e04nq solves sparse linear programming or convex quadratic programming problems. The initialization method (E04NPF not in this release) must have been called before calling e04nq.

Syntax

C#
public static void e04nq( string start, E04..::..E04NQ_QPHX qphx, int m, int n, int ne, int nname, int lenc, int ncolh, int iobj, double objadd, string prob, double[] acol, int[] inda, int[] loca, double[] bl, double[] bu, double[] c, string[] names, int[] helast, int[] hs, double[] x, double[] pi, double[] rc, ref int ns, out int ninf, out double sinf, out double obj, E04..::..e04nqOptions options, out int ifail )

public static void e04nq(
	string start,
	E04..::..E04NQ_QPHX qphx,
	int m,
	int n,
	int ne,
	int nname,
	int lenc,
	int ncolh,
	int iobj,
	double objadd,
	string prob,
	double[] acol,
	int[] inda,
	int[] loca,
	double[] bl,
	double[] bu,
	double[] c,
	string[] names,
	int[] helast,
	int[] hs,
	double[] x,
	double[] pi,
	double[] rc,
	ref int ns,
	out int ninf,
	out double sinf,
	out double obj,
	E04..::..e04nqOptions options,
	out int ifail
)

Visual Basic
Public Shared Sub e04nq ( _ start As String, _ qphx As E04..::..E04NQ_QPHX, _ m As Integer, _ n As Integer, _ ne As Integer, _ nname As Integer, _ lenc As Integer, _ ncolh As Integer, _ iobj As Integer, _ objadd As Double, _ prob As String, _ acol As Double(), _ inda As Integer(), _ loca As Integer(), _ bl As Double(), _ bu As Double(), _ c As Double(), _ names As String(), _ helast As Integer(), _ hs As Integer(), _ x As Double(), _ pi As Double(), _ rc As Double(), _ ByRef ns As Integer, _ <OutAttribute> ByRef ninf As Integer, _ <OutAttribute> ByRef sinf As Double, _ <OutAttribute> ByRef obj As Double, _ options As E04..::..e04nqOptions, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub e04nq ( _
	start As String, _
	qphx As E04..::..E04NQ_QPHX, _
	m As Integer, _
	n As Integer, _
	ne As Integer, _
	nname As Integer, _
	lenc As Integer, _
	ncolh As Integer, _
	iobj As Integer, _
	objadd As Double, _
	prob As String, _
	acol As Double(), _
	inda As Integer(), _
	loca As Integer(), _
	bl As Double(), _
	bu As Double(), _
	c As Double(), _
	names As String(), _
	helast As Integer(), _
	hs As Integer(), _
	x As Double(), _
	pi As Double(), _
	rc As Double(), _
	ByRef ns As Integer, _
	<OutAttribute> ByRef ninf As Integer, _
	<OutAttribute> ByRef sinf As Double, _
	<OutAttribute> ByRef obj As Double, _
	options As E04..::..e04nqOptions, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void e04nq( String^ start, E04..::..E04NQ_QPHX^ qphx, int m, int n, int ne, int nname, int lenc, int ncolh, int iobj, double objadd, String^ prob, array<double>^ acol, array<int>^ inda, array<int>^ loca, array<double>^ bl, array<double>^ bu, array<double>^ c, array<String^>^ names, array<int>^ helast, array<int>^ hs, array<double>^ x, array<double>^ pi, array<double>^ rc, int% ns, [OutAttribute] int% ninf, [OutAttribute] double% sinf, [OutAttribute] double% obj, E04..::..e04nqOptions^ options, [OutAttribute] int% ifail )

Visual C++

public:
static void e04nq(
	String^ start, 
	E04..::..E04NQ_QPHX^ qphx, 
	int m, 
	int n, 
	int ne, 
	int nname, 
	int lenc, 
	int ncolh, 
	int iobj, 
	double objadd, 
	String^ prob, 
	array<double>^ acol, 
	array<int>^ inda, 
	array<int>^ loca, 
	array<double>^ bl, 
	array<double>^ bu, 
	array<double>^ c, 
	array<String^>^ names, 
	array<int>^ helast, 
	array<int>^ hs, 
	array<double>^ x, 
	array<double>^ pi, 
	array<double>^ rc, 
	int% ns, 
	[OutAttribute] int% ninf, 
	[OutAttribute] double% sinf, 
	[OutAttribute] double% obj, 
	E04..::..e04nqOptions^ options, 
	[OutAttribute] int% ifail
)

F#
static member e04nq : start : string * qphx : E04..::..E04NQ_QPHX * m : int * n : int * ne : int * nname : int * lenc : int * ncolh : int * iobj : int * objadd : float * prob : string * acol : float[] * inda : int[] * loca : int[] * bl : float[] * bu : float[] * c : float[] * names : string[] * helast : int[] * hs : int[] * x : float[] * pi : float[] * rc : float[] * ns : int byref * ninf : int byref * sinf : float byref * obj : float byref * options : E04..::..e04nqOptions * ifail : int byref -> unit

static member e04nq : 
        start : string * 
        qphx : E04..::..E04NQ_QPHX * 
        m : int * 
        n : int * 
        ne : int * 
        nname : int * 
        lenc : int * 
        ncolh : int * 
        iobj : int * 
        objadd : float * 
        prob : string * 
        acol : float[] * 
        inda : int[] * 
        loca : int[] * 
        bl : float[] * 
        bu : float[] * 
        c : float[] * 
        names : string[] * 
        helast : int[] * 
        hs : int[] * 
        x : float[] * 
        pi : float[] * 
        rc : float[] * 
        ns : int byref * 
        ninf : int byref * 
        sinf : float byref * 
        obj : float byref * 
        options : E04..::..e04nqOptions * 
        ifail : int byref -> unit

Parameters

start

Type: System..::..String

On entry: indicates how a starting basis (and certain other items) will be obtained.

$start = "C"$: Requests that an internal Crash procedure be used to choose an initial basis, unless a Basis file is provided via optional parameters Old Basis File, Insert File or Load File.
$start = "B"$: Is the same as $start = "C"$ but is more meaningful when a Basis file is given.
$start = "W"$: Means that a basis is already defined in hs and a start point is already defined in x (probably from an earlier call).

Constraint:

start = "B"

"C"

"W"

qphx

Type: NagLibrary..::..E04..::..E04NQ_QPHX

For QP problems, you must supply a version of qphx to compute the matrix product

H x

for a given vector

x

. If

H

has rows and columns of zeros, it is most efficient to order

x

so that the nonlinear variables appear first. For example, if

x = {(y, z)}^{T}

and only

y

enters the objective quadratically then

H x = (\begin{matrix} H_{1} & 0 \\ 0 & 0 \end{matrix}) (\begin{matrix} y \\ z \end{matrix}) = (\begin{matrix} H_{1} y \\ 0 \end{matrix}) .

(2)

In this case, ncolh should be the dimension of

y

, and qphx should compute

H_{1} y

. For FP and LP problems, qphx will never be called by e04nq and hence qphx may be the dummy method E04NSH.

A delegate of type E04NQ_QPHX.

m: Type: System..::..Int32
On entry: $m$ , the number of general linear constraints (or slacks). This is the number of rows in the linear constraint matrix $A$ , including the free row (if any; see iobj). Note that $A$ must have at least one row. If your problem has no constraints, or only upper or lower bounds on the variables, then you must include a dummy row with sufficiently wide upper and lower bounds (see also acol, inda and loca).

Constraint: $m \geq 1$ .

n: Type: System..::..Int32
On entry: $n$ , the number of variables (excluding slacks). This is the number of columns in the linear constraint matrix $A$ .

Constraint: $n \geq 1$ .

ne: Type: System..::..Int32
On entry: the number of nonzero elements in $A$ .

Constraint: $1 \leq ne \leq n \times m$ .

nname

Type: System..::..Int32

On entry: the number of column (i.e., variable) and row names supplied in the array names.

$nname = 1$: There are no names. Default names will be used in the printed output.
$nname = n + m$: All names must be supplied.

Constraint:

nname = 1

n + m

lenc: Type: System..::..Int32
On entry: the number of elements in the constant objective vector $c$ .
If $lenc > 0$ , the first lenc elements of $x$ belong to variables corresponding to the constant objective term $c$ .

Constraint: $0 \leq lenc \leq n$ .

ncolh: Type: System..::..Int32
On entry: $n_{H}$ , the number of leading nonzero columns of the Hessian matrix $H$ . For FP and LP problems, ncolh must be set to zero.
The first ncolh elements of $x$ belong to variables corresponding to the nonzero block of the QP Hessian.

Constraint: $0 \leq ncolh \leq n$ .

iobj: Type: System..::..Int32
On entry: if $iobj > 0$ , row iobj of $A$ is a free row containing the nonzero elements of the vector $c$ appearing in the linear objective term $c^{T} x$ .
If $iobj = 0$ , there is no free row, and the linear objective vector should be supplied in array c.

Constraint: $0 \leq iobj \leq m$ .

objadd: Type: System..::..Double
On entry: the constant $q$ , to be added to the objective for printing purposes. Typically $objadd = 0.0E0$ .

prob: Type: System..::..String
On entry: the name for the problem. It is used in the printed solution and in some methods that output Basis files. A blank name may be used.

acol: Type: array<System..::..Double>[]()[][]
An array of size [ne]
On entry: the nonzero elements of $A$ , ordered by increasing column index. Note that all elements must be assigned a value in the calling program.

inda: Type: array<System..::..Int32>[]()[][]
An array of size [ne]
On entry: $inda [i - 1]$ must contain the row index of the nonzero element stored in $acol [i - 1]$ , for $i = 1, 2, \dots, ne$ . Thus a pair of values $(acol [i - 1], inda [i - 1])$ contains a matrix element and its corresponding row index.
Note that the row indices for a column may be supplied in any order.

Constraint: $1 \leq inda [i - 1] \leq m$ , for $i = 1, 2, \dots, ne$ .

loca

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

An array of size [

n + 1

]

On entry:

loca [j - 1]

must contain the index in acol and inda of the start of the

j

th column, for

j = 1, 2, \dots, n

. Thus for

j = 1 : n

, the entries of column

j

are held in

acol [k - 1 : l - 1]

and their corresponding row indices are in

inda [k - 1 : l - 1]

, where

k = loca [j - 1]

and

l = loca [j] - 1

. To specify the

j

th column as empty, set

loca [j - 1] = loca [j]

. Note that the first and last elements of loca must be

loca [0] = 1

and

loca [n] = ne + 1

. If your problem has no constraints, or just bounds on the variables, you may include a dummy ‘free’ row with a single (zero) element by setting

ne = 1

acol [0] = 0.0

inda [0] = 1

loca [0] = 1

, and

loca [j - 1] = 2

, for

j = 2 : n + 1

. This row is made ‘free’ by setting its bounds to be

bl [n] = - bigbnd

and

bu [n] = bigbnd

, where

bigbnd

is the value of the optional parameter Infinite Bound Size.

Constraints:

$loca [0] = 1$ ;
$loca [j - 1] \geq 1$ , for $j = 2, 3, \dots, n$ ;
$loca [n] = ne + 1$ ;
$0 \leq loca [j] - loca [j - 1] \leq m$ , for $j = 1, 2, \dots, n$ .

bl: Type: array<System..::..Double>[]()[][]
An array of size [ $n + m$ ]
On entry: $l$ , the lower bounds for all the variables and general constraints, in the following order. The first n elements of bl must contain the bounds on the variables $x$ , and the next m elements the bounds for the general linear constraints $A x$ (which, equivalently, are the bounds for the slacks, $s$ ) and the free row (if any). To fix the $j$ th variable, set $bl [j - 1] = bu [j - 1] = β$ , say, where $|β| < bigbnd$ . To specify a nonexistent lower bound (i.e., $l_{j} = - \infty$ ), set $bl [j - 1] \leq - bigbnd$ . Here, $bigbnd$ is the value of the optional parameter Infinite Bound Size. To specify the $j$ th constraint as an equality, set $bl [n + j - 1] = bu [n + j - 1] = β$ , say, where $|β| < bigbnd$ . Note that the lower bound corresponding to the free row must be set to $- \infty$ and stored in $bl [n + iobj - 1]$ .

Constraint: if $iobj > 0$ , $bl [n + iobj - 1] \leq - bigbnd$
(See also the description for bu.)

bu

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

An array of size [

n + m

]

On entry:

u

, the upper bounds for all the variables and general constraints, in the following order. The first n elements of bu must contain the bounds on the variables

x

, and the next m elements the bounds for the general linear constraints

A x

(which, equivalently, are the bounds for the slacks,

s

) and the free row (if any). To specify a nonexistent upper bound (i.e.,

u_{j} = + \infty

), set

bu [j - 1] \geq bigbnd

. Note that the upper bound corresponding to the free row must be set to

+ \infty

and stored in

bu [n + iobj - 1]

Constraints:

if $iobj > 0$ , $bu [n + iobj - 1] \geq bigbnd$ ;
otherwise $bl [i - 1] \leq bu [i - 1]$ .

c: Type: array<System..::..Double>[]()[][]
An array of size [ $\max (1, lenc)$ ]
Note: the dimension of the array c must be at least $\max (1, lenc)$ if $iobj \neq 0$ , and at least $1$ otherwise.
On entry: contains the explicit objective vector $c$ (if any). If the problem is of type FP, or if $lenc = 0$ , then c is not referenced. (In that case, c may be dimensioned (1), or it could be any convenient array.)

names: Type: array<System..::..String>[]()[][]
An array of size [nname]
On entry: the optional column and row names, respectively.
If $nname = 1$ , names is not referenced and the printed output will use default names for the columns and rows.

If $nname = n + m$ , the first n elements must contain the names for the columns and the next m elements must contain the names for the rows. Note that the name for the free row (if any) must be stored in $names [n + iobj - 1]$ .

helast

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

An array of size [

n + m

]

On entry: defines which variables are to be treated as being elastic in elastic mode. The allowed values of helast are:

$helast [j - 1]$	Status in elastic mode
$0$	Variable $j$ is non-elastic and cannot be infeasible
$1$	Variable $j$ can violate its lower bound
$2$	Variable $j$ can violate its upper bound
$3$	Variable $j$ can violate either its lower or upper bound

helast need not be assigned if optional parameter

Elastic Mode = 0

Constraint: if

Elastic Mode \neq 0

helast [j - 1] = 0, 1, 2, 3

, for

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

hs

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

An array of size [

n + m

]

On entry: if

start = "C"

"B"

, and a Basis file of some sort is to be input (see the description of the optional parameters Old Basis File, Insert File or Load File), then hs and x need not be set at all.

start = "C"

"B"

and there is no Basis file, the first n elements of hs and x must specify the initial states and values, respectively, of the variables

x

. (The slacks

s

need not be initialized.) An internal Crash procedure is then used to select an initial basis matrix

B

. The initial basis matrix will be triangular (neglecting certain small elements in each column). It is chosen from various rows and columns of

(\begin{matrix} A & - I \end{matrix})

. Possible values for

hs [j - 1]

are as follows:

$hs [j - 1]$	State of $x [j - 1]$ during Crash procedure
$0$ or $1$	Eligible for the basis
$2$	Ignored
$3$	Eligible for the basis (given preference over $0$ or $1$ )
$4$ or $5$	Ignored

If nothing special is known about the problem, or there is no wish to provide special information, you may set

hs [j - 1] = 0

and

x [j - 1] = 0.0

, for

j = 1, 2, \dots, n

. All variables will then be eligible for the initial basis. Less trivially, to say that the

j

th variable will probably be equal to one of its bounds, set

hs [j - 1] = 4

and

x [j - 1] = bl [j - 1]

hs [j - 1] = 5

and

x [j - 1] = bu [j - 1]

as appropriate.

Following the Crash procedure, variables for which

hs [j - 1] = 2

are made superbasic. Other variables not selected for the basis are then made nonbasic at the value

x [j - 1]

bl [j - 1] \leq x [j - 1] \leq bu [j - 1]

, or at the value

bl [j - 1]

bu [j - 1]

closest to

x [j - 1]

start = "W"

, hs and x must specify the initial states and values, respectively, of the variables and slacks

(x, s)

. If e04nq has been called previously with the same values of n and m, hs already contains satisfactory information.

Constraints:

if $start = "C"$ or $"B"$ , $0 \leq hs [j - 1] \leq 5$ , for $j = 1, 2, \dots, n$ ;
if $start = "W"$ , $0 \leq hs [j - 1] \leq 3$ , for $j = 1, 2, \dots, n + m$ .

On exit: the final states of the variables and slacks

(x, s)

. The significance of each possible value of

hs [j - 1]

is as follows:

$hs [j - 1]$	State of variable $j$	Normal value of $x [j - 1]$
$0$	Nonbasic	$bl [j - 1]$
$1$	Nonbasic	$bu [j - 1]$
$2$	Superbasic	Between $bl [j - 1]$ and $bu [j - 1]$
$3$	Basic	Between $bl [j - 1]$ and $bu [j - 1]$

ninf = 0

, basic and superbasic variables may be outside their bounds by as much as the value of the optional parameter Feasibility Tolerance. Note that unless the optional parameter

Scale Option = 0

is specified, the optional parameter Feasibility Tolerance applies to the variables of the scaled problem. In this case, the variables of the original problem may be as much as

0.1

outside their bounds, but this is unlikely unless the problem is very badly scaled.

Very occasionally some nonbasic variables may be outside their bounds by as much as the optional parameter Feasibility Tolerance, and there may be some nonbasic variables for which

x [j - 1]

lies strictly between its bounds.

ninf > 0

, some basic and superbasic variables may be outside their bounds by an arbitrary amount (bounded by sinf if

Scale Option = 0

x: Type: array<System..::..Double>[]()[][]
An array of size [ $n + m$ ]
On entry: the initial values of the variables $x$ , and, if $start = "W"$ , the slacks $s$ , i.e., $(x, s)$ . (See the description for parameter hs.)
On exit: the final values of the variables and slacks $(x, s)$ .

pi: Type: array<System..::..Double>[]()[][]
An array of size [m]
On exit: contains the dual variables $π$ (a set of Lagrange multipliers (shadow prices) for the general constraints).

rc: Type: array<System..::..Double>[]()[][]
An array of size [ $n + m$ ]
On exit: contains the reduced costs, $g - {(\begin{matrix} A & - I \end{matrix})}^{T} π$ . The vector $g$ is the gradient of the objective if x is feasible, otherwise it is the gradient of the Phase 1 objective. In the former case, $g (i) = 0$ , for $i = n + 1 : m$ , hence $rc [n + 1 : m - 1] = π$ .

ns: Type: System..::..Int32%
On entry: $n_{S}$ , the number of superbasics. For QP problems, ns need not be specified if $start = "C"$ , but must retain its value from a previous call when $start = "W"$ . For FP and LP problems, ns need not be initialized.
On exit: the final number of superbasics. This will be zero for FP and LP problems.

ninf: Type: System..::..Int32%
On exit: the number of infeasibilities.

sinf: Type: System..::..Double%
On exit: the sum of the scaled infeasibilities. This will be zero if $ninf = 0$ , and is most meaningful when $Scale Option = 0$ .

obj: Type: System..::..Double%
On exit: the value of the objective function.
If $ninf = 0$ , obj includes the quadratic objective term $\frac{1}{2} x^{T} H x$ (if any).

If $ninf > 0$ , obj is just the linear objective term $c^{T} x$ (if any).

For FP problems, obj is set to zero.

Note that obj does not include contributions from the constant term objadd or the objective row, if any.

options: Type: NagLibrary..::..E04..::..e04nqOptions
An Object of type E04.e04nqOptions. 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

e04nq is designed to solve large-scale linear or quadratic programming problems of the form:

\underset{x \in R^{n}}{minimize} f (x) subject to ​ l \leq (\begin{matrix} x \\ A x \end{matrix}) \leq u,

(1)

where

x

is an

n

-vector of variables,

l

and

u

are constant lower and upper bounds,

A

is an

m

n

sparse matrix and

f (x)

is a linear or quadratic objective function that may be specified in a variety of ways, depending upon the particular problem being solved. The optional parameter Maximize may be used to specify a problem in which

f (x)

is maximized instead of minimized.

Upper and lower bounds are specified for all variables and constraints. This form allows full generality in specifying various types of constraint. In particular, the

j

th constraint may be defined as an equality by setting

l_{j} = u_{j}

. If certain bounds are not present, the associated elements of

l

u

may be set to special values that are treated as

- \infty

+ \infty

The possible forms for the function

f (x)

are summarised in Table 1. The most general form for

f (x)

f (x) = q + c^{T} x + \frac{1}{2} x^{T} H x = q + \sum_{j = 1}^{n} c_{j} x_{j} + \frac{1}{2} \sum_{i = 1}^{n} \sum_{j = 1}^{n} x_{i} H_{i j} x_{j}

where

q

is a constant,

c

is a constant

n

-vector and

H

is a constant symmetric

n

n

matrix with elements

\{H_{i j}\}

. In this form,

f

is a quadratic function of

x

and (1) is known as a quadratic program (QP). e04nq is suitable for all convex quadratic programs. The defining feature of a convex QP is that the matrix

H

must be positive semidefinite, i.e., it must satisfy

x^{T} H x \geq 0

for all

x

. If not,

f (x)

is nonconvex and e04nq will terminate with the error indicator

ifail = 11

. If

f (x)

is nonconvex it may be more appropriate to call e04vh instead.

Problem type	Objective function $f (x)$	Hessian matrix $H$
FP	Not applicable	$q = c = H = 0$
LP	$q + c^{T} x$	$H = 0$
QP	$q + c^{T} x + \frac{1}{2} x^{T} H x$	Symmetric positive semidefinite

Table 1
Choices for the objective function

f (x)

H = 0

, then

f (x) = q + c^{T} x

and the problem is known as a linear program (LP). In this case, rather than defining an

H

with zero elements, you can define

H

to have no columns by setting

ncolh = 0

(see [Parameters]).

H = 0

q = 0

, and

c = 0

, there is no objective function and the problem is a feasible point problem (FP), which is equivalent to finding a point that satisfies the constraints on

x

. In the situation where no feasible point exists, several options are available for finding a point that minimizes the constraint violations (see the description of the optional parameter Elastic Mode).

e04nq is suitable for large LPs and QPs in which the matrix

A

is sparse, i.e., when the number of zero elements is sufficiently large that it is worthwhile using algorithms which avoid computations and storage involving zero elements. The matrix

A

is input to e04nq by means of the three array parameters acol, inda and loca. This allows you to specify the pattern of nonzero elements in

A

e04nq exploits structure in

H

by requiring

H

to be defined implicitly in a method that computes the product

H x

for any given vector

x

. In many cases, the product

H x

can be computed very efficiently for any given

x

, e.g.,

H

may be a sparse matrix, or a sum of matrices of rank-one.

For problems in which

A

can be treated as a dense matrix, it is usually more efficient to use e04mf, e04nc or e04nf.

There is considerable flexibility allowed in the definition of

f (x)

in Table 1. The vector

c

defining the linear term

c^{T} x

can be input in three ways: as a sparse row of

A

; as an explicit dense vector

c

; or as both a sparse row and an explicit vector (in which case,

c^{T} x

will be the sum of two linear terms). When stored in

A

c

is the iobjth row of

A

, which is known as the objective row. The objective row must always be a free row of

A

in the sense that its lower and upper bounds must be

- \infty

and

+ \infty

. Storing

c

as part of

A

is recommended if

c

is a sparse vector. Storing

c

as an explicit vector is recommended for a sequence of problems, each with a different objective (see parameters c and lenc).

The upper and lower bounds on the

m

elements of

A x

are said to define the general constraints of the problem. Internally, e04nq converts the general constraints to equalities by introducing a set of slack variables

s

, where

s = {(s_{1}, s_{2}, \dots, s_{m})}^{T}

. For example, the linear constraint

5 \leq 2 x_{1} + 3 x_{2} \leq + \infty

is replaced by

2 x_{1} + 3 x_{2} - s_{1} = 0

, together with the bounded slack

5 \leq s_{1} \leq + \infty

. The problem defined by (1) can therefore be re-written in the following equivalent form:

\underset{x \in R^{n}, s \in R^{m}}{minimize} f (x) subject to ​ A x - s = 0, l \leq (\begin{matrix} x \\ s \end{matrix}) \leq u .

Since the slack variables

s

are subject to the same upper and lower bounds as the elements of

A x

, the bounds on

x

and

A x

can simply be thought of as bounds on the combined vector

(x, s)

. (In order to indicate their special role in QP problems, the original variables

x

are sometimes known as ‘column variables’, and the slack variables

s

are known as ‘row variables’.)

Each LP or QP problem is solved using a two-phase iterative procedure (in which the general constraints are satisfied throughout): a feasibility phase (Phase 1), in which the sum of infeasibilities with respect to the bounds on

x

and

s

is minimized to find a feasible point that satisfies all constraints within a specified feasibility tolerance; and an optimality phase (Phase 2), in which

f (x)

is minimized (or maximized) by constructing a sequence of iterates that lies within the feasible region.

Phase 1 involves solving a linear program of the form

Phase 1
	$\underset{x, s, v, w}{minimize} \sum_{j = 1}^{n + m} (v_{j} + w_{j})$
	$subject to A x - s = 0, l \leq (\begin{matrix} x \\ s \end{matrix}) - v + w \leq u, v \geq 0, w \geq 0$

which is equivalent to minimizing the sum of the constraint violations. If the constraints are feasible (i.e., at least one feasible point exists), eventually a point will be found at which both

v

and

w

are zero. Then the associated value of

(x, s)

satisfies the original constraints and is used as the starting point for the Phase 2 iterations for minimizing

f (x)

If the constraints are infeasible (i.e.,

v \neq 0

w \neq 0

at the end of Phase 1), no solution exists for (1) and you have the option of either terminating or continuing in so-called elastic mode (see the discussion of the optional parameter Elastic Mode). In elastic mode, a ‘relaxed’ or ‘perturbed’ problem is solved in which

f (x)

is minimized while allowing some of the bounds to become ‘elastic’, i.e., to change from their specified values. Variables subject to elastic bounds are known as elastic variables. An elastic variable is free to violate one or both of its original upper or lower bounds. You are able to assign which bounds will become elastic if elastic mode is ever started (see the parameter helast in [Parameters]).

To make the relaxed problem meaningful, e04nq minimizes

f (x)

while (in some sense) finding the ‘smallest’ violation of the elastic variables. In the situation where all the variables are elastic, the relaxed problem has the form

Phase 2 ( $γ$ )
	$\underset{x, s, v, w}{minimize} f (x) + γ \sum_{j = 1}^{n + m} (v_{j} + w_{j})$
	$subject to A x - s = 0, l \leq (\begin{matrix} x \\ s \end{matrix}) - v + w \leq u, v \geq 0, w \geq 0$ ,

where

γ

is a non-negative parameter known as the elastic weight (see the description of the optional parameter Elastic Weight), and

f (x) + γ \sum_{j} (v_{j} + w_{j})

is called the composite objective. In the more general situation where only a subset of the bounds are elastic, the

v

's and

w

's for the non-elastic bounds are fixed at zero.

The elastic weight can be chosen to make the composite objective behave like the original objective

f (x)

, the sum of infeasibilities, or anything in-between. If

γ = 0

, e04nq will attempt to minimize

f

subject to the (true) upper and lower bounds on the non-elastic variables (and declare the problem infeasible if the non-elastic variables cannot be made feasible).

At the other extreme, choosing

γ

sufficiently large will have the effect of minimizing the sum of the violations of the elastic variables subject to the original constraints on the non-elastic variables. Choosing a large value of the elastic weight is useful for defining a ‘least-infeasible’ point for an infeasible problem.

In Phase 1 and elastic mode, all calculations involving

v

and

w

are done implicitly in the sense that an elastic variable

x_{j}

is allowed to violate its lower bound (say) and an explicit value of

v

can be recovered as

v_{j} = l_{j} - x_{j}

A constraint is said to be active or binding at

x

if the associated element of either

x

A x

is equal to one of its upper or lower bounds. Since an active constraint in

A x

has its associated slack variable at a bound, the status of both simple and general upper and lower bounds can be conveniently described in terms of the status of the variables

(x, s)

. A variable is said to be nonbasic if it is temporarily fixed at its upper or lower bound. It follows that regarding a general constraint as being active is equivalent to thinking of its associated slack as being nonbasic.

At each iteration of an active-set method, the constraints

A x - s = 0

are (conceptually) partitioned into the form

B x_{B} + S x_{S} + N x_{N} = 0,

where

x_{N}

consists of the nonbasic elements of

(x, s)

and the basis matrix

B

is square and nonsingular. The elements of

x_{B}

and

x_{S}

are called the basic and superbasic variables respectively; with

x_{N}

they are a permutation of the elements of

x

and

s

. At a QP solution, the basic and superbasic variables will lie somewhere between their upper or lower bounds, while the nonbasic variables will be equal to one of their bounds. At each iteration,

x_{S}

is regarded as a set of independent variables that are free to move in any desired direction, namely one that will improve the value of the objective function (or sum of infeasibilities). The basic variables are then adjusted in order to ensure that

(x, s)

continues to satisfy

A x - s = 0

. The number of superbasic variables (

n_{S}

say) therefore indicates the number of degrees of freedom remaining after the constraints have been satisfied. In broad terms,

n_{S}

is a measure of how nonlinear the problem is. In particular,

n_{S}

will always be zero for FP and LP problems.

If it appears that no improvement can be made with the current definition of

B

S

and

N

, a nonbasic variable is selected to be added to

S

, and the process is repeated with the value of

n_{S}

increased by one. At all stages, if a basic or superbasic variable encounters one of its bounds, the variable is made nonbasic and the value of

n_{S}

is decreased by one.

Associated with each of the

m

equality constraints

A x - s = 0

is a dual variable

π_{i}

. Similarly, each variable in

(x, s)

has an associated reduced gradient

d_{j}

(also known as a reduced cost). The reduced gradients for the variables

x

are the quantities

g - A^{T} π

, where

g

is the gradient of the QP objective function, and the reduced gradients for the slack variables

s

are the dual variables

π

. The QP subproblem is optimal if

d_{j} \geq 0

for all nonbasic variables at their lower bounds,

d_{j} \leq 0

for all nonbasic variables at their upper bounds and

d_{j} = 0

for all superbasic variables. In practice, an approximate QP solution is found by slightly relaxing these conditions on

d_{j}

(see the description of the optional parameter Optimality Tolerance).

The process of computing and comparing reduced gradients is known as pricing (a term first introduced in the context of the simplex method for linear programming). To ‘price’ a nonbasic variable

x_{j}

means that the reduced gradient

d_{j}

associated with the relevant active upper or lower bound on

x_{j}

is computed via the formula

d_{j} = g_{j} - a_{j}^{T} π

, where

a_{j}

is the

j

th column of

(\begin{matrix} A & - I \end{matrix})

. (The variable selected by such a process and the corresponding value of

d_{j}

(i.e., its reduced gradient) are the quantities +SBS and dj in the monitoring file output; see [Description of the Printed Output].) If

A

has significantly more columns than rows (i.e.,

n ≫ m

), pricing can be computationally expensive. In this case, a strategy known as partial pricing can be used to compute and compare only a subset of the

d_{j}

e04nq is based on SQOPT, which is part of the SNOPT package described in Gill et al. (2005a). It uses stable numerical methods throughout and includes a reliable basis package (for maintaining sparse

L U

factors of the basis matrix

B

), a practical anti-degeneracy procedure, efficient handling of linear constraints and bounds on the variables (by an active-set strategy), as well as automatic scaling of the constraints. Further details can be found in [Algorithmic Details].

References

Fourer R (1982) Solving staircase linear programs by the simplex method Math. Programming 23 274–313

Gill P E and Murray W (1978) Numerically stable methods for quadratic programming Math. Programming 14 349–372

Gill P E, Murray W and Saunders M A (1995) User's guide for QPOPT 1.0: a Fortran package for quadratic programming Report SOL 95-4 Department of Operations Research, Stanford University

Gill P E, Murray W and Saunders M A (2005a) Users' guide for SQOPT 7: a Fortran package for large-scale linear and quadratic programming Report NA 05-1 Department of Mathematics, University of California, San Diego http://www.ccom.ucsd.edu/~peg/papers/sqdoc7.pdf

Gill P E, Murray W and Saunders M A (2005b) Users' guide for SNOPT 7.1: a Fortran package for large-scale linear nonlinear programming Report NA 05-2 Department of Mathematics, University of California, San Diego http://www.ccom.ucsd.edu/~peg/papers/sndoc7.pdf

Gill P E, Murray W, Saunders M A and Wright M H (1987) Maintaining LU factors of a general sparse matrix Linear Algebra and its Applics. 88/89 239–270

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

Hall J A J and McKinnon K I M (1996) The simplest examples where the simplex method cycles and conditions where EXPAND fails to prevent cycling Report MS 96–100 Department of Mathematics and Statistics, University of Edinburgh

Error Indicators and Warnings

Note: e04nq may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

$ifail = 1$: The initialization method (E04NPF not in this release) has not been called or at least one of lencw, leniw and lenrw is less than $600$ .

$ifail = 2$: An input parameter is invalid.

$ifail = 3$: The requested accuracy could not be achieved.

$ifail = 4$

Weak QP solution found. The final

x

is not unique.

This exit will occur when the following are true:

(i)	the problem is feasible;
(ii)	the reduced gradient is negligible;
(iii)	the Lagrange multipliers are optimal; and
(iv)	the reduced Hessian is singular or there are some very small multipliers.

This exit cannot occur if

H

is positive definite (i.e.,

f (x)

is strictly convex).

$ifail = 5$: The problem is infeasible. The general constraints cannot all be satisfied simultaneously to within the value of the optional parameter Feasibility Tolerance.
Feasibility is measured with respect to the upper and lower bounds on the variables and slacks. The message tells us that among all the points satisfying the general constraints $A x - s = 0$ , there is apparently no point that satisfies the bounds on $x$ and $s$ . Violations as small as the Feasibility Tolerance are ignored, but at least one component of $x$ or $s$ violates a bound by more than the tolerance.
Note: although the objective function is the sum of infeasibilities (when $ninf > 0$ ), this sum will not necessarily have been minimized when $Elastic Mode = 1$ .
If $Elastic Mode \neq 0$ , e04nq will optimize the QP objective and the sum of infeasibilities, suitably weighted using the optional parameter Elastic Mode. The method will tend to determine a ‘good’ infeasible point if the elastic weight is sufficiently large.

$ifail = 6$: The problem is unbounded (or badly scaled). For a minimization problem, the objective function is not bounded below in the feasible region.
For linear problems, unboundedness is detected by the simplex method when a nonbasic variable can be increased or decreased by an arbitrary amount without causing a basic variable to violate a bound. Consider adding an upper or lower bound to the variable. Also, examine the constraints that have nonzeros in the associated column, to see if they have been formulated as intended.
Very rarely, the scaling of the problem could be so poor that numerical error will give an erroneous indication of unboundedness. Consider using the optional parameter Scale Option.

$ifail = 7$: Too many iterations. The value of the optional parameter Iterations Limit is too small.
The Iterations limit was exceeded before the required solution could be found. Check the iteration log to be sure that progress was being made. If so, restart the run using a Basis file that was saved at the end of the run.

$ifail = 8$: The value of the optional parameter Superbasics Limit is too small. The current set of basic and superbasic variables have been optimized as much as possible and a pricing operation is necessary to continue, but there are already Superbasics Limit superbasics (and no room for any more).
In general, raise the Superbasics Limit $s$ by a reasonable amount, bearing in mind the storage needed for reduced Hessian (see [Definition of the Working Set and Search Direction]). (The Reduced Hessian Dimension $h$ will also increase to $s$ unless specified otherwise, and the associated storage will be about $\frac{1}{2} s^{2}$ words.) In some cases you may have to set $h < s$ to conserve storage, but beware that the rate of convergence will probably fall off severely.

$ifail = 9$: The basis is singular after several attempts to factorize it (adding slacks where necessary). Either the problem is badly scaled or the value of the optional parameter LU Factor Tolerance is too large.

$ifail = 10$: Numerical error in trying to satisfy the general constraints. The basis is very ill-conditioned.
An $L U$ factorization of the basis has just been obtained and used to recompute the basic variables $x_{B}$ , given the present values of the superbasic and nonbasic variables. However, a row check has revealed that the resulting solution does not satisfy the current constraints $A x - s = 0$ sufficiently well.
This probably means that the current basis is very ill-conditioned. Request the Scale Option if there are any linear constraints and variables.
For certain highly structured basis matrices (notably those with band structure), a systematic growth may occur in the factor $U$ . Consult the description of Umax, Umin and Growth in [Description of Monitoring Information], and set the optional parameter LU Factor Tolerance to $2.0$ (or possibly even smaller, but not less than $1.0$ ).

$ifail = 11$: An indefinite matrix was detected during the computation of the reduced Hessian factor $R$ (see [Definition of the Working Set and Search Direction]). This may be caused by $H$ being indefinite. Check also that qphx has been coded correctly and that all relevant elements of $H x$ have been assigned their correct values. If qphx is coded correctly and $H$ is positive semidefinite, the failure may be caused by ill conditioning. Try reducing the values of the optional parameters LU Factor Tolerance and LU Update Tolerance. If there are very large values in $H$ , check the scaling of the variables and constraints.

$ifail = 12$: Internal memory allocation failed when attempting to obtain the required workspace. Please contact NAG.

$ifail = 13$: Internal memory allocation was insufficient. Please contact NAG.

$ifail = 14$: An error has occurred in the basis package, perhaps indicating incorrect setup of arrays inda and loca. Set the optional parameter Print File and examine the output carefully for further information.

$ifail = 15$: An unexpected error has occurred. Set the optional parameter Print File and examine the output carefully for further information.

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

Accuracy

e04nq 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 a description of the printed output.

Description of the Printed Output

Print Level > 0

, one line of information is output to the Print File every

k

th iteration, where

k

is the specified Print Frequency. A heading is printed before the first such line following a basis factorization. The heading contains the items described below. In this description, a pricing operation is defined to be the process by which one or more nonbasic variables are selected to become superbasic (in addition to those already in the superbasic set). The variable selected will be denoted by jq. If the problem is purely linear, variable jq will usually become basic immediately (unless it should happen to reach its opposite bound and return to the nonbasic set).

If optional parameter Partial Price is in effect, variable jq is selected from

A_{pp}

I_{pp}

, the ppth segments of the constraint matrix

(\begin{matrix} A & - I \end{matrix})

Label	Description
Itn	is the iteration count.
pp	is the partial-price indicator. The variable selected by the last pricing operation came from the ppth partition of $A$ and $- I$ . Note that pp is reset to zero whenever the basis is refactorized.
dj	is the value of the reduced gradient (or reduced cost) for the variable selected by the pricing operation at the start of the current iteration. Algebraically, dj is $d_{j} = g_{j} - π^{T} a_{j}$ , for $j = jq$ , where $g_{j}$ is the gradient of the current objective function, $π$ is the vector of dual variables, and $a_{j}$ is the $j$ th column of the constraint matrix $(\begin{matrix} A & - I \end{matrix})$ . Note that dj is the norm of the reduced-gradient vector at the start of the iteration, just after the pricing operation.
+SBS	is the variable jq selected by the pricing operation to be added to the superbasic set.
-SBS	is the variable chosen to leave the superbasic set. It has become basic if the entry under -B is nonzero, otherwise it becomes nonbasic.
-BS	is the variable removed from the basis to become nonbasic.
Step	is the value of the step length $α$ taken along the current search direction $p$ . The variables $x$ have just been changed to $x + α p$ . If a variable is made superbasic during the current iteration (i.e., +SBS is positive), Step will be the step to the nearest bound. During the optimality phase, the step can be greater than unity only if the reduced Hessian is not positive definite.
Pivot	is the $r$ th element of a vector $y$ satisfying $B y = a_{q}$ whenever $a_{q}$ (the $q$ th column of the constraint matrix $(\begin{matrix} A & - I \end{matrix})$ replaces the $r$ th column of the basis matrix $B$ . Wherever possible, Step is chosen so as to avoid extremely small values of Pivot (since they may cause the basis to be nearly singular). In extreme cases, it may be necessary to increase the value of the optional parameter Pivot Tolerance to exclude very small elements of $y$ from consideration during the computation of Step.
nInf	is the number of violated constraints (infeasibilities) before the present iteration. This number will not increase unless iterations are in elastic mode.
sInf	is the sum of infeasibilities before the present iteration. It will usually decrease at each nonzero step, but if nInf decreases by $2$ or more, sInf may occasionally increase. However, in elastic mode it will decrease monotonically.
Objective	is the value of the current objective function after the present iteration. Note, if Elastic Mode is $2$ , the heading is Composite Obj.
L+U	L is the number of nonzeros in the basis factor $L$ . Immediately after a basis factorization $B = L U$ , L contains lenL (see [Description of Monitoring Information]). Further nonzeros are added to L when various columns of $B$ are later replaced. (Thus, L increases monotonically.) U is the number of nonzeros in the basis factor $U$ . Immediately after a basis factorization $B = L U$ , U contains lenU (see [Description of Monitoring Information]). As columns of $B$ are replaced, the matrix $U$ is maintained explicitly (in sparse form). The value of U may fluctuate up or down; in general, it will tend to increase.
ncp	is the number of compressions required to recover workspace in the data structure for $U$ . This includes the number of compressions needed during the previous basis factorization. Normally, ncp should increase very slowly.

The following will be output if the problem is QP or if the superbasic set is non-empty.

Label	Description
rgNorm	is the largest reduced-gradient among the superbasic variables after the current iteration. During the optimality phase, this will be approximately zero after a unit step.
nS	is the current number of superbasic variables.
condHz	is a lower bound on the condition number of the reduced Hessian (see [Definition of the Working Set and Search Direction]). The larger this number, the more difficult the problem. Attention should be given to the scaling of the variables and the constraints to guard against high values of condHz.

Example

This example minimizes the quadratic function

f (x) = c^{T} x + \frac{1}{2} x^{T} H x

, where

c = {(- 200.0, - 2000.0, - 2000.0, - 2000.0, - 2000.0, 400.0, 400.0)}^{T}

H = (\begin{matrix} 2 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 & 2 \\ 0 & 0 & 0 & 0 & 0 & 2 & 2 \end{matrix})

subject to the bounds

\begin{matrix} 0 \leq x_{1} \leq 200 \\ 0 \leq x_{2} \leq 2500 \\ 400 \leq x_{3} \leq 800 \\ 100 \leq x_{4} \leq 700 \\ 0 \leq x_{5} \leq 1500 \\ 0 \leq x_{6} \\ 0 \leq x_{7} \end{matrix}

and to the linear constraints

\begin{matrix} x_{1} & + & x_{2} & + & x_{3} & + & x_{4} & + & x_{5} & + & x_{6} & + & x_{7} & = & 2000 \\ 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 & 60 \\ 0.03 x_{1} & + & 0.05 x_{2} & + & 0.08 x_{3} & + & 0.02 x_{4} & + & 0.06 x_{5} & + & 0.01 x_{6} & + & 0.03 x_{7} & \leq & 100 \\ 0.02 x_{1} & + & 0.04 x_{2} & + & 0.01 x_{3} & + & 0.02 x_{4} & + & 0.02 x_{5} & \leq & 40 \\ 0.02 x_{1} & + & 0.03 x_{2} & + & 0.01 x_{5} & \leq & 30 \\ 1500 & \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} \\ 250 & \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 & 300 \end{matrix}

The initial point, which is infeasible, is

x_{0} = {(0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0)}^{T} .

The optimal solution (to five figures) is

x^{*} = {(0.0, 349.40, 648.85, 172.85, 407.52, 271.36, 150.02)}^{T} .

One bound constraint and four linear constraints are active at the solution. Note that the Hessian matrix

H

is positive semidefinite.

Example program (C#): e04nqe.cs

Example program data: e04nqe.d

Example program results: e04nqe.r

Algorithmic Details

This section contains a detailed description of the method used by e04nq.

Overview

e04nq is based on an inertia-controlling method that maintains a Cholesky factorization of the reduced Hessian (see below). The method is similar to that of 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 the method or appear in the printed output.

The method used has two distinct phases: finding an initial feasible point by minimizing the sum of infeasibilities (the feasibility phase), and minimizing the quadratic objective function within the feasible region (the optimality phase). The computations in both phases are performed by the same methods. The two-phase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities (the printed quantity sInf; see [Description of the Printed Output]) to the quadratic objective function (the printed quantity Objective; see [Description of the Printed Output]).

In general, an iterative process is required to solve a quadratic program. Given an iterate

(x, s)

in both the original variables

x

and the slack variables

s

, a new iterate

(\bar{x}, \bar{s})

is defined by

(\begin{matrix} \bar{x} \\ \bar{s} \end{matrix}) = (\begin{matrix} x \\ s \end{matrix}) + α p,

(3)

where the step length

α

is a non-negative scalar (the printed quantity Step; see [Description of Monitoring Information]), and

p

is called the search direction. (For simplicity, we shall consider a typical iteration and avoid reference to the index of the iteration.) Once an iterate is feasible (i.e., satisfies the constraints), all subsequent iterates remain feasible.

Definition of the Working Set and Search Direction

At each iterate

(x, s)

, a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied ‘exactly’ (to within the value of the optional parameter Feasibility Tolerance). The working set is the current prediction of the constraints that hold with equality at a solution of the LP or QP problem. Let

m_{W}

denote the number of constraints in the working set (including bounds), and let

W

denote the associated

m_{W}

(n + m)

working set matrix consisting of the

m_{W}

gradients of the working set constraints.

The search direction is defined so that constraints in the working set remain unaltered for any value of the step length. It follows that

p

must satisfy the identity

W p = 0 .

(4)

This characterisation allows

p

to be computed using any

n

n_{Z}

full-rank matrix

Z

that spans the null space of

W

. (Thus,

n_{Z} = n - m_{W}

and

W Z = 0

.) The null space matrix

Z

is defined from a sparse

L U

factorization of part of

W

(see (7) and (8)). The direction

p

will satisfy (4) if

p = Z p_{Z},

(5)

where

p_{Z}

is any

n_{Z}

-vector.

The working set contains the constraints

A x - s = 0

and a subset of the upper and lower bounds on the variables

(x, s)

. Since the gradient of a bound constraint

x_{j} \geq l_{j}

x_{j} \leq u_{j}

is a vector of all zeros except for

\pm 1

in position

j

, it follows that the working set matrix contains the rows of

(\begin{matrix} A & - I \end{matrix})

and the unit rows associated with the upper and lower bounds in the working set.

The working set matrix

W

can be represented in terms of a certain column partition of the matrix

(\begin{matrix} A & - I \end{matrix})

by (conceptually) partitioning the constraints

A x - s = 0

so that

B x_{B} + S x_{S} + N x_{N} = 0,

(6)

where

B

is a square nonsingular basis and

x_{B}

x_{S}

and

x_{N}

are the basic, superbasic and nonbasic variables respectively. The nonbasic variables are equal to their upper or lower bounds at

(x, s)

, and the superbasic variables are independent variables that are chosen to improve the value of the current objective function. The number of superbasic variables is

n_{S}

(the printed quantity nS; see [Description of the Printed Output]). Given values of

x_{N}

and

x_{S}

, the basic variables

x_{B}

are adjusted so that

(x, s)

satisfies (6).

P

is a permutation matrix such that

(\begin{matrix} A & - I \end{matrix}) P = (\begin{matrix} B & S & N \end{matrix})

, then

W

satisfies

W P = (\begin{matrix} B & S & N \\ 0 & 0 & I_{N} \end{matrix}),

(7)

where

I_{N}

is the identity matrix with the same number of columns as

N

The null space matrix

Z

is defined from a sparse

L U

factorization of part of

W

. In particular,

Z

is maintained in ‘reduced gradient’ form, using the LUSOL package (see Gill et al. (1991)) to maintain sparse

L U

factors of the basis matrix

B

as the

B S N

partition changes. Given the permutation

P

, the null space basis is given by

Z = P (\begin{matrix} - B^{- 1} S \\ I \\ 0 \end{matrix}) .

(8)

This matrix is used only as an operator, i.e., it is never computed explicitly. Products of the form

Z v

and

Z^{T} g

are obtained by solving with

B

B^{T}

. This choice of

Z

implies that

n_{Z}

, the number of ‘degrees of freedom’ at

(x, s)

, is the same as

n_{S}

, the number of superbasic variables.

Let

g_{Z}

and

H_{Z}

denote the reduced gradient and reduced Hessian of the objective function:

g_{Z} = Z^{T} g and H_{Z} = Z^{T} H Z,

(9)

where

g

is the objective gradient at

(x, s)

. Roughly speaking,

g_{Z}

and

H_{Z}

describe the first and second derivatives of an

n_{S}

-dimensional unconstrained problem for the calculation of

p_{Z}

. (The condition estimator of

H_{Z}

is the quantity condHz in the monitoring file output; see [Description of the Printed Output].)

At each iteration, an upper triangular factor

R

is available such that

H_{Z} = R^{T} R

. Normally,

R

is computed from

R^{T} R = Z^{T} H Z

at the start of the optimality phase and then updated as the QP working set changes. For efficiency, the dimension of

R

should not be excessive (say,

n_{S} \leq 1000

). This is guaranteed if the number of nonlinear variables is ‘moderate’.

If the QP problem contains linear variables,

H

is positive semidefinite and

R

may be singular with at least one zero diagonal element. However, an inertia-controlling strategy is used to ensure that only the last diagonal element of

R

can be zero. (See Gill et al. (1991) for a discussion of a similar strategy for indefinite quadratic programming.)

If the initial

R

is singular, enough variables are fixed at their current value to give a nonsingular

R

. This is equivalent to including temporary bound constraints in the working set. Thereafter,

R

can become singular only when a constraint is deleted from the working set (in which case no further constraints are deleted until

R

becomes nonsingular).

Main Iteration

If the reduced gradient is zero,

(x, s)

is a constrained stationary point on the working set. During the feasibility phase, the reduced gradient will usually be zero only at a vertex (although it may be zero elsewhere in the presence of constraint dependencies). During the optimality phase, a zero reduced gradient implies that

x

minimizes the quadratic objective function when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers

λ

are defined from the equations

W^{T} λ = g (x) .

(10)

A Lagrange multiplier,

λ_{j}

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

λ_{j} \leq σ

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

λ_{j} \geq - σ

when the associated constraint is at its lower bound, where

σ

depends on the value of the optional parameter Optimality Tolerance. If a multiplier is nonoptimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by continuing the minimization with the corresponding constraint excluded from the working set. (This step is sometimes referred to as ‘deleting’ a constraint from the working set.) If optimal multipliers occur during the feasibility phase but the sum of infeasibilities is nonzero, there is no feasible point and the method terminates immediately with

ifail = 3

The special form (7) of the working set allows the multiplier vector

λ

, the solution of (10), to be written in terms of the vector

d = (\begin{matrix} g \\ 0 \end{matrix}) - (\begin{matrix} A^{T} \\ - I \end{matrix}) π = (\begin{matrix} g - A^{T} π \\ π \end{matrix}),

(11)

where

π

satisfies the equations

B^{T} π = g_{B}

, and

g_{B}

denotes the basic elements of

g

. The elements of

π

are the Lagrange multipliers

λ_{j}

associated with the equality constraints

A x - s = 0

. The vector

d_{N}

of nonbasic elements of

d

consists of the Lagrange multipliers

λ_{j}

associated with the upper and lower bound constraints in the working set. The vector

d_{S}

of superbasic elements of

d

is the reduced gradient

g_{Z}

in (9). The vector

d_{B}

of basic elements of

d

is zero, by construction. (The Euclidean norm of

d_{S}

and the final values of

d_{S}

g

and

π

are the quantities rgNorm, Reduced Gradnt, Obj Gradient and Dual Activity in the monitoring file output; see [Description of Monitoring Information].)

If the reduced gradient is not zero, Lagrange multipliers need not be computed and the search direction is given by

p = Z p_{Z}

(see (8) and (12)). The step length is chosen to maintain feasibility with respect to the satisfied constraints.

There are two possible choices for

p_{Z}

, depending on whether or not

H_{Z}

is singular. If

H_{Z}

is nonsingular,

R

is nonsingular and

p_{Z}

in (5) is computed from the equations

R^{T} R p_{Z} = - g_{Z},

(12)

where

g_{Z}

is the reduced gradient at

x

. In this case,

(x, s) + p

is the minimizer of the objective function subject to the working set constraints being treated as equalities. If

(x, s) + p

is feasible,

α

is defined to be unity. In this case, the reduced gradient at

(\bar{x}, \bar{s})

will be zero, and Lagrange multipliers are computed at the next iteration. Otherwise,

α

is set to

α_{N}

, the step to the ‘nearest’ constraint along

p

. This constraint is then added to the working set at the next iteration.

H_{Z}

is singular, then

R

must also be singular, and an inertia-controlling strategy is used to ensure that only the last diagonal element of

R

is zero. (See Gill et al. (1991) for a discussion of a similar strategy for indefinite quadratic programming.) In this case,

p_{Z}

satisfies

p_{Z}^{T} H_{Z} p_{Z} = 0 and g_{Z}^{T} p_{Z} \leq 0,

(13)

which allows the objective function to be reduced by any step of the form

(x, s) + α p

, where

α > 0

. The vector

p = Z p_{Z}

is a direction of unbounded descent for the QP problem in the sense that the QP objective is linear and decreases without bound along

p

. If no finite step of the form

(x, s) + α p

(where

α > 0

) reaches a constraint not in the working set, the QP problem is unbounded and the method terminates immediately with

ifail = 6

. Otherwise,

α

is defined as the maximum feasible step along

p

and a constraint active at

(x, s) + α p

is added to the working set for the next iteration.

e04nq makes explicit allowance for infeasible constraints. Infeasible linear constraints are detected first by solving a problem of the form

\begin{matrix} \underset{x, v, w}{minimize} e^{T} (v + w) subject to ​ l \leq (\begin{matrix} x \\ G x - v + w \end{matrix}) \leq u, v \geq 0, w \geq 0, \end{matrix}

(14)

where

e^{T} = (1, 1, \dots, 1)

. This is equivalent to minimizing the sum of the general linear constraint violations subject to the simple bounds. (In the linear programming literature, the approach is often called elastic programming.)

Miscellaneous

If the basis matrix is not chosen carefully, the condition of the null space matrix

Z

in (8) could be arbitrarily high. To guard against this, the method implements a ‘basis repair’ feature in which the LUSOL package (see Gill et al. (1991)) is used to compute the rectangular factorization

{(\begin{matrix} B & S \end{matrix})}^{T} = L U,

(15)

returning just the permutation

P

that makes

P L P^{T}

unit lower triangular. The pivot tolerance is set to require

{|P L P^{T}|}_{i j} \leq 2

, and the permutation is used to define

P

in (7). It can be shown that

‖Z‖

is likely to be little more than unity. Hence,

Z

should be well-conditioned regardless of the condition of

W

. This feature is applied at the beginning of the optimality phase if a potential

B - S

ordering is known.

The EXPAND procedure (see Gill et al. (1989)) is used to reduce the possibility of cycling at a point where the active constraints are nearly linearly dependent. Although there is no absolute guarantee that cycling will not occur, the probability of cycling is extremely small (see Hall and McKinnon (1996)). The main feature of EXPAND is that the feasibility tolerance is increased at the start of every iteration. This allows a positive step to be taken at every iteration, perhaps at the expense of violating the bounds on

(x, s)

by a small amount.

Suppose that the value of the optional parameter Feasibility Tolerance is

δ

. Over a period of

K

iterations (where

K

is the value of the optional parameter Expand Frequency), the feasibility tolerance actually used by the method (i.e., the working feasibility tolerance) increases from

0.5 δ

δ

(in steps of

0.5 δ / K

At certain stages the following ‘resetting procedure’ is used to remove small constraint infeasibilities. First, all nonbasic variables are moved exactly onto their bounds. A count is kept of the number of nontrivial adjustments made. If the count is nonzero, the basic variables are recomputed. Finally, the working feasibility tolerance is reinitialized to

0.5 δ

If a problem requires more than

K

iterations, the resetting procedure is invoked and a new cycle of 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 the method reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any nontrivial adjustments are made, iterations are continued.

The EXPAND procedure not only allows a positive step to be taken at every iteration, but also provides a potential choice of constraints to be added to the working set. All constraints at a distance

α

(where

α \leq α_{N}

) 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. This strategy helps keep the basis matrix

B

well-conditioned.

Description of Monitoring Information

This section describes the intermediate printout and final printout which constitutes the monitoring information produced by e04nq. (See also the description of the optional parameters Print File and Print Level.) You can control the level of printed output.

Crash Statistics

When

Print Level \geq 10

Print File > 0

and System Information Yes has been specified, the following lines of intermediate printout (less than

120

characters) are produced on the unit number specified by optional parameter Print File whenever

start = "C"

(see [Parameters]). They refer to the number of columns selected by the Crash procedure during each of several passes through

A

, whilst searching for a triangular basis matrix.

Label	Description
Slacks	is the number of slacks selected initially.
Free cols	is the number of free columns in the basis, including those whose bounds are rather far apart.
Preferred	is the number of ‘preferred’ columns in the basis (i.e., $hs [j - 1] = 3$ for some $j \leq n$ ). It will be a subset of the columns for which $hs [j - 1] = 3$ was specified.
Unit	is the number of unit columns in the basis.
Double	is the number of double columns in the basis.
Triangle	is the number of triangular columns in the basis.
Pad	is the number of slacks used to pad the basis (to make it a nonsingular triangle).

Basis Factorization Statistics

When

Print Level \geq 10

and

Print File > 0

, the first seven items of intermediate printout in the list below are produced on the unit number specified by optional parameter Print File whenever the matrix

B

B_{S} = {(\begin{matrix} B & S \end{matrix})}^{T}

is factorized. Gaussian elimination is used to compute an

L U

factorization of

B

B_{S}

, where

P L P^{T}

is a lower triangular matrix and

P U Q

is an upper triangular matrix for some permutation matrices

P

and

Q

. The factorization is stabilized in the manner described under the optional parameter LU Factor Tolerance. In addition, if System Information Yes has been specified, the entries from Elems onwards are also output.

Label

Description

Factor

the number of factorizations since the start of the run.

Demand

a code giving the reason for the present factorization, as follows:

Code	Meaning
0	First $L U$ factorization.
1	The number of updates reached the Factorization Frequency.
2	The nonzeros in the updated factors have increased significantly.
7	Not enough storage to update factors.
10	Row residuals are too large (see the description of the optional parameter Check Frequency).
11	Ill-conditioning has caused inconsistent results.

Itn

is the current minor iteration number.

Nonlin

is the number of nonlinear variables in the current basis

B

Linear

is the number of linear variables in

B

Slacks

is the number of slack variables in

B

B, BR, BS or BT factorize

is the type of

L U

factorization.

B	periodic factorization of the basis $B$ .
BR	more careful rank-revealing factorization of $B$ using threshold rook pivoting. This occurs mainly at the start, if the first basis factors seem singular or ill-conditioned. Followed by a normal B factorize.
BS	$B_{S}$ is factorized to choose a well-conditioned $B$ from the current $(B S)$ . Followed by a normal B factorize.
BT	same as BS except the current $B$ is tried first and accepted if it appears to be not much more ill-conditioned than after the previous BS factorize.

is the number of rows in

B

B_{S}

is the number of columns in

B

B_{S}

. Preceded by ‘=’ or ‘>’ respectively.

Elems

is the number of nonzero elements in

B

B_{S}

Amax

is the largest nonzero in

B

B_{S}

Density

is the percentage nonzero density of

B

B_{S}

Merit/MerRP/MerCP

Merit is the average Markowitz merit count for the elements chosen to be the diagonals of

P U Q

. Each merit count is defined to be

(c - 1) (r - 1)

where

c

and

r

are the number of nonzeros in the column and row containing the element at the time it is selected to be the next diagonal. Merit is the average of n such quantities. It gives an indication of how much work was required to preserve sparsity during the factorization. If LU Complete Pivoting or LU Rook Pivoting has been selected, this heading is changed to MerCP, respectively MerRP.

lenL

is the number of nonzeros in

L

L+U

is the number of nonzeros representing the basis factors

L

and

U

. Immediately after a basis factorization

B = L U

, this is lenL+lenU, the number of subdiagonal elements in the columns of a lower triangular matrix and the number of diagonal and superdiagonal elements in the rows of an upper-triangular matrix. Further nonzeros are added to L when various columns of

B

are later replaced. As columns of

B

are replaced, the matrix

U

is maintained explicitly (in sparse form). The value of L will steadily increase, whereas the value of U may fluctuate up or down. Thus the value of L+U may fluctuate up or down (in general, it will tend to increase).

Cmpressns

is the number of times the data structure holding the partially factored matrix needed to be compressed to recover unused storage. Ideally this number should be zero. If it is more than

3

4

, the amount of workspace available to e04nq should be increased for efficiency.

Incres

is the percentage increase in the number of nonzeros in

L

and

U

relative to the number of nonzeros in

B

B_{S}

Utri

is the number of triangular rows of

B

B_{S}

at the top of

U

lenU

the number of nonzeros in

U

, including its diagonals.

Ltol

is the largest subdiagonal element allowed in

L

. This is the specified LU Factor Tolerance or a smaller value that is currently being used for greater stability.

Umax

the maximum nonzero element in

U

Ugrwth

is the ratio

Umax / Amax

, which ideally should not be substantially larger than

10.0

100.0

. If it is orders of magnitude larger, it may be advisable to reduce the LU Factor Tolerance to

5.0

4.0

3.0

2.0

, say (but bigger than

1.0

As long as Lmax is not large (say

5.0

or less),

\max (Amax, Umax) / DUmin

gives an estimate of the condition number

B

. If this is extremely large, the basis is nearly singular. Slacks are used to replace suspect columns of

B

and the modified basis is refactored.

Ltri

is the number of triangular columns of

B

B_{S}

at the left of

L

dense1

is the number of columns remaining when the density of the basis matrix being factorized reached

0.3

Lmax

is the actual maximum subdiagonal element in

L

(bounded by Ltol).

Akmax

is the largest nonzero generated at any stage of the

L U

factorization. (Values much larger than Amax indicate instability.) Akmax is not printed if LU Partial Pivoting is selected.

Agrwth

is the ratio

Akmax / Amax

. Values much larger than

100

(say) indicate instability. Agrwth is not printed if LU Partial Pivoting is selected.

bump

is the size of the block to be factorized nontrivially after the triangular rows and columns of

B

B_{S}

have been removed.

dense2

is the number of columns remaining when the density of the basis matrix being factorized reached

0.6

. (The Markowitz pivot strategy searches fewer columns at that stage.)

DUmax

is the largest diagonal of

P U Q

DUmin

is the smallest diagonal of

P U Q

condU

the ratio

DUmax / DUmin

, which estimates the condition number of

U

(and of

B

if Ltol is less than

5.0

, say).

Basis Map

When

Print Level \geq 10

and

Print File > 0

, the following lines of intermediate printout (less than

80

characters) are produced on the unit number specified by optional parameter Print File. They refer to the elements of the names array (see [Parameters]).

Label	Description
Name	gives the name for the problem (blank if problem unnamed).
Infeasibilities	gives the number of infeasibilities. Printed only if the final point is infeasible.
Objective Value	gives the objective value at the final point (or the value of the sum of infeasibilities). Printed only if the final point is feasible.
Status	gives the exit status for the problem (i.e., Optimal soln, Weak soln, Unbounded, Infeasible, Excess itns, Error condn or Feasble soln) followed by details of the direction of the optimization (i.e., (Min) or (Max)).
Iteration	gives the iteration number when the file was created.
Superbasics	gives the number of superbasic variables.
Objective	gives the name of the free row for the problem (blank if objective unnamed).
RHS	gives the name of the constraint right-hand side for the problem (blank if objective unnamed).
Ranges	gives the name of the ranges for the problem (blank if objective unnamed).
Bounds	gives the name of the bounds for the problem (blank if objective unnamed).

Solution Output

At the end of a run, the final solution will be output to the Print file. Some header information appears first to identify the problem and the final state of the optimization procedure. A ROWS section and a COLUMNS section then follow, giving one line of information for each row and column.

The ROWS section

General constraints take the form

l \leq A x \leq u

. The

i

th constraint is therefore of the form

α \leq ν_{i} x \leq β,

where

ν_{i}

is the

i

th row of

A

Internally, the constraints take the form

A x - s = 0

, where

s

is the set of slack variables (which happen to satisfy the bounds

l \leq s \leq u

). For the

i

th constraint, the slack variable

s_{i}

is directly available, and it is sometimes convenient to refer to its state. It should satisfy

α \leq s_{i} \leq β

. A fullstop (.) is printed for any numerical value that is exactly zero.

Label

Description

Number

is the value of

n + i

. (This is used internally to refer to

s_{i}

in the intermediate output.)

Row

gives the name of

ν_{i}

State

the state of

ν_{i}

(the state of

s_{i}

relative to the bounds

α

and

β

). The various states possible are as follows:

LL	$s_{i}$ is nonbasic at its lower limit, $α$ .
UL	$s_{i}$ is nonbasic at its upper limit, $β$ .
EQ	$s_{i}$ is nonbasic and fixed at the value $α = β$ .
FR	$s_{i}$ is nonbasic and currently zero, even though it is free to take any value between its bounds $α$ and $β$ .
BS	$s_{i}$ is basic.
SBS	$s_{i}$ is superbasic.

A key is sometimes printed before State. Note that unless the optional parameter

Scale Option = 0

is specified, the tests for assigning a key are applied to the variables of the scaled problem.

A	Alternative optimum possible. The variable is nonbasic, but its reduced gradient is essentially zero. This means that if the variable were allowed to start moving away from its bound, there would be no change in the value of 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 basic or superbasic, but it is equal (or very close) to one of its bounds.
I	Infeasible. The variable is basic or superbasic and is currently violating one of its bounds by more than the value of the Feasibility Tolerance.
N	Not precisely optimal. If the slack is superbasic, the dual variable $π_{i}$ is not sufficiently small, as measured by the Optimality Tolerance. If the slack is nonbasic, $π_{i}$ is not sufficiently positive or negative. If a loose Optimality Tolerance has been used, or if iterations were terminated before optimality, this key might be helpful in deciding whether or not to restart the run.

Activity

is the value of

ν_{i} x

at the final iterate.

Slack Activity

is the value by which the row differs from its nearest bound. (For the free row (if any), it is set to Activity.)

Lower Limit

α

, the lower bound specified for the variable

s_{i}

. None indicates that

bl [j - 1] \leq - infbnd

Upper Limit

β

, the upper bound specified for the variable

s_{i}

. None indicates that

bu [j - 1] \geq infbnd

Dual Activity

is the value of the dual variable

π_{i}

(the Lagrange multiplier for

ν_{i}

; see [Main Iteration]). For FP problems,

π_{i}

is set to zero.

gives the index

i

of the

i

th row.

The COLUMNS Section

Let the

j

th component of

x

be the variable

x_{j}

and assume that it satisfies the bounds

α \leq x_{j} \leq β

. A fullstop (.) is printed for any numerical value that is exactly zero.

Label

Description

Number

is the column number

j

. (This is used internally to refer to

x_{j}

in the intermediate output.)

Column

gives the name of

x_{j}

State

the state of

x_{j}

relative to the bounds

α

and

β

. The various states possible are as follows:

LL	$x_{j}$ is nonbasic at its lower limit, $α$ .
UL	$x_{j}$ is nonbasic at its upper limit, $β$ .
EQ	$x_{j}$ is nonbasic and fixed at the value $α = β$ .
FR	$x_{j}$ is nonbasic and currently zero, even though it is free to take any value between its bounds $α$ and $β$ .
BS	$x_{j}$ is basic.
SBS	$x_{j}$ is superbasic.

A key is sometimes printed before State. Note that unless the optional parameter

Scale Option = 0

is specified, the tests for assigning a key are applied to the variables of the scaled problem.

A	Alternative optimum possible. The variable is nonbasic, but its reduced gradient is essentially zero. This means that if the variable were allowed to start moving away from its bound, there would be no change in the value of 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 basic or superbasic, but it is equal (or very close) to one of its bounds.
I	Infeasible. The variable is basic or superbasic and is currently violating one of its bounds by more than the value of the Feasibility Tolerance.
N	Not precisely optimal. If the slack is superbasic, the dual variable $π_{i}$ is not sufficiently small, as measured by the Optimality Tolerance. If the slack is nonbasic, $π_{i}$ is not sufficiently positive or negative. If a loose Optimality Tolerance has been used, or if iterations were terminated before optimality, this key might be helpful in deciding whether or not to restart the run.

Activity

is the value of

x_{j}

at the final iterate.

Obj Gradient

is the value of

g_{j}

at the final iterate. For FP problems,

g_{j}

is set to zero.

Lower Limit

is the lower bound specified for the variable. None indicates that

bl [j - 1] \leq - infbnd

Upper Limit

is the upper bound specified for the variable. None indicates that

bu [j - 1] \geq infbnd

Reduced Gradnt

is the value of

d_{j}

at the final iterate (see [Main Iteration]). For FP problems,

d_{j}

is set to zero.

m + j

is the value of

m + j

Note: if two problems are the same except that one minimizes

f (x)

and the other maximizes

- f (x)

, their solutions will be the same but the signs of the dual variables

π_{i}

and the reduced gradients

d_{j}

will be reversed.

The Solution File

If a positive Solution File is specified, the information contained in a printed solution may also be output to the relevant file (which may be the Print file if so desired). Infinite Upper and Lower limits appear as

\pm 10^{20}

rather than None. Other real values are output with format 1pe16.6. The maximum line length is

111

characters, including what would be the carriage-control character if the file were printed.

A Solution file is intended to be read from disk by a self-contained program that extracts and saves certain values as required for possible further computation. Typically the first

14

lines would be ignored. Each subsequent line may be read using adapted to suit the occasion. The end of the ROWS section is marked by a line that starts with a

1

and is otherwise blank. If this and the next

4

lines are skipped, the COLUMNS section (see [The COLUMNS Section]) can then be read under the same format. (There should be no need to use any BACKSPACE statements.)

The Summary File

If Summary File

f

is specified with

f > 0

, certain brief information will be output to unit

f

. When e04nq is run interactively, unit

f

will usually be the terminal. For batch jobs a disk file should be used, to retain a concise log of each run if desired. (A Summary File is more easily perused than the associated Print file).

A Summary file (like the Print file) is not rewound after a problem has been processed. The maximum line length is

72

characters, including a carriage-control character in column 1.

The following information is included:

1.	The optional parameters supplied via the option setting methods, if any;
2.	The Basis file loaded, if any;
3.	The status of the solution after each basis factorization (whether feasible; the objective value; the number of function calls so far);
4.	The same information every $k$ th iteration, where $k$ is the specified Summary Frequency;
5.	Warnings and error messages;
6.	The exit condition and a summary of the final solution.

Item 4. is preceded by a blank line, but item 5. is not.

The meaning of the printout for linear constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’,

n

replaced by

m

names [j - 1]

replaced by

names [n + j - 1]

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:

Constrnt

gives the name 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 Residual column to become positive.

Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Description of the Printed Output

Example

Algorithmic Details

Overview

Definition of the Working Set and Search Direction

Main Iteration

Miscellaneous

Description of Monitoring Information

Crash Statistics

Basis Factorization Statistics

Basis Map

Solution Output

The ROWS section

The COLUMNS Section

The Solution File

The Summary File

See Also