e04mf: FL CL CPP AD PY MB

NAG CL Interface
e04mfc (lp_solve)

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1 Purpose

e04mfc solves general linear programming problems. It is not intended for large sparse problems.

2 Specification

#include <nag.h>

void	e04mfc (Integer n, Integer nclin, const double a[], Integer tda, const double bl[], const double bu[], const double cvec[], double x[], double objf, Nag_E04_Opt options, Nag_Comm comm, NagError fail)

The function may be called by the names: e04mfc, nag_opt_lp_solve or nag_opt_lp.

3 Description

e04mfc is designed to solve Linear Programming (LP) problems of the form

\underset{x \in R^{n}}{minimize} c^{T} x subject to l \leq {\begin{matrix} x \\ A x \end{matrix}} \leq u,

where

c

is an

n

element vector and

A

is an

m_{lin} \times n

matrix.

The function allows the linear objective function to be omitted in which case a feasible point (FP) for the set of constraints is sought.

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

u

can be set to special values that will be treated as

- \infty

+ \infty

. (See the description of the optional parameter

options . inf_bound

in Section 12.2).

You must supply an initial estimate of the solution.

Details about the algorithm are described in Section 11, but it is not necessary to read this more advanced section before using e04mfc.

4 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 86-1 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 anti-cycling procedure for linearly constrained optimization Math. Programming 45 437–474

Gill P E, Murray W, Saunders M A and Wright M H (1991) Inertia-controlling methods for general quadratic programming SIAM Rev. 33 1–36

Gill P E, Murray W and Wright M H (1991) Numerical Linear Algebra and Optimization (Volume 1) Addison Wesley, Redwood City, California.

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the number of variables.

Constraint: $n > 0$ .
2: $nclin$ – Integer Input: On entry: $m_{lin}$ , the number of general linear constraints.

Constraint: $nclin \geq 0$ .
3: $a [nclin \times tda]$ – const double Input: Note: the $(i, j)$ th element of the matrix $A$ is stored in $a [(i - 1) \times tda + j - 1]$ .

On entry: the $i$ th row of a must contain the coefficients of the $i$ th general linear constraint (the $i$ th row of $A$ ), for $i = 1, 2, \dots, m_{lin}$ .
If $nclin = 0$ then the array a is not referenced.
4: $tda$ – Integer Input: On entry: the stride separating matrix column elements in the array a.

Constraint: if $nclin > 0$ , $tda \geq n$
5: $bl [n + nclin]$ – const double Input
6: $bu [n + nclin]$ – const double Input: On entry: bl must contain the lower bounds and bu the upper bounds, for all the constraints in the following order. The first $n$ elements of each array must contain the bounds on the variables, and the next $m_{lin}$ elements the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e., $l_{j} = - \infty$ ), set $bl [j - 1] \leq - options . inf_bound$ , and to specify a nonexistent upper bound (i.e., $u_{j} = + \infty$ ), set $bu [j - 1] \geq options . inf_bound$ ; here $options . inf_bound$ is the value of the optional parameter $options . inf_bound$ , whose default value is $10^{20}$ (see Section 12.2). To specify the $j$ th constraint as an equality, set $bl [j - 1] = bu [j - 1] = β$ , say, where $| β | < options . inf_bound$ .

Constraint: $bl [j - 1] \leq bu [j - 1]$ , for $j = 1, 2, \dots, n + nclin - 1$ .
7: $cvec [n]$ – const double Input: On entry: the coefficients of the objective function when the problem is of type $options . prob = Nag_LP$ . The problem type is specified by the optional parameter $options . prob$ (see Section 12.2) and the values $options . prob = Nag_LP$ or $Nag_FP$ represent linear programming problem and feasible point problem respectively. $options . prob = Nag_LP$ is the default problem type for e04mfc.
If the problem type $options . prob = Nag_FP$ is specified then cvec is not referenced and a NULL pointer may be given.
8: $x [n]$ – double Input/Output: On entry: an initial estimate of the solution.

On exit: the point at which e04mfc terminated. If $fail . code = NE_NOERROR$ , NW_SOLN_NOT_UNIQUE or NW_NOT_FEASIBLE, x contains an estimate of the solution.
9: $objf$ – double * Output: On exit: the value of the objective function at $x$ if $x$ is feasible, or the sum of infeasibilities at $x$ otherwise. If the problem is of type $options . prob = Nag_FP$ and $x$ is feasible, objf is set to zero.
10: $options$ – Nag_E04_Opt * Input/Output: On entry/exit: a pointer to a structure of type Nag_E04_Opt whose members are optional parameters for e04mfc. These structure members offer the means of adjusting some of the argument values of the algorithm and on output will supply further details of the results. A description of the members of options is given below in Section 12.2. Some of the results returned in options can be used by e04mfc to perform a ‘warm start’ if it is re-entered (see the member $options . start$ in Section 12.2).
If any of these optional parameters are required, then the structure options should be declared and initialized by a call to e04xxc immediately before being supplied as an argument to e04mfc.
11: $comm$ – Nag_Comm * Input/Output: Note: comm is a NAG defined type (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

On entry/exit: structure containing pointers for user communication with an optional user-defined printing function. See Section 12.3.1 for details. If you do not need to make use of this communication feature then the null pointer NAGCOMM_NULL may be used in the call to e04mfc.
12: $fail$ – NagError * Input/Output: The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

If one of NE_INT_ARG_LT, NE_2_INT_ARG_LT, NE_OPT_NOT_INIT, NE_BAD_PARAM, NE_INVALID_INT_RANGE_1, NE_INVALID_INT_RANGE_2, NE_INVALID_REAL_RANGE_FF, NE_INVALID_REAL_RANGE_F, NE_CVEC_NULL, NE_WARM_START, NE_BOUND, NE_BOUND_LCON, NE_STATE_VAL and NE_ALLOC_FAIL occurs, no values will have been assigned to objf, or to $options . ax$ and $options . lambda$ . x and $options . state$ will be unchanged.
NE_2_INT_ARG_LT: On entry, $tda = ⟨ value ⟩$ while $n = ⟨ value ⟩$ . These arguments must satisfy $tda \geq n$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $options . print_level$ had an illegal value.

On entry, argument $options . prob$ had an illegal value.

On entry, argument $options . start$ had an illegal value.
NE_BOUND: The lower bound for variable $⟨ value ⟩$ (array element $bl [⟨ value ⟩]$ ) is greater than the upper bound.
NE_BOUND_LCON: The lower bound for linear constraint $⟨ value ⟩$ (array element $bl [⟨ value ⟩]$ ) is greater than the upper bound.
NE_CVEC_NULL: $options . prob = ⟨ value ⟩$ but argument $cvec =$ NULL.
NE_INT_ARG_LT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

On entry, $nclin = ⟨ value ⟩$ .
Constraint: $nclin \geq 0$ .
NE_INVALID_INT_RANGE_1: Value $⟨ value ⟩$ given to $options . fcheck$ not valid. Correct range is $options . fcheck \geq 1$ .

Value $⟨ value ⟩$ given to $options . max_iter$ not valid. Correct range is $options . max_iter \geq 0$ .
NE_INVALID_INT_RANGE_2: Value $⟨ value ⟩$ given to $options . reset_ftol$ not valid. Correct range is $0 < options . reset_ftol < 10000000$ .
NE_INVALID_REAL_RANGE_F: Value $⟨ value ⟩$ given to $options . ftol$ not valid. Correct range is $options . ftol > 0.0$ .

Value $⟨ value ⟩$ given to $options . inf_bound$ not valid. Correct range is $options . inf_bound > 0.0$ .

Value $⟨ value ⟩$ given to $options . inf_step$ not valid. Correct range is $options . inf_step > 0.0$ .
NE_INVALID_REAL_RANGE_FF: Value $⟨ value ⟩$ given to $options . crash_tol$ not valid. Correct range is $0.0 \leq options . crash_tol \leq 1.0$ .
NE_NOT_APPEND_FILE: Cannot open file $⟨ string ⟩$ for appending.
NE_NOT_CLOSE_FILE: Cannot close file $⟨ string ⟩$ .
NE_OPT_NOT_INIT: options structure not initialized.
NE_STATE_VAL: $options . state [⟨ value ⟩]$ is out of range. $options . state [⟨ value ⟩] = ⟨ value ⟩$ .
NE_UNBOUNDED: Solution appears to be unbounded.
This value of $fail . code$ implies that a step as large as $options . inf_step$ would have to be taken in order to continue the algorithm. This situation can occur only when the problem is of type $options . prob = Nag_LP$ and at least one variable has no upper or lower bound.
NE_WARM_START: $options . start = Nag_Warm$ but pointer $options . state =$ NULL.
NE_WRITE_ERROR: Error occurred when writing to file $⟨ string ⟩$ .
NW_NOT_FEASIBLE: No feasible point was found for the linear constraints.
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, if the feasibility tolerance for each violated constraint exceeds its Residual at the final point. You should check that there are no constraint redundancies. If the data for the constraints are accurate only to the absolute precision $σ$ , you should ensure that the value of the optional parameter $options . ftol$ is greater than $σ$ . For example, if all elements of $A$ are of order unity and are accurate only to three decimal places, the optional parameter $options . ftol$ should be at least $10^{−3}$ .
NW_OVERFLOW_WARN: Serious ill-conditioning in the working set after adding constraint $⟨ value ⟩$ . Overflow may occur in subsequent iterations.
If overflow occurs preceded by this warning then serious ill-conditioning has probably occurred in the working set when adding a constraint. It may be possible to avoid the difficulty by increasing the magnitude of the optional parameter $options . ftol$ and re-running the program. If the message recurs even after this change, the offending linearly dependent constraint $j$ must be removed from the problem.
NW_SOLN_NOT_UNIQUE: Optimal solution is not unique.
$x$ is a weak local minimum (the projected gradient is negligible, the Lagrange multipliers are optimal but there is a small multiplier). This means that the solution $x$ is not unique.
NW_TOO_MANY_ITER: The maximum number of iterations, $⟨ value ⟩$ , have been performed.
The value of the optional parameter $options . max_iter$ may be too small. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), increase the value of $options . max_iter$ and rerun e04mfc (possibly using the $options . start = Nag_Warm$ facility to specify the initial working set).

7 Accuracy

e04mfc implements a numerically stable active set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

e04mfc is not threaded in any implementation.

9 Further Comments

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. (1986) for further information and advice.

10 Example

This example is a portfolio investment problem taken from Gill et al. (1991). The objective function to be minimized is

- 5 x_{1} - 2 x_{3}

subject to the bounds

\begin{matrix} x_{1} \geq −75 \\ x_{2} \geq −1000 \\ x_{3} \geq −25 \end{matrix}

and the general constraints

\begin{matrix} 20 x_{1} + 2 x_{2} + 100 x_{3} = 0 \\ 18 x_{1} + 3 x_{2} + 102 x_{3} \geq −600 \\ 15 x_{1} - \frac{1}{2} x_{2} - 25 x_{3} \geq 0 \\ −5 x_{1} + \frac{3}{2} x_{2} - 25 x_{3} \geq −500 \\ −5 x_{1} - \frac{1}{2} x_{2} + 75 x_{3} \geq −1000 \end{matrix}

The initial point, which is feasible, is

x_{0} = {(10.0, 20.0, 100.0)}^{T} .

The solution is

x^{*} = {(75.0, - 250.0, - 10.0)}^{T} .

Three general constraints are active at the solution, the bound constraints are all inactive.

The options structure is declared and initialized by e04xxc, a value is assigned directly to option

options . inf_bound

and e04mfc is then called. On successful return two further options are read from a data file by use of e04xyc and the problem is re-run. The memory freeing function e04xzc is used to free the memory assigned to the pointers in the options structure. You must not use the standard C function free() for this purpose.

11 Further Description

This section gives a detailed description of the algorithm used in e04mfc. This, and possibly the next section, Section 12, may be omitted if the more sophisticated features of the algorithm and software are not currently of interest.

11.1 Overview

e04mfc is based on an inertia-controlling method due to Gill and Murray (1978) and is described in detail by Gill et al. (1991). Here the main features of the method are summarised. Where possible, explicit reference is made to the names of variables that are arguments of e04mfc or appear in the printed output. e04mfc 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 two-phase 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 LP case when

m_{lin} \leq 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

\bar{x}

is defined by

\bar{x} = x + α p,

(1)

where the steplength

α

is a non-negative 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

options . ftol

; see Section 12.2). 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 component of the search direction to zero. Thus, the associated variable is fixed and the 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 components of the search direction are zero, the columns of

A

corresponding to fixed variables may be ignored.

Let

m_{w}

denote the number of general constraints in the working set and let

n_{f x}

denote the number of variables fixed at one of their bounds (

m_{w}

and

n_{f x}

are the quantities Lin and Bnd in the printed output from e04mfc). Similarly, let

n_{f r} (n_{f r} = n - n_{f x})

denote the number of free variables. At every iteration, the variables are re-ordered so that the last

n_{f x}

variables are fixed, with all other relevant vectors and matrices ordered accordingly.

11.2 Definition of the Search Direction

Let

A_{f r}

denote the

m_{w} \times n_{f r}

sub-matrix of general constraints in the working set corresponding to the free variables, and let

p_{f r}

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

A_{f r} p_{f r} = 0 .

(2)

In order to compute

p_{f r}

, the

T Q

factorization of

A_{f r}

is used:

A_{f r} Q_{f r} = (\begin{matrix} 0 & T \end{matrix}),

(3)

where

T

is a nonsingular

m_{w} \times m_{w}

upper triangular matrix (i.e.,

t_{i j} = 0

i > j

), and the nonsingular

n_{f r} \times n_{f r}

matrix

Q_{f r}

is the product of orthogonal transformations (see Gill et al. (1984)). If the columns of

Q_{f r}

are partitioned so that

Q_{f r} = (\begin{matrix} Z & Y \end{matrix}),

where

Y

n_{f r} \times m_{w}

, then the

n_{z}

(n_{z} = n_{f r} - m_{w})

columns of

Z

form a basis for the null space of

A_{f r}

. Let

n_{r}

be an integer such that

0 \leq n_{r} \leq 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 Nrz in the printed output from e04mfc. In many cases,

Z_{r}

will include all the columns of

Z

.) The direction

p_{f r}

will satisfy (2) if

p_{f r} = Z_{r} p_{r},

(4)

where

p_{r}

is any

n_{r}

-vector.

11.3 The Main Iteration

Let

Q

denote the

n \times n

matrix

Q = (\begin{matrix} Q_{f r} \\ I_{f x} \end{matrix}),

where

I_{f x}

is the identity matrix of order

n_{f x}

. Let

g_{q}

denote the transformed gradient

g_{q} = Q^{T} c

and let the vector of first

n_{r}

elements of

g_{q}

be denoted by

g_{r}

. The quantity

g_{r}

is known as the reduced gradient of

c^{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 non-vertices 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

λ_{c}

and

λ_{b}

for the general and bound constraints are defined from the equations

A_{f r}^{T} λ_{c} = g_{f r} and λ_{b} = g_{f x} - A_{f x}^{T} λ_{c} .

(5)

Given a positive constant

δ

of the order of the machine precision, a Lagrange multiplier

λ_{j}

corresponding to an inequality constraint in the working set is said to be optimal if

λ_{j} \leq δ

when the associated constraint is at its upper bound, or if

λ_{j} \geq - δ

when the associated constraint is at its lower bound. If a multiplier is non-optimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint (with index Jdel; see Section 12.3) 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 e04mfc will continue until the minimum value of the sum of infeasibilities has been found. At this point, the Lagrange multiplier

λ_{j}

corresponding to an inequality constraint in the working set will be such that

- (1 + δ) \leq λ_{j} \leq δ

when the associated constraint is at its upper bound, and

- δ \leq λ_{j} \leq (1 + δ)

when the associated constraint is at its lower bound. Lagrange multipliers for equality constraints will satisfy

| λ_{j} | \leq 1 + δ

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_{f r}

: if the status of a general constraint changes, a row of

A_{f r}

is altered; if a bound constraint enters or leaves the working set, a column of

A_{f r}

changes. Explicit representations are recurred of the matrices

T

and

Q_{f r}

and of vectors

Q^{T} g

, and

Q^{T} c

One of the most important features of e04mfc 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

T Q

factor

T

(the printed value Cond T; see Section 12.3). In constructing the initial working set, constraints are excluded that would result in a large value of Cond T.

e04mfc 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 anti-cycling 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

α_{m}

denote the maximum step at which

x + α_{m} p

does not violate any constraint by more than its feasibility tolerance. All constraints at a distance

α (α \leq α_{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.

11.4 Choosing the Initial Working Set

Let

Z

be partitioned as

Z = (Z_{r} Z_{a})

. A working set for which

Z_{r}

defines the null space can be obtained by including the rows of

Z_{a}^{T}

as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by

Z_{r}

, as described in Section 11.2.

The artificially augmented working set is given by

{\bar{A}}_{f r} = (\begin{matrix} Z_{a}^{T} \\ A_{fr} \end{matrix}),

(6)

so that

p_{f r}

will satisfy

A_{f r} p_{f r} = 0

and

Z_{a}^{T} p_{f r} = 0

. By definition of the

T Q

factorization,

{\bar{A}}_{f r}

automatically satisfies the following:

{\bar{A}}_{FR} Q_{FR} = (\begin{matrix} Z_{A}^{T} \\ A_{FR} \end{matrix}) Q_{FR} = (\begin{matrix} Z_{A}^{T} \\ A_{FR} \end{matrix}) (\begin{matrix} Z_{R} & Z_{A} & Y \end{matrix}) = (\begin{matrix} 0 & \bar{T} \end{matrix}),

where

\bar{T} = (\begin{matrix} I & 0 \\ 0 & T \end{matrix}),

and hence the

T Q

factorization of (6) is available trivially from

T

and

Q_{f r}

without additional expense.

The matrix

Z_{a}

is not kept fixed, since its role is purely to define an appropriate null space; the

T Q

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}^{T} g_{f r} = 0

, since this simply involves repartitioning

Q_{f r}

. The ‘artificial’ multiplier vector associated with the rows of

Z_{a}^{T}

is equal to

Z_{a}^{T} g_{f r}

, 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 printed output (see Section 12.3).

The number of columns in

Z_{a}

and

Z_{r}

and the Euclidean norm of

Z_{r}^{T} g_{f r}

, appear in the printed output as Nart, Nrz and Norm Gz (see Section 12.3).

Under some circumstances, a different type of artificial constraint is used when solving a linear program. Although the algorithm of e04mfc 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_{lin} \leq 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 printed output (see Section 12.3).

12 Optional Parameters

A number of optional input and output arguments to e04mfc are available through the structure argument options, type Nag_E04_Opt. An argument may be selected by assigning an appropriate value to the relevant structure member; those arguments not selected will be assigned default values. If no use is to be made of any of the optional parameters you should use the NAG defined null pointer, E04_DEFAULT, in place of options when calling e04mfc; the default settings will then be used for all arguments.

Before assigning values to options directly the structure must be initialized by a call to the function e04xxc. Values may then be assigned to the structure members in the normal C manner.

After return from e04mfc, the options structure may only be re-used for future calls of e04mfc if the dimensions of the new problem are the same. Otherwise, the structure must be cleared by a call of e04xzc) and re-initialized by a call of e04xxc before future calls. Failure to do this will result in unpredictable behaviour.

Option settings may also be read from a file using the function e04xyc in which case initialization of the options structure will be performed automatically if not already done. Any subsequent direct assignment to the options structure must not be preceded by initialization.

If assignment of functions and memory to pointers in the options structure is required, this must be done directly in the calling program; they cannot be assigned using e04xyc.

12.1 Optional Parameter Checklist and Default Values

For easy reference, the following list shows the members of options which are valid for e04mfc together with their default values where relevant. The number

ε

is a generic notation for machine precision (see X02AJC).

Nag_ProblemType prob	$Nag_LP$
Nag_Start start	$Nag_Cold$
Boolean list	Nag_TRUE
Nag_PrintType print_level	Nag_Soln_Iter
char outfile[512]	stdout
void (*print_fun)()	NULL
Integer max_iter	$\max (50, 5 (n + nclin))$
double crash_tol	0.01
double ftol	$\sqrt{ε}$
double optim_tol	$ε^{0.8}$
Integer reset_ftol	10000
Integer fcheck	50
double inf_bound	$10^{20}$
double inf_step	$\max (options . inf_bound, 10^{20})$
Integer *state	size $n + nclin$
double *ax	size nclin
double *lambda	size $n + nclin$
Integer iter

12.2 Description of the Optional Parameters

prob – Nag_ProblemType

Default

= Nag_LP

On entry: specifies the problem type. The following are the two possible values of

options . prob

and the size of the array cvec that is required to define the objective function:

$Nag_FP$	cvec not accessed;
$Nag_LP$	$cvec [n]$ required;

Nag_FP

denotes a feasible point problem and

Nag_LP

a linear programming problem.

Constraint:

options . prob = Nag_FP

Nag_LP

start – Nag_Start

Default

= Nag_Cold

On entry: specifies how the initial working set is chosen. With

options . start = Nag_Cold

, e04mfc 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

options . crash_tol

; see below).

With

options . start = Nag_Warm

, you must provide a valid definition of every element of the array pointer

options . state

(see below for the definition of this member of options). e04mfc will override your specification of

options . state

if necessary, so that a poor choice of the working set will not cause a fatal error.

options . start = Nag_Warm

will be advantageous if a good estimate of the initial working set is available – for example, when e04mfc is called repeatedly to solve related problems.

Constraint:

options . start = Nag_Cold

Nag_Warm

list – Nag_Boolean

Default

= Nag_TRUE

On entry: if

options . list = Nag_TRUE

the argument settings in the call to e04mfc will be printed.

print_level – Nag_PrintType

Default

= Nag_Soln_Iter

On entry: the level of results printout produced by e04mfc. The following values are available:

$Nag_NoPrint$	No output.
$Nag_Soln$	The final solution.
$Nag_Iter$	One line of output for each iteration.
$Nag_Iter_Long$	A longer line of output for each iteration with more information (line exceeds 80 characters).
$Nag_Soln_Iter$	The final solution and one line of output for each iteration.
$Nag_Soln_Iter_Long$	The final solution and one long line of output for each iteration (line exceeds 80 characters).
$Nag_Soln_Iter_Const$	As $Nag_Soln_Iter_Long$ with the Lagrange multipliers, the variables $x$ , the constraint values $A x$ and the constraint status also printed at each iteration.
$Nag_Soln_Iter_Full$	As $Nag_Soln_Iter_Const$ with the diagonal elements of the upper triangular matrix $T$ associated with the $T Q$ factorization 3 of the working set.

Details of each level of results printout are described in Section 12.3.

Constraint:

options . print_level = Nag_NoPrint

Nag_Soln

Nag_Iter

Nag_Soln_Iter

Nag_Iter_Long

Nag_Soln_Iter_Long

Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

outfile – const char[512]

Default

= stdout

On entry: the name of the file to which results should be printed. If

options . outfile [0] =' \0'

then the stdout stream is used.

print_fun – pointer to function

Default

=

NULL

On entry: printing function defined by you; the prototype of

options . print_fun

void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);

See Section 12.3.1 below for further details.

max_iter – Integer

Default

= \max (50, 5 (n + nclin))

On entry:

options . max_iter

specifies the maximum number of iterations to be performed by e04mfc.

If you wish to check that a call to e04mfc is correct before attempting to solve the problem in full then

options . max_iter

may be set to

0

. No iterations will then be performed but the initialization stages prior to the first iteration will be processed and a listing of argument settings output if

options . list = Nag_TRUE

(the default setting).

Constraint:

options . max_iter \geq 0

crash_tol – double

Default

= 0.01

On entry:

options . crash_tol

is used in conjunction with the optional parameter

options . start

. When

options . start

has the default setting, i.e.,

options . start = Nag_Cold

, e04mfc selects an initial working set. The initial working set will include bounds or general inequality constraints that lie within

options . crash_tol

of their bounds. In particular, a constraint of the form

a_{j}^{T} x \geq l

will be included in the initial working set if

| a_{j}^{T} x - l | \leq options . crash_tol \times (1 + | l |)

Constraint:

0.0 \leq options . crash_tol \leq 1.0

ftol – double

Default

= \sqrt{ε}

On entry:

options . ftol

defines the maximum acceptable 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

options . ftol

10^{−6}

e04mfc attempts to find a feasible solution before optimizing the objective function. If the sum of infeasibilities cannot be reduced to zero, e04mfc finds 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

options . ftol

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

options . ftol

Constraint:

options . ftol > 0.0

optim_tol – double

ε^{0.8}

On entry:

options . optim_tol

defines the tolerance used to determine whether the bounds and generated constraints have the correct sign for the solution to be judged optimal.

Constraint:

options . optim_tol \neq ε

reset_ftol – Integer

Default

= 10000

On entry: this option is part of an anti-cycling 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

options . ftol

δ

. Over a period of

options . reset_ftol

iterations, the feasibility tolerance actually used by e04mfc increases from

0.5 δ

δ

(in steps of

0.5 δ / options . reset_ftol

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 current feasibility tolerance is reinitialized to

0.5 δ

If a problem requires more than

options . reset_ftol

iterations, the resetting procedure is invoked and a new cycle of

options . reset_ftol

iterations is started. (The decision to resume the feasibility phase or optimality phase is based on comparing any constraint infeasibilities with

δ

The resetting procedure is also invoked when e04mfc reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any nontrivial adjustments are made, iterations are continued.

Constraint:

0 < options . reset_ftol < 10000000

fcheck – Integer

Default

= 50

On entry: every

options . fcheck

iterations, 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 re-factorized and the variables are recomputed to satisfy the constraints more accurately.

Constraint:

options . fcheck \geq 1

inf_bound – double

Default

= 10^{20}

On entry:

options . inf_bound

defines the ‘infinite’ bound in the definition of the problem constraints. Any upper bound greater than or equal to

options . inf_bound

will be regarded as

+ \infty

(and similarly for a lower bound less than or equal to

- options . inf_bound

Constraint:

options . inf_bound > 0.0

inf_step – double

Default

= \max (options . inf_bound, 10^{20})

On entry:

options . inf_step

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

options . prob = Nag_LP

). If the change in

x

during an iteration would exceed the value of

options . inf_step

, the objective function is considered to be unbounded below in the feasible region.

Constraint:

options . inf_step > 0.0

state – Integer *

Default memory

= n + nclin

On entry:

options . state

need not be set if the default option of

options . start = Nag_Cold

is used as

n + nclin

values of memory will be automatically allocated by e04mfc.

If the option

options . start = Nag_Warm

has been chosen,

options . state

must point to a minimum of

n + nclin

elements of memory. This memory will already be available if the options structure has been used in a previous call to e04mfc from the calling program, using the same values of n and nclin and

options . start = Nag_Cold

. If a previous call has not been made sufficient memory must be allocated to

options . state

by you.

When a warm start is chosen

options . state

should specify the desired status of the constraints at the start of the feasibility phase. More precisely, the first

n

elements of

options . state

refer to the upper and lower bounds on the variables, and the next

m_{lin}

elements refer to the general linear constraints (if any). Possible values for

options . state [j - 1]

are as follows:

$options . state [j - 1]$	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 should only be specified if $bl [j - 1] = bu [j - 1]$ . The values $1$ , 2 or 3 all have the same effect when $bl [j - 1] = bu [j - 1]$ .

The values

−2

−1

and 4 are also acceptable but will be reset to zero by the function. In particular, if e04mfc has been called previously with the same values of n and nclin,

options . state

already contains satisfactory information. (See also the description of the optional parameter

options . start

). The function also adjusts (if necessary) the values supplied in x to be consistent with the values supplied in

options . state

On exit: if e04mfc exits with

fail . code = NE_NOERROR

, NW_SOLN_NOT_UNIQUE or NW_NOT_FEASIBLE, the values in

options . state

indicate the status of the constraints in the working set at the solution. Otherwise,

options . state

indicates the composition of the working set at the final iterate. The significance of each possible value of

options . state [j - 1]

is as follows:

$options . state [j - 1]$	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.
$0$	The constraint is satisfied to within the feasibility tolerance, but is not in the working set.
$1$	This inequality constraint is included in the working set at its lower bound.
$2$	This inequality constraint is included in the working set at its upper bound.
$3$	This constraint is included in the working set as an equality. This value of $options . state$ can occur only when $bl [j - 1] = bu [j - 1]$ .
$4$	This corresponds to optimality being declared with $x [j - 1]$ being temporarily fixed at its current value. This value of $options . state$ can only occur when $fail . code = NW_SOLN_NOT_UNIQUE$ .

ax – double *

Default memory

= nclin

On entry: nclin values of memory will be automatically allocated by e04mfc and this is the recommended method of use of

options . ax

. However you may supply memory from the calling program.

On exit: if

nclin > 0

options . ax

points to the final values of the linear constraints

A x

lambda – double *

Default memory

= n + nclin

On entry:

n + nclin

values of memory will be automatically allocated by e04mfc and this is the recommended method of use of

options . lambda

. However you may supply memory from the calling program.

On exit: 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_{lin}

elements contain the multipliers for the general linear constraints (if any). If

options . state [j - 1] = 0

(i.e., constraint

j

is not in the working set),

options . lambda [j - 1]

is zero. If

x

is optimal,

options . lambda [j - 1]

should be non-negative if

options . state [j - 1] = 1

, non-positive if

options . state [j - 1] = 2

and zero if

options . state [j - 1] = 4

iter – Integer

On exit: the total number of iterations performed in the feasibility phase and (if appropriate) the optimality phase.

12.3 Description of Printed Output

The level of printed output can be controlled with the structure members

options . list

and

options . print_level

(see Section 12.2). If

options . list = Nag_TRUE

then the argument values to e04mfc are listed, whereas the printout of results is governed by the value of

options . print_level

. The default of

options . print_level = Nag_Soln_Iter

provides a single line of output at each iteration and the final result. This section describes all of the possible levels of results printout available from e04mfc.

The convention for numbering the constraints in the iteration results is that indices 1 to

n

refer to the bounds on the variables, and indices

n + 1

n + m_{lin}

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

options . print_level = Nag_Iter

Nag_Soln_Iter

the following line of output is produced on completion of each iteration.

Itn	the iteration count.
Jdel	the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted.
Jadd	the index of the constraint added to the working set. If Jadd is zero, no constraint was added.
Step	the step taken along the computed search direction. If a constraint is added during the current iteration (i.e., Jadd is positive), Step will be the step to the nearest constraint. During the optimality phase, the step can be greater than $1$ only if the reduced Hessian is not positive definite.
Ninf	the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Obj	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, Obj is the value of the objective function. 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 non-increasing. 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	the number of simple bound constraints in the current working set.
Lin	the number of general linear constraints in the current working set.
Nart	the number of artificial constraints in the working set, i.e., the number of columns of $Z_{a}$ (see Section 11). At the start of the optimality phase, Nart provides an estimate of the number of non-positive eigenvalues in the reduced Hessian.
Nrz	is the number of columns of $Z_{r}$ (see Section 11). Nrz is the dimension of the subspace in which the objective function is currently being minimized. The value of Nrz is the number of variables minus the number of constraints in the working set; i.e., $Nrz = n - (Bnd + Lin + Nart)$ .
	The value of $n_{z}$ , the number of columns of $Z$ (see Section 11) can be calculated as $n_{z} = n - (Bnd + Lin)$ . A zero value of $n_{z}$ implies that $x$ lies at a vertex of the feasible region.
Norm Gz	$‖ Z_{r}^{T} g_{f r} ‖$ , 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.

options . print_level = Nag_Iter_Long

Nag_Soln_Iter_Long

Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

the line of printout is extended to give the following information. (Note this longer line extends over more than 80 characters).

NOpt	is the number of non-optimal 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.

When

options . print_level = Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

more detailed results are given at each iteration. For the setting

options . print_level = Nag_Soln_Iter_Const

additional values output are:

Value of x	the value of $x$ currently held in x.
State	the current value of $options . state$ associated with $x$ .
Value of Ax	the value of $A x$ currently held in $options . ax$ .
State	the current value of $options . state$ associated with $A x$ .

Also printed are the Lagrange Multipliers for the bound constraints, linear constraints and artificial constraints.

options . print_level = Nag_Soln_Iter_Full

then the diagonal of

T

and

Z_{r}

are also output at each iteration.

When

options . print_level = Nag_Soln

Nag_Soln_Iter

Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

the final printout from e04mfc includes a listing of the status of every variable and constraint. The following describes the printout for each variable.

Varbl	the name (V) and index $j$ , for $j = 1, 2, \dots, n$ , of the variable.
State	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.
Value	the value of the variable at the final iteration.
Lower bound	the lower bound specified for the variable. (None indicates that $bl [j - 1] \leq - options . inf_bound$ .)
Upper bound	the upper bound specified for the variable. (None indicates that $bu [j - 1] \geq options . inf_bound$ .)
Lagr mult	the value of the Lagrange multiplier for the associated bound constraint. This will be zero if State is FR. If $x$ is optimal, the multiplier should be non-negative if State is LL, and non-positive if State is UL.
Residual	the difference between the variable Value and the nearer of its bounds $bl [j - 1]$ and $bu [j - 1]$ .

The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, and with the following change in the heading:

LCon	the name (L) and index $j$ , for $j = 1, 2, \dots, m_{lin}$ of the constraint.

12.3.1 Output of results via a user-defined printing function

You may also specify your own print function for output of iteration results and the final solution by use of the

options . print_fun

function pointer, which has prototype

void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);

The rest of this section can be skipped if you wish to use the default printing facilities.

When a user-defined function is assigned to

options . print_fun

this will be called in preference to the internal print function of e04mfc. Calls to the user-defined function are again controlled by means of the

options . print_level

member. Information is provided through st and comm, the two structure arguments to

options . print_fun

comm \to it_prt = Nag_TRUE

then the results from the last iteration of e04mfc are set in the following members of st:

first – Nag_Boolean: Nag_TRUE on the first call to $options . print_fun$ .

iter – Integer: The number of iterations performed.

n – Integer: The number of variables.

nclin – Integer: The number of linear constraints.

jdel – Integer: Index of constraint deleted.

jadd – Integer: Index of constraint added.

step – double: The step taken along the current search direction.

ninf – Integer: The number of infeasibilities.

f – double: The value of the current objective function.

bnd – Integer: Number of bound constraints in the working set.

lin – Integer: Number of general linear constraints in the working set.

nart – Integer: Number of artificial constraints in the working set.

nrz – Integer: Number of columns of $Z_{r}$ .

norm_gz – double: Euclidean norm of the reduced gradient, $‖ Z_{r}^{T} g_{f r} ‖$ .

nopt – Integer: Number of non-optimal Lagrange multipliers.

min_lm – double: Value of the Lagrange multiplier associated with the deleted constraint.

condt – double: A lower bound on the condition number of the working set.

x – double *: x points to the n memory locations holding the current point $x$ .

ax – double *: $options . ax$ points to the nclin memory locations holding the current values $A x$ .

state – Integer *: $options . state$ points to the $n + nclin$ memory locations holding the status of the variables and general linear constraints. See Section 12.2 for a description of the possible status values.

t – double *: The upper triangular matrix $T$ with $st \to lin$ columns. Matrix element $i, j$ is held in $st \to t [(i - 1) * st \to tdt + j - 1]$ .

tdt – Integer: The trailing dimension for $st \to t$ .

comm \to new_lm = Nag_TRUE

then the Lagrange multipliers have been updated and the following members are set:

kx – Integer *: Indices of the bound constraints with associated multipliers. Value of $st \to kx [i - 1]$ is the index of the constraint with multiplier $st \to lambda [i - 1]$ , for $i = 1, 2, \dots, st \to bnd$ .

kactive – Integer *: Indices of the linear constraints with associated multipliers. Value of $st \to kactive [i - 1]$ is the index of the constraint with multiplier $st \to lambda [st \to bnd + i - 1]$ , for $i = 1, 2, \dots, st \to lin$ .

lambda – double *: The multipliers for the constraints in the working set. $options . lambda [i - 1]$ , for $i = 1, 2, \dots, st \to bnd$ hold the multipliers for the bound constraints while the multipliers for the linear constraints are held at indices $i - 1 = st \to bnd, \dots, st \to bnd + st \to lin$ .

gq – double *: $st \to gq [i - 1]$ , for $i = 1, 2, \dots, st \to nart$ hold the multipliers for the artificial constraints.

The following members of st are also relevant and apply when

comm \to it_prt

comm \to new_lm

is Nag_TRUE.

refactor – Nag_Boolean: Nag_TRUE if iterative refinement performed. See Section 11.3 and optional parameter $options . reset_ftol$ .

jmax – Integer: If $st \to refactor = Nag_TRUE$ then $st \to jmax$ holds the index of the constraint with the maximum violation.

errmax – double: If $st \to refactor = Nag_TRUE$ then $st \to errmax$ holds the value of the maximum violation.

moved – Nag_Boolean: Nag_TRUE if some variables moved to their bounds. See the optional parameter $options . reset_ftol$ .

nmoved – Integer: If $st \to moved = Nag_TRUE$ then $st \to nmoved$ holds the number of variables which were moved to their bounds.

rowerr – Nag_Boolean: Nag_TRUE if some constraints are not satisfied to within $options . ftol$ .

feasible – Nag_Boolean: Nag_TRUE when a feasible point has been found.

comm \to sol_prt = Nag_TRUE

then the final result from e04mfc is available and the following members of st are set:

iter – Integer: The number of iterations performed.

n – Integer: The number of variables.

nclin – Integer: The number of linear constraints.

x – double *: x points to the n memory locations holding the final point $x$ .

f – double *: The final objective function value or, if $x$ is not feasible, the sum of infeasibilities. If the problem is of type $options . prob = Nag_FP$ and $x$ is feasible then $st \to f$ is set to zero.

ax – double *: $options . ax$ points to the nclin memory locations holding the final values $A x$ .

state – Integer *: $st \to state$ points to the $n + nclin$ memory locations holding the final status of the variables and general linear constraints. See Section 12.2 for a description of the possible status values.

lambda – double *: $st \to lambda$ points to the $n + nclin$ final values of the Lagrange multipliers.

bl – double *: bl points to the $n + nclin$ lower bound values.

bu – double *: bu points to the $n + nclin$ upper bound values.

endstate – Nag_EndState

The state of termination of e04mfc. Possible values of

st \to endstate

and their correspondence to the exit value of

fail . code

are:

Value of $st \to endstate$	Value of $fail . code$
$Nag_Feasible$ and $Nag_Optimal$	NE_NOERROR
$Nag_Weakmin$	NW_SOLN_NOT_UNIQUE
$Nag_Unbounded$	NE_UNBOUNDED
$Nag_Infeasible$	NW_NOT_FEASIBLE
$Nag_Too_Many_Iter$	NW_TOO_MANY_ITER

The relevant members of the structure comm are:

it_prt – Nag_Boolean: Will be Nag_TRUE when the print function is called with the result of the current iteration.

sol_prt – Nag_Boolean: Will be Nag_TRUE when the print function is called with the final result.

new_lm – Nag_Boolean: Will be Nag_TRUE when the Lagrange multipliers have been updated.

user – double
iuser – Integer
p – Pointer: Pointers for communication of user information. If used they must be allocated memory either before entry to e04mfc or during a call to $options . print_fun$ . The type Pointer will be void * with a C compiler that defines void *.