e04mf solves general linear programming problems. It is not intended for large sparse problems.

Syntax

C#
public static void e04mf( int n, int nclin, double[,] a, double[] bl, double[] bu, double[] cvec, int[] istate, double[] x, out int iter, out double obj, double[] ax, double[] clamda, E04..::..e04mfOptions options, out int ifail )

public static void e04mf(
	int n,
	int nclin,
	double[,] a,
	double[] bl,
	double[] bu,
	double[] cvec,
	int[] istate,
	double[] x,
	out int iter,
	out double obj,
	double[] ax,
	double[] clamda,
	E04..::..e04mfOptions options,
	out int ifail
)

Visual Basic
Public Shared Sub e04mf ( _ n As Integer, _ nclin As Integer, _ a As Double(,), _ bl As Double(), _ bu As Double(), _ cvec As Double(), _ istate As Integer(), _ x As Double(), _ <OutAttribute> ByRef iter As Integer, _ <OutAttribute> ByRef obj As Double, _ ax As Double(), _ clamda As Double(), _ options As E04..::..e04mfOptions, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub e04mf ( _
	n As Integer, _
	nclin As Integer, _
	a As Double(,), _
	bl As Double(), _
	bu As Double(), _
	cvec As Double(), _
	istate As Integer(), _
	x As Double(), _
	<OutAttribute> ByRef iter As Integer, _
	<OutAttribute> ByRef obj As Double, _
	ax As Double(), _
	clamda As Double(), _
	options As E04..::..e04mfOptions, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void e04mf( int n, int nclin, array<double,2>^ a, array<double>^ bl, array<double>^ bu, array<double>^ cvec, array<int>^ istate, array<double>^ x, [OutAttribute] int% iter, [OutAttribute] double% obj, array<double>^ ax, array<double>^ clamda, E04..::..e04mfOptions^ options, [OutAttribute] int% ifail )

Visual C++

public:
static void e04mf(
	int n, 
	int nclin, 
	array<double,2>^ a, 
	array<double>^ bl, 
	array<double>^ bu, 
	array<double>^ cvec, 
	array<int>^ istate, 
	array<double>^ x, 
	[OutAttribute] int% iter, 
	[OutAttribute] double% obj, 
	array<double>^ ax, 
	array<double>^ clamda, 
	E04..::..e04mfOptions^ options, 
	[OutAttribute] int% ifail
)

F#
static member e04mf : n : int * nclin : int * a : float[,] * bl : float[] * bu : float[] * cvec : float[] * istate : int[] * x : float[] * iter : int byref * obj : float byref * ax : float[] * clamda : float[] * options : E04..::..e04mfOptions * ifail : int byref -> unit

static member e04mf : 
        n : int * 
        nclin : int * 
        a : float[,] * 
        bl : float[] * 
        bu : float[] * 
        cvec : float[] * 
        istate : int[] * 
        x : float[] * 
        iter : int byref * 
        obj : float byref * 
        ax : float[] * 
        clamda : float[] * 
        options : E04..::..e04mfOptions * 
        ifail : int byref -> unit

Parameters

n: Type: System..::..Int32
On entry: $n$ , the number of variables.

Constraint: $n > 0$ .

nclin: Type: System..::..Int32
On entry: $m_{L}$ , the number of general linear constraints.

Constraint: $nclin \geq 0$ .

a: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, dim2]
Note: dim1 must satisfy the constraint: $dim1 \geq \max (1, nclin)$
Note: the second dimension of the array a must be at least $n$ if $nclin > 0$ and at least $1$ if $nclin = 0$ .
On entry: the $i$ th row of a must contain the coefficients of the $i$ th general linear constraint, for $i = 1, 2, \dots, m_{L}$ .
If $nclin = 0$ , a is not referenced.

bl

Type: array<System..::..Double>[]()[][]

An array of size [

n + nclin

]

On entry: 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

bl [j - 1] \leq - bigbnd

, and to specify a nonexistent upper bound (i.e.,

u_{j} = + \infty

), set

bu [j - 1] \geq bigbnd

; the default value of

bigbnd

10^{20}

, but this may be changed by the optional parameter Infinite Bound Size. To specify the

j

th constraint as an equality, set

bl [j - 1] = bu [j - 1] = β

, say, where

|β| < bigbnd

Constraints:

$bl [j - 1] \leq bu [j - 1]$ , for $j = 1, 2, \dots, n + nclin$ ;
if $bl [j - 1] = bu [j - 1] = β$ , $|β| < bigbnd$ .

bu

Type: array<System..::..Double>[]()[][]

An array of size [

n + nclin

]

On entry: 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

bl [j - 1] \leq - bigbnd

, and to specify a nonexistent upper bound (i.e.,

u_{j} = + \infty

), set

bu [j - 1] \geq bigbnd

; the default value of

bigbnd

10^{20}

, but this may be changed by the optional parameter Infinite Bound Size. To specify the

j

th constraint as an equality, set

bl [j - 1] = bu [j - 1] = β

, say, where

|β| < bigbnd

Constraints:

$bl [j - 1] \leq bu [j - 1]$ , for $j = 1, 2, \dots, n + nclin$ ;
if $bl [j - 1] = bu [j - 1] = β$ , $|β| < bigbnd$ .

cvec: Type: array<System..::..Double>[]()[][]
An array of size [dim1]
Note: the dimension of the array cvec must be at least $n$ if the problem is of type LP (the default), and at least $1$ otherwise.
On entry: the coefficients of the objective function when the problem is of type LP.
If the problem is of type FP, cvec is not referenced.

istate

Type: array<System..::..Int32>[]()[][]

An array of size [

n + nclin

]

On entry: 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

istate [j - 1]

are as follows:

$istate [j - 1]$	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 must not be specified unless $bl [j - 1] = bu [j - 1]$ .

The values

- 2

- 1

and

4

are also acceptable but will be reset to zero by the method. If 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 method also adjusts (if necessary) the values supplied in x to be consistent with istate.

Constraint:

- 2 \leq istate [j - 1] \leq 4

, for

j = 1, 2, \dots, n + nclin

On exit: the status of the constraints in the working set at the point returned in x. The significance of each possible value of

istate [j - 1]

is as follows:

$istate [j - 1]$	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 istate 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 istate can occur only when $ifail = 1$ on exit.

x: Type: array<System..::..Double>[]()[][]
An array of size [n]
On entry: an initial estimate of the solution.
On exit: the point at which e04mf terminated. If $ifail = 0$ , $1$ or $4$ , x contains an estimate of the solution.

iter: Type: System..::..Int32%
On exit: the total number of iterations performed.

obj: Type: System..::..Double%
On exit: 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.

ax: Type: array<System..::..Double>[]()[][]
An array of size [ $\max (1, nclin)$ ]
On exit: the final values of the linear constraints $A x$ .
If $nclin = 0$ , ax is not referenced.

clamda: Type: array<System..::..Double>[]()[][]
An array of size [ $n + nclin$ ]
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 $m_{L}$ elements contain the multipliers for the general linear constraints (if any). If $istate [j - 1] = 0$ (i.e., constraint $j$ is not in the working set), $clamda [j - 1]$ is zero. If $x$ is optimal, $clamda [j - 1]$ should be non-negative if $istate [j - 1] = 1$ , non-positive if $istate [j - 1] = 2$ and zero if $istate [j - 1] = 4$ .

options: Type: NagLibrary..::..E04..::..e04mfOptions
An Object of type E04.e04mfOptions. Used to configure optional parameters to this method.

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

e04mf 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_{L}

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 method 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

u

can be set to special values that will be treated as

- \infty

+ \infty

. (See the description of the optional parameter Infinite Bound Size.)

You must supply an initial estimate of the solution.

The method used by 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 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 (1981) Practical Optimization Academic Press

Error Indicators and Warnings

Note: e04mf may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

Some error messages may refer to parameters that are dropped from this interface (LDA) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.

$ifail = 1$: $x$ 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.

$ifail = 2$: 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 ( $default value = 10^{20}$ ) 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 ( $default value = 10^{20}$ ).

$ifail = 3$: 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 [Description of the 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 $σ$ , you should ensure that the value of the optional parameter Feasibility Tolerance ( $default value = \sqrt{ε}$ , where $ε$ is the machine precision) is greater than $σ$ . 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 $10^{- 3}$ .

$ifail = 4$: The limiting number of iterations was reached before normal termination occurred.
The value of the optional parameter Iteration Limit ( $default value = \max (50, 5 (n + m_{L}))$ ) may be too small. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), either rerun e04mf with a larger value of Iteration Limit or, alternatively, rerun e04mf using the Warm Start facility to specify the initial working set.

$ifail = 5$: Not used by this method.

$ifail = 6$: An input parameter is invalid.

$ifail = 7$: The designated problem type was not FP or LP. Rerun e04mf with the optional parameter Problem Type set to one of these values.

$Overflow$: If the printed output before the overflow error contains a warning about serious ill-conditioning 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 ( $default value = \sqrt{ε}$ , where $ε$ 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.

$ifail = -9000$: An error occured, see message report.
$ifail = -6000$: Invalid Parameters $〈value〉$
$ifail = -4000$: Invalid dimension for array $〈value〉$
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

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.

Parallelism and Performance

None.

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 class and Gill et al. (1981) for further information and advice.

Description of the Printed Output

This section describes the intermediate printout and final printout produced by e04mf. The intermediate printout is a subset of the monitoring information produced by the method at every iteration (see [Description of Monitoring Information]). You can control the level of printed output (see the description of the optional parameter Print Level). Note that the intermediate printout and final printout are produced only if

Print Level \geq 10

(the default for e04mf, by default no output is produced by ).

The following line of summary output ( $< 80$ 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 $‖Z_{R}^{T} g_{FR}‖$ , 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 $j$ , for $j = 1, 2, \dots, n$ , of the variable.
State gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the Feasibility Tolerance, State will be ++ or -- respectively.
A key is sometimes printed before State.
A Alternative optimum possible. The variable is active at one of its bounds, but its Lagrange multiplier is essentially zero. This means that if the variable were allowed to start moving away from its bound then there would be no change to the objective function. The values of the other free variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case the values of the Lagrange multipliers might also change.
D Degenerate. The variable is free, but it is equal to (or very close to) one of its bounds.
I Infeasible. The variable is currently violating one of its bounds by more than the Feasibility Tolerance.

Value is the value of the variable at the final iteration.
Lower Bound is the lower bound specified for the variable. None indicates that $bl [j - 1] \leq - bigbnd$ .
Upper Bound is the upper bound specified for the variable. None indicates that $bu [j - 1] \geq bigbnd$ .
Lagr Mult is the Lagrange multiplier for the associated bound. This will be zero if State is FR unless $bl [j - 1] \leq - bigbnd$ and $bu [j - 1] \geq bigbnd$ , in which case the entry will be blank. If $x$ is optimal, the multiplier should be non-negative if State is LL and non-positive if State is UL.
Slack is the difference between the variable Value and the nearer of its (finite) bounds $bl [j - 1]$ and $bu [j - 1]$ . A blank entry indicates that the associated variable is not bounded (i.e., $bl [j - 1] \leq - bigbnd$ and $bu [j - 1] \geq bigbnd$ ).

The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, $bl [j - 1]$ and $bu [j - 1]$ are replaced by $bl [n + j - 1]$ and $bu [n + j - 1]$ respectively, and with the following change in the heading:
L Con gives the name (L) and index $j$ , for $j = 1, 2, \dots, n_{L}$ , 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
$- 0.02 x_{1} - 0.2 x_{2} - 0.2 x_{3} - 0.2 x_{4} - 0.2 x_{5} + 0.04 x_{6} + 0.04 x_{7}$
subject to the bounds
$\begin{matrix} - 0.01 \leq x_{1} \leq 0.01 \\ - 0.1 \leq x_{2} \leq 0.15 \\ - 0.01 \leq x_{3} \leq 0.03 \\ - 0.04 \leq x_{4} \leq 0.02 \\ - 0.1 \leq x_{5} \leq 0.05 \\ - 0.01 \leq x_{6} \\ - 0.01 \leq x_{7} \end{matrix}$
and the general constraints
$\begin{matrix} x_{1} & + & x_{2} & + & x_{3} & + & x_{4} & + & x_{5} & + & x_{6} & + & x_{7} & = & - 0.13 \\ 0.15 x_{1} & + & 0.04 x_{2} & + & 0.02 x_{3} & + & 0.04 x_{4} & + & 0.02 x_{5} & + & 0.01 x_{6} & + & 0.03 x_{7} & \leq & - 0.0049 \\ 0.03 x_{1} & + & 0.05 x_{2} & + & 0.08 x_{3} & + & 0.02 x_{4} & + & 0.06 x_{5} & + & 0.01 x_{6} & \leq & - 0.0064 \\ 0.02 x_{1} & + & 0.04 x_{2} & + & 0.01 x_{3} & + & 0.02 x_{4} & + & 0.02 x_{5} & \leq & - 0.0037 \\ 0.02 x_{1} & + & 0.03 x_{2} & + & 0.01 x_{5} & \leq & - 0.0012 \\ - 0.0992 & \leq & 0.70 x_{1} & + & 0.75 x_{2} & + & 0.80 x_{3} & + & 0.75 x_{4} & + & 0.80 x_{5} & + & 0.97 x_{6} \\ - 0.003 & \leq & 0.02 x_{1} & + & 0.06 x_{2} & + & 0.08 x_{3} & + & 0.12 x_{4} & + & 0.02 x_{5} & + & 0.01 x_{6} & + & 0.97 x_{7} & \leq & 0.002 \end{matrix}$

The initial point, which is infeasible, is
$x_{0} = {(- 0.01, - 0.03, 0.0, - 0.01, - 0.1, 0.02, 0.01)}^{T} .$
The optimal solution (to five figures) is
$x^{*} = {(- 0.01, - 0.1, 0.03, 0.02, - 0.067485, - 0.0022801, - 0.00023453)}^{T} .$
Four bound constraints and three general constraints are active at the solution.
Example program (C#): e04mfe.cs
Example program data: e04mfe.d
Example program results: e04mfe.r

Algorithmic Details

This section contains a detailed description of the method used by e04mf.

Overview

e04mf is based on an inertia-controlling 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 parameters of e04mf or appear in the printed output. 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 methods. 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 case when $m_{L} \leq n$ . 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 step length $α$ 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 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 $m_{W}$ denote the number of general constraints in the working set and let $n_{FX}$ denote the number of variables fixed at one of their bounds ( $m_{W}$ and $n_{FX}$ are the quantities Lin and Bnd in the monitoring file output from e04mf; see [Description of Monitoring Information]). Similarly, let $n_{FR}$ ( $n_{FR} = n - n_{FX}$ ) denote the number of free variables. At every iteration, the variables are reordered so that the last $n_{FX}$ variables are fixed, with all other relevant vectors and matrices ordered accordingly.

Definition of Search Direction

Let $A_{FR}$ denote the $m_{W}$ by $n_{FR}$ sub-matrix of general constraints in the working set corresponding to the free variables, and let $p_{FR}$ 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
$A_{FR} p_{FR} = 0 .$ (2)
In order to compute $p_{FR}$ , the $T Q$ factorization of $A_{FR}$ is used:
$A_{FR} Q_{FR} = (0 T),$ (3)
where $T$ is a nonsingular $m_{W}$ by $m_{W}$ upper triangular matrix (i.e., $t_{i j} = 0$ if $i > j$ ), and the nonsingular $n_{FR}$ by $n_{FR}$ matrix $Q_{FR}$ is the product of orthogonal transformations (see Gill et al. (1984)). If the columns of $Q_{FR}$ are partitioned so that
$Q_{FR} = (Z Y),$
where $Y$ is $n_{FR}$ by $m_{W}$ , then the $n_{Z}$ ( $n_{Z} = n_{FR} - m_{W}$ ) columns of $Z$ form a basis for the null space of $A_{FR}$ . 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 Zr in the monitoring file output from e04mf. In many cases, $Z_{R}$ will include all the columns of $Z$ .) The direction $p_{FR}$ will satisfy (2) if
$p_{FR} = Z_{R} p_{R}$ (4)
where $p_{R}$ is any $n_{R}$ -vector.

Main Iteration

Let $Q$ denote the $n$ by $n$ matrix
$Q = (\begin{matrix} Q_{FR} \\ I_{FX} \end{matrix}),$
where $I_{FX}$ is the identity matrix of order $n_{FX}$ . Let $g_{Q}$ denote the transformed gradient
$g_{Q} = Q^{T} c$
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 $c^{T} x$ . 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
$A_{FR}^{T} λ_{C} = g_{FR} and λ_{B} = g_{FX} - A_{FX}^{T} λ_{C} .$ (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 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 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 $λ_{j}$ 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 $A_{FR}$ : if the status of a general constraint changes, a row of $A_{FR}$ is altered; if a bound constraint enters or leaves the working set, a column of $A_{FR}$ changes. Explicit representations are recurred of the matrices $T$ and $Q_{FR}$ ; and of vectors $Q^{T} g$ , and $Q^{T} c$ .
One of the most important features of 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 $T Q$ 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.
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 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.

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 [Definition of Search Direction].
The artificially augmented working set is given by
${\bar{A}}_{FR} = (\begin{matrix} Z_{A}^{T} \\ A_{FR} \end{matrix}),$ (6)
so that $p_{FR}$ will satisfy $A_{FR} p_{FR} = 0$ and $Z_{A}^{T} p_{FR} = 0$ . By definition of the $T Q$ factorization, ${\bar{A}}_{FR}$ 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 $Q_{FR}$ 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 $Z_{R}^{T} g_{FR} = 0$ , since this simply involves repartitioning $Q_{FR}$ . The ‘artificial’ multiplier vector associated with the rows of $Z_{A}^{T}$ is equal to $Z_{A}^{T} g_{FR}$ , 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 $Z_{R}^{T} g_{FR}$ , 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 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., $m_{L} \leq n$ ). 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 monitoring file output (see [Description of Monitoring Information]).

Description of Monitoring Information

This section describes the long line of output ( $> 80$ characters) which forms part of the monitoring information produced by 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 $Print Level \geq 5$ and $Monitoring File \geq 0$ , 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., $Zr = n - (Bnd + Lin + Art)$ .
The value of $n_{Z}$ , the number of columns of $Z$ (see [Definition of Search Direction]) 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 is $‖Z_{R}^{T} g_{FR}‖$ , 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.

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 $‖Z_{R}^{T} g_{FR}‖$ , 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].)

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., $Zr = n - (Bnd + Lin + Art)$ . The value of $n_{Z}$ , the number of columns of $Z$ (see [Definition of Search Direction]) 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	is $‖Z_{R}^{T} g_{FR}‖$ , 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.

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Scaling

Description of the Printed Output

Example

Algorithmic Details

Overview

Definition of Search Direction

Main Iteration

Choosing the Initial Working Set

Description of Monitoring Information

See Also