$Default $\text{}=\sqrt{\epsilon}$
If $r\ge \epsilon $, $r$ defines the maximum acceptable absolute violation in each constraint at a ‘feasible’ point. For example, if the variables and the coefficients in the general constraints are of order unity, and the latter are correct to about $6$ decimal digits, it would be appropriate to specify $r$ as ${10}^{6}$. If $0\le r<\epsilon $, the default value is used.
nag_opt_lp_solve (e04mf) attempts to find a feasible solution before optimizing the objective function. If the sum of infeasibilities cannot be reduced to zero, the optional parameter
Minimum Sum of Infeasibilities can be used to find the minimum value of the sum. Let
Sinf be the corresponding sum of infeasibilities. If
Sinf is quite small, it may be appropriate to raise
$r$ by a factor of
$10$ or
$100$. Otherwise, some error in the data should be suspected.
Note that a ‘feasible solution’ is a solution that satisfies the current constraints to within the tolerance $r$.
Infinite Bound Size $r$Default $\text{}={10}^{20}$
If $r>0$, $r$ defines the ‘infinite’ bound $\mathit{bigbnd}$ in the definition of the problem constraints. Any upper bound greater than or equal to $\mathit{bigbnd}$ will be regarded as $+\infty $ (and similarly any lower bound less than or equal to $\mathit{bigbnd}$ will be regarded as $\infty $). If $r<0$, the default value is used.
Infinite Step Size $r$Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\mathit{bigbnd},{10}^{20}\right)$
If $r>0$, $r$ specifies the magnitude of the change in variables that will be considered a step to an unbounded solution. (Note that an unbounded solution can occur only when the problem is of type LP.) If the change in $x$ during an iteration would exceed the value of $r$, the objective function is considered to be unbounded below in the feasible region. If $r\le 0$, the default value is used.
Iteration Limit $i$Default $\text{}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(50,5\left(n+{m}_{L}\right)\right)$
Iters
Itns
The value of $i$ specifies the maximum number of iterations allowed before termination. With $i=0$ and ${\mathbf{Print\; Level}}>0$, the workspace needed will be computed and printed, but no iterations will be performed. If $i<0$, the default value is used.
List
Nolist Default for nag_opt_lp_solve (e04mf) $\text{}={\mathbf{Nolist}}$
Normally each optional parameter specification is printed as it is supplied. Optional parameter
Nolist may be used to suppress the printing and optional parameter
List may be used to restore printing.
Minimum Sum of Infeasibilities $\overline{)\mathbf{N}}\mathbf{o}$Default $=\mathrm{NO}$
If no feasible point exists for the constraints, this option is used to control whether or not nag_opt_lp_solve (e04mf) will calculate a point that minimizes the constraint violations. If ${\mathbf{Minimum\; Sum\; of\; Infeasibilities}}=\mathrm{NO}$, nag_opt_lp_solve (e04mf) will terminate as soon as it is evident that no feasible point exists for the constraints. The final point will generally not be the point at which the sum of infeasibilities is minimized. If ${\mathbf{Minimum\; Sum\; of\; Infeasibilities}}=\mathrm{YES}$, nag_opt_lp_solve (e04mf) will continue until the sum of infeasibilities is minimized.
Monitoring File $i$Default $\text{}=1$
If $i\ge 0$ and ${\mathbf{Print\; Level}}\ge 5$, monitoring information produced by nag_opt_lp_solve (e04mf) at every iteration is sent to a file with logical unit number $i$. If $i<0$ and/or ${\mathbf{Print\; Level}}<5$, no monitoring information is produced.
Optimality Tolerance $r$Default $\text{}={\epsilon}^{0.8}$
If $r\ge \epsilon $, $r$ defines the tolerance used to determine if the bounds and general constraints have the right ‘sign’ for the solution to be judged to be optimal.
If $0\le r<\epsilon $, the default value is used.
Print Level $i$
The value of
$i$ controls the amount of printout produced by
nag_opt_lp_solve (e04mf), as indicated below. A detailed description of the printed output is given in
Printed Output (summary output at each iteration and the final solution) and
Description of Monitoring Information (monitoring information at each iteration).
The following printout is sent to the current advisory message unit (as defined by
nag_file_set_unit_advisory (x04ab)):
$i$ 
Output 
$\phantom{\ge 0}0$ 
No output. 
$\phantom{\ge 0}1$ 
The final solution only. 
$\phantom{\ge 0}5$ 
One line of summary output ($\text{}<80$ characters; see Printed Output) for each iteration (no printout of the final solution). 
$\text{}\ge 10$ 
The final solution and one line of summary output for each iteration. 
The following printout is sent to the logical unit number defined by the optional parameter
Monitoring File:
$i$ 
Output 
$\text{}<\phantom{0}5$ 
No output. 
$\text{}\ge \phantom{0}5$ 
One long line of output ($\text{}>80$ characters; see Description of Monitoring Information) for each iteration (no printout of the final solution). 
$\text{}\ge 20$ 
At each iteration, the Lagrange multipliers, the variables $x$, the constraint values $Ax$ and the constraint status (see istate). 
$\text{}\ge 30$ 
At each iteration, the diagonal elements of the upper triangular matrix $T$ associated with the $TQ$ factorization (3) (see Definition of Search Direction) of the working set. 
If
${\mathbf{Print\; Level}}\ge 5$ and the unit number defined by the optional parameter
Monitoring File is the same as that defined by
nag_file_set_unit_advisory (x04ab), then the summary output is suppressed.
Problem Type $a$Default $=$ LP
This option specifies the type of objective function to be minimized during the optimality phase. The following is the optional keyword and the dimensions of the array that must be specified in order to define the objective function:
LP 
${\mathbf{cvec}}\left({\mathbf{n}}\right)$ required. 
For problems of type FP, the objective function is omitted and
cvec is not referenced. The following keywords are also acceptable. The minimum abbreviation of each keyword is underlined.
$a$ 
Option 
Linear 
LP 
Feasible 
FP 
Description of Monitoring Information
This section describes the long line of output (
$\text{}>80$ characters) which forms part of the monitoring information produced by
nag_opt_lp_solve (e04mf). (See also the description of the optional parameters
Monitoring File and
Print Level.) You can control the level of printed output.
To aid interpretation of the printed results, the following convention is used for numbering the constraints: indices $1$ through $n$ refer to the bounds on the variables, and indices $n+1$ through $n+{m}_{L}$ refer to the general constraints. When the status of a constraint changes, the index of the constraint is printed, along with the designation L (lower bound), U (upper bound), E (equality), F (temporarily fixed variable) or A (artificial constraint).
When
${\mathbf{Print\; Level}}\ge 5$ and
${\mathbf{Monitoring\; File}}\ge 0$, the following line of output is produced at every iteration on the unit number specified by optional parameter
Monitoring File. In all cases, the values of the quantities printed are those in effect
on completion of the given iteration.
Itn 
is the iteration count.

Jdel 
is the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted.

Jadd 
is the index of the constraint added to the working set. If Jadd is zero, no constraint was added.

Step 
is the step taken along the computed search direction. If a constraint is added during the current iteration, Step will be the step to the nearest constraint. When the problem is of type LP, the step can be greater than one during the optimality phase.

Ninf 
is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.

Sinf/Objective 
is the value of the current objective function. If $x$ is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If $x$ is feasible, Objective is the value of the objective function of (1). The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point. During the optimality phase the value of the objective function will be nonincreasing. During the feasibility phase the number of constraint infeasibilities will not increase until either a feasible point is found or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.

Bnd 
is the number of simple bound constraints in the current working set.

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

Art 
is the number of artificial constraints in the working set, i.e., the number of columns of ${Z}_{A}$ (see Choosing the Initial Working Set).

Zr 
is the number of columns of ${Z}_{R}$ (see Definition of Search Direction). Zr is the dimension of the subspace in which the objective function is currently being minimized. The value of Zr is the number of variables minus the number of constraints in the working set; i.e., $\mathtt{Zr}=n\left(\mathtt{Bnd}+\mathtt{Lin}+\mathtt{Art}\right)$.The value of ${n}_{Z}$, the number of columns of $Z$ (see Definition of Search Direction) can be calculated as ${n}_{Z}=n\left(\mathtt{Bnd}+\mathtt{Lin}\right)$. A zero value of ${n}_{Z}$ implies that $x$ lies at a vertex of the feasible region.

Norm Gz 
is $\Vert {Z}_{R}^{\mathrm{T}}{g}_{\mathrm{FR}}\Vert $, the Euclidean norm of the reduced gradient with respect to ${Z}_{R}$. During the optimality phase, this norm will be approximately zero after a unit step.

NOpt 
is the number of nonoptimal Lagrange multipliers at the current point. NOpt is not printed if the current $x$ is infeasible or no multipliers have been calculated. At a minimizer, NOpt will be zero.

Min Lm 
is the value of the Lagrange multiplier associated with the deleted constraint. If Min Lm is negative, a lower bound constraint has been deleted, if Min Lm is positive, an upper bound constraint has been deleted. If no multipliers are calculated during a given iteration Min Lm will be zero.

Cond T 
is a lower bound on the condition number of the working set.

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