e04nk solves sparse linear programming or convex quadratic programming problems.

Syntax

C#
public static void e04nk( int n, int m, int nnz, int iobj, int ncolh, E04..::..E04NK_QPHX qphx, double[] a, int[] ha, int[] ka, double[] bl, double[] bu, string start, string[] names, int nname, string[] crname, ref int ns, double[] xs, int[] istate, out int miniz, out int minz, out int ninf, out double sinf, out double obj, double[] clamda, int[] iz, double[] z, E04..::..e04nkOptions options, out int ifail )

public static void e04nk(
	int n,
	int m,
	int nnz,
	int iobj,
	int ncolh,
	E04..::..E04NK_QPHX qphx,
	double[] a,
	int[] ha,
	int[] ka,
	double[] bl,
	double[] bu,
	string start,
	string[] names,
	int nname,
	string[] crname,
	ref int ns,
	double[] xs,
	int[] istate,
	out int miniz,
	out int minz,
	out int ninf,
	out double sinf,
	out double obj,
	double[] clamda,
	int[] iz,
	double[] z,
	E04..::..e04nkOptions options,
	out int ifail
)

Visual Basic
Public Shared Sub e04nk ( _ n As Integer, _ m As Integer, _ nnz As Integer, _ iobj As Integer, _ ncolh As Integer, _ qphx As E04..::..E04NK_QPHX, _ a As Double(), _ ha As Integer(), _ ka As Integer(), _ bl As Double(), _ bu As Double(), _ start As String, _ names As String(), _ nname As Integer, _ crname As String(), _ ByRef ns As Integer, _ xs As Double(), _ istate As Integer(), _ <OutAttribute> ByRef miniz As Integer, _ <OutAttribute> ByRef minz As Integer, _ <OutAttribute> ByRef ninf As Integer, _ <OutAttribute> ByRef sinf As Double, _ <OutAttribute> ByRef obj As Double, _ clamda As Double(), _ iz As Integer(), _ z As Double(), _ options As E04..::..e04nkOptions, _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub e04nk ( _
	n As Integer, _
	m As Integer, _
	nnz As Integer, _
	iobj As Integer, _
	ncolh As Integer, _
	qphx As E04..::..E04NK_QPHX, _
	a As Double(), _
	ha As Integer(), _
	ka As Integer(), _
	bl As Double(), _
	bu As Double(), _
	start As String, _
	names As String(), _
	nname As Integer, _
	crname As String(), _
	ByRef ns As Integer, _
	xs As Double(), _
	istate As Integer(), _
	<OutAttribute> ByRef miniz As Integer, _
	<OutAttribute> ByRef minz As Integer, _
	<OutAttribute> ByRef ninf As Integer, _
	<OutAttribute> ByRef sinf As Double, _
	<OutAttribute> ByRef obj As Double, _
	clamda As Double(), _
	iz As Integer(), _
	z As Double(), _
	options As E04..::..e04nkOptions, _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void e04nk( int n, int m, int nnz, int iobj, int ncolh, E04..::..E04NK_QPHX^ qphx, array<double>^ a, array<int>^ ha, array<int>^ ka, array<double>^ bl, array<double>^ bu, String^ start, array<String^>^ names, int nname, array<String^>^ crname, int% ns, array<double>^ xs, array<int>^ istate, [OutAttribute] int% miniz, [OutAttribute] int% minz, [OutAttribute] int% ninf, [OutAttribute] double% sinf, [OutAttribute] double% obj, array<double>^ clamda, array<int>^ iz, array<double>^ z, E04..::..e04nkOptions^ options, [OutAttribute] int% ifail )

Visual C++

public:
static void e04nk(
	int n, 
	int m, 
	int nnz, 
	int iobj, 
	int ncolh, 
	E04..::..E04NK_QPHX^ qphx, 
	array<double>^ a, 
	array<int>^ ha, 
	array<int>^ ka, 
	array<double>^ bl, 
	array<double>^ bu, 
	String^ start, 
	array<String^>^ names, 
	int nname, 
	array<String^>^ crname, 
	int% ns, 
	array<double>^ xs, 
	array<int>^ istate, 
	[OutAttribute] int% miniz, 
	[OutAttribute] int% minz, 
	[OutAttribute] int% ninf, 
	[OutAttribute] double% sinf, 
	[OutAttribute] double% obj, 
	array<double>^ clamda, 
	array<int>^ iz, 
	array<double>^ z, 
	E04..::..e04nkOptions^ options, 
	[OutAttribute] int% ifail
)

F#
static member e04nk : n : int * m : int * nnz : int * iobj : int * ncolh : int * qphx : E04..::..E04NK_QPHX * a : float[] * ha : int[] * ka : int[] * bl : float[] * bu : float[] * start : string * names : string[] * nname : int * crname : string[] * ns : int byref * xs : float[] * istate : int[] * miniz : int byref * minz : int byref * ninf : int byref * sinf : float byref * obj : float byref * clamda : float[] * iz : int[] * z : float[] * options : E04..::..e04nkOptions * ifail : int byref -> unit

static member e04nk : 
        n : int * 
        m : int * 
        nnz : int * 
        iobj : int * 
        ncolh : int * 
        qphx : E04..::..E04NK_QPHX * 
        a : float[] * 
        ha : int[] * 
        ka : int[] * 
        bl : float[] * 
        bu : float[] * 
        start : string * 
        names : string[] * 
        nname : int * 
        crname : string[] * 
        ns : int byref * 
        xs : float[] * 
        istate : int[] * 
        miniz : int byref * 
        minz : int byref * 
        ninf : int byref * 
        sinf : float byref * 
        obj : float byref * 
        clamda : float[] * 
        iz : int[] * 
        z : float[] * 
        options : E04..::..e04nkOptions * 
        ifail : int byref -> unit

Parameters

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: $n \geq 1$ .

m: Type: System..::..Int32
On entry: $m$ , the number of general linear constraints (or slacks). This is the number of rows in $A$ , including the free row (if any; see iobj).

Constraint: $m \geq 1$ .

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

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

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, i.e., the problem is either an FP problem (in which case iobj must be set to zero), or a QP problem with $c = 0$ .

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

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.

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

qphx

Type: NagLibrary..::..E04..::..E04NK_QPHX

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

H x

. If

H

has zero rows and columns, it is most efficient to order the variables

x = {(\begin{matrix} y & z \end{matrix})}^{T}

so that

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}),

where the nonlinear variables

y

appear first as shown. For FP and LP problems, qphx will never be called by e04nk and hence qphx may be the dummy method E04NKU/E54NKU.

A delegate of type E04NK_QPHX.

a: Type: array<System..::..Double>[]()[][]
An array of size [nnz]
On entry: the nonzero elements of $A$ , ordered by increasing column index. Note that elements with the same row and column indices are not allowed.

ha: Type: array<System..::..Int32>[]()[][]
An array of size [nnz]
On entry: $ha [i - 1]$ must contain the row index of the nonzero element stored in $a [i - 1]$ , for $i = 1, 2, \dots, nnz$ . Note that the row indices for a column may be supplied in any order.

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

ka

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

An array of size [

n + 1

]

On entry:

ka [j - 1]

must contain the index in a of the start of the

j

th column, for

j = 1, 2, \dots, n

ka [n]

must be set to

nnz + 1

. To specify the

j

th column as empty, set

ka [j - 1] = ka [j]

. As a consequence

ka [0]

is always

1

Constraints:

$ka [0] = 1$ ;
$ka [j - 1] \geq 1$ , for $j = 2, 3, \dots, n$ ;
$ka [n] = nnz + 1$ ;
$0 \leq ka [j] - ka [j - 1] \leq m$ , 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 $A x$ (or slacks $s$ ) and the free row (if any). To specify a nonexistent lower bound (i.e., $l_{j} = - \infty$ ), set $bl [j - 1] \leq - bigbnd$ , where $bigbnd$ is the value of 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 $|β| < bigbnd$ . 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

(or 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 $iobj > 0$ , $bu [n + iobj - 1] \geq bigbnd$ ;
$bl [j - 1] \leq bu [j - 1]$ , for $j = 1, 2, \dots, n + m$ ;
if $bl [j - 1] = bu [j - 1] = β$ , $|β| < bigbnd$ .

start

Type: System..::..String

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

$start = "C"$: An internal Crash procedure will be used to choose an initial basis matrix $B$ .
$start = "W"$: A basis is already defined in istate (probably from a previous call).

Constraint:

start = "C"

"W"

names

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

An array of size [

5

]

On entry: a set of names associated with the so-called MPSX form of the problem, as follows:

$names [0]$: Must contain the name for the problem (or be blank).
$names [1]$: Must contain the name for the free row (or be blank).
$names [2]$: Must contain the name for the constraint right-hand side (or be blank).
$names [3]$: Must contain the name for the ranges (or be blank).
$names [4]$: Must contain the name for the bounds (or be blank).

(These names are used in the monitoring file output; see [Description of Monitoring Information].)

nname

Type: System..::..Int32

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

$nname = 1$: 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

crname: Type: array<System..::..String>[]()[][]
An array of size [nname]
On entry: the optional column and row names, respectively.
If $nname = 1$ , crname 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 $crname [n + iobj - 1]$ .

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.

xs: Type: array<System..::..Double>[]()[][]
An array of size [ $n + m$ ]
On entry: the initial values of the variables and slacks $(x, s)$ . (See the description for istate.)
On exit: the final values of the variables and slacks $(x, s)$ .

istate

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

An array of size [

n + m

]

On entry: if

start = "C"

, the first n elements of istate and xs 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

istate [j - 1]

are as follows:

$istate [j - 1]$	State of $xs [j - 1]$ 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

istate [j - 1] = 0

and

xs [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

istate [j - 1] = 4

and

xs [j - 1] = bl [j - 1]

istate [j - 1] = 5

and

xs [j - 1] = bu [j - 1]

as appropriate.

Following the Crash procedure, variables for which

istate [j - 1] = 2

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

xs [j - 1]

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

, or at the value

bl [j - 1]

bu [j - 1]

closest to

xs [j - 1]

start = "W"

, istate and xs must specify the initial states and values, respectively, of the variables and slacks

(x, s)

. If e04nk has been called previously with the same values of n and m, istate already contains satisfactory information.

Constraints:

if $start = "C"$ , $0 \leq istate [j - 1] \leq 5$ , for $j = 1, 2, \dots, n$ ;
if $start = "W"$ , $0 \leq istate [j - 1] \leq 3$ , 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

istate [j - 1]

is as follows:

$istate [j - 1]$	State of variable $j$	Normal value of $xs [j - 1]$
$0$	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

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

xs [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

miniz: Type: System..::..Int32%
On exit: the minimum value of _leniz required to start solving the problem. If $ifail = 12$ , e04nk may be called again with _leniz suitably larger than miniz. (The bigger the better, since it is not certain how much workspace the basis factors need.)

minz: Type: System..::..Int32%
On exit: the minimum value of _lenz required to start solving the problem. If $ifail = 13$ , e04nk may be called again with _lenz suitably larger than minz. (The bigger the better, since it is not certain how much workspace the basis factors need.)

ninf: Type: System..::..Int32%
On exit: the number of infeasibilities. This will be zero if $ifail = 0$ or $1$ .

sinf: Type: System..::..Double%
On exit: the sum of infeasibilities. This will be zero if $ninf = 0$ . (Note that e04nk does not attempt to compute the minimum value of sinf if $ifail = 3$ .)

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.

clamda: Type: array<System..::..Double>[]()[][]
An array of size [ $n + m$ ]
On exit: a set of Lagrange multipliers for the bounds on the variables and the general constraints. More precisely, the first n elements contain the multipliers (reduced costs) for the bounds on the variables, and the next m elements contain the multipliers (shadow prices) for the general linear constraints.

iz: Type: array<System..::..Int32>[]()[][]
An array of size [dim1]
Note: dim1 must satisfy the constraint: $_leniz \geq 1$

z: Type: array<System..::..Double>[]()[][]
An array of size [dim1]
Note: dim1 must satisfy the constraint: $_lenz \geq 1$
the dimension of the array z.
On entry: the dimension of the array z as declared in the (sub)program from which e04nk is called.

Constraint: $_lenz \geq 1$ .
The amounts of workspace provided and required (i.e., miniz and minz) are (by default for e04nk) output. Since the minimum values of _leniz and _lenz required to start solving the problem are returned in miniz and minz, respectively, you may prefer to obtain appropriate values from the output of a preliminary run with the length of z and iz set to $1$ . (e04nk will then terminate with $ifail = 12$ .)

options: Type: NagLibrary..::..E04..::..e04nkOptions
An Object of type E04.e04nkOptions. 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

e04nk is designed to solve a class of quadratic programming problems that are assumed to be stated in the following general 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 a set of variables,

A

is an

m

n

matrix and the objective function

f (x)

may be specified in a variety of ways depending upon the particular problem to be solved. The optional parameter Maximize may be used to specify an alternative problem in which

f (x)

is maximized. The possible forms for

f (x)

are listed in Table 1, in which the prefixes FP, LP and QP stand for ‘feasible point’, ‘linear programming’ and ‘quadratic programming’ respectively,

c

is an

n

-element vector and

H

is the

n

n

second-derivative matrix

\nabla^{2} f (x)

(the Hessian matrix).

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

Table 1

For LP and QP problems, the unique global minimum value of

f (x)

is found. For FP problems,

f (x)

is omitted and the method attempts to find a feasible point for the set of constraints. For QP problems, you must also provide a method that computes

H x

for any given vector

x

. (

H

need not be stored explicitly.) If

H

is the zero matrix, the method will still solve the resulting LP problem; however, this can be accomplished more efficiently by setting

ncolh = 0

(see [Parameters]).

The defining feature of a convex QP problem is that the matrix

H

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

x^{T} H x \geq 0

for all

x

. Otherwise,

f (x)

is said to be nonconvex and it may be more appropriate to call e04ug instead.

e04nk is intended to solve large-scale linear and quadratic programming problems in which the constraint 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 method also takes advantage of sparsity in

c

. (Sparsity in

H

can be exploited in the method that computes

H x

.) For problems in which

A

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

The upper and lower bounds on the

m

elements of

A x

are said to define the general constraints of the problem. Internally, e04nk 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

A x

and

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 an active-set method. This is an iterative procedure with two phases: a feasibility phase, in which the sum of infeasibilities is minimized to find a feasible point; and an optimality phase, in which

f (x)

is minimized by constructing a sequence of iterations that lies within the feasible region.

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^{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 +S and dj in the monitoring file output; see [Description of Monitoring Information].) 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}

's.

e04nk is based on SQOPT, which is part of the SNOPT package described in Gill et al. (2002), which in turn utilizes methods from the MINOS package (see Murtagh and Saunders (1995)). 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 (2002) SNOPT: An SQP Algorithm for Large-scale Constrained Optimization 12 979–1006 SIAM J. Optim.

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

Murtagh B A and Saunders M A (1995) MINOS 5.4 users' guide Report SOL 83-20R Department of Operations Research, Stanford University

Error Indicators and Warnings

Note: e04nk 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 (LENIZ, LENZ) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.

$ifail = 1$: Weak solution found. The final $x$ is not unique, although $x$ gives the global minimum value of the objective function.

$ifail = 2$: The problem is unbounded (or badly scaled). The objective function is not bounded below in the feasible region.

$ifail = 3$: The problem is infeasible. The general constraints cannot all be satisfied simultaneously to within the value of the optional parameter Feasibility Tolerance ( $default value = \max (10^{- 6}, \sqrt{ε})$ , where $ε$ is the machine precision).

$ifail = 4$: Too many iterations. The value of the optional parameter Iteration Limit ( $default value = \max (50, 5 (n + m))$ ) is too small.

$ifail = 5$: The reduced Hessian matrix $Z^{T} H Z$ (see [Definition of the Working Set and Search Direction]) exceeds its assigned dimension. The value of the optional parameter Superbasics Limit ( $default value = \min (n_{H} + 1, n)$ ) is too small.

$ifail = 6$: The Hessian matrix $H$ appears to be indefinite. This sometimes occurs because the values of the optional parameters LU Factor Tolerance ( $default value = 100.0$ ) and LU Update Tolerance ( $default value = 10.0$ ) are too large. Check also that qphx has been coded correctly and that all relevant elements of $H x$ have been assigned their correct values.

$ifail = 7$: An input parameter is invalid.

$ifail = 8$: Numerical error in trying to satisfy the general constraints. The basis is very ill-conditioned.

$ifail = 9$: Not enough integer workspace for the basis factors. Increase _leniz and rerun e04nk.

$ifail = 10$: Not enough real workspace for the basis factors. Increase _lenz and rerun e04nk.

$ifail = 11$: The basis is singular after $15$ attempts to factorize it (adding slacks where necessary). Either the problem is badly scaled or the value of the optional parameter LU Factor Tolerance ( $default value = 100.0$ ) is too large.

$ifail = 12$: Not enough integer workspace to start solving the problem. Increase _leniz to at least miniz and rerun e04nk.

$ifail = 13$: Not enough real workspace to start solving the problem. Increase _lenz to at least minz and rerun e04nk.

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

Accuracy

e04nk 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

This section describes the intermediate printout and final printout produced by e04nk. 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 e04nk, 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.
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 the sum of the magnitudes of constraint violations. If $x$ is feasible, Objective is the value of the objective function. The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point.
During the optimality phase, the value of the objective function will be 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.

Norm rg is $‖d_{S}‖$ , the Euclidean norm of the reduced gradient (see [Main Iteration]). During the optimality phase, this norm will be approximately zero after a unit step. For FP and LP problems, Norm rg is not printed.

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.
Variable gives the name of the variable. If $nname = 1$ , a default name is assigned to the $j$ th variable, for $j = 1, 2, \dots, n$ . If $nname = n + m$ , the name supplied in $crname [j - 1]$ is assigned to the $j$ th variable.
State gives the state of the variable (LL if nonbasic on its lower bound, UL if nonbasic on its upper bound, EQ if nonbasic and fixed, FR if nonbasic and strictly between its bounds, BS if basic and SBS if superbasic).
A key is sometimes printed before State. Note that unless the optional parameter $Scale Option = 0$ ( $default value = 2$ ) 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. The variable is nonbasic or superbasic. If the value of the reduced gradient for the variable exceeds the value of the optional parameter Optimality Tolerance, the solution would not be declared optimal because the reduced gradient for the variable would not be considered negligible.

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. If $x$ is optimal, the multiplier should be non-negative if State is LL, non-positive if State is UL and zero if State is BS or SBS.
Residual 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 linear constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, $n$ replaced by $m$ , $crname [j - 1]$ replaced by $crname [n + j - 1]$ , $bl [j - 1]$ 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.

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
$\begin{matrix} x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} = 2000 \\ 0.15 x_{1} + 0.04 x_{2} + 0.02 x_{3} + 0.04 x_{4} + 0.02 x_{5} + 0.01 x_{6} + 0.03 x_{7} \leq 60 \\ 0.03 x_{1} + 0.05 x_{2} + 0.08 x_{3} + 0.02 x_{4} + 0.06 x_{5} + 0.01 x_{6} \leq 100 \\ 0.02 x_{1} + 0.04 x_{2} + 0.01 x_{3} + 0.02 x_{4} + 0.02 x_{5} \leq 40 \\ 0.02 x_{1} + 0.03 x_{2} + 0.01 x_{5} \leq 30 \\ 1500 \leq 0.70 x_{1} + 0.75 x_{2} + 0.80 x_{3} + 0.75 x_{4} + 0.80 x_{5} + 0.97 x_{6} \\ 250 \leq 0.02 x_{1} + 0.06 x_{2} + 0.08 x_{3} + 0.12 x_{4} + 0.02 x_{5} + 0.01 x_{6} + 0.97 x_{7} \leq 300 \end{matrix}$
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#): e04nke.cs
Example program data: e04nke.d
Example program results: e04nke.r

Algorithmic Details

This section contains a detailed description of the method used by e04nk.

Overview

e04nk 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 Monitoring Information]) to the quadratic objective function (the printed quantity Objective; see [Description of Monitoring Information]).
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
$(\begin{matrix} \bar{x} \\ \bar{s} \end{matrix}) = (\begin{matrix} x \\ s \end{matrix}) + α p,$ (2)
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 $m_{W}$ by $(n + m)$ 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
$W p = 0 .$ (3)
This characterisation allows $p$ to be computed using any $n$ by $n_{Z}$ 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 (6) and (7)). The direction $p$ will satisfy (3) if
$p = Z p_{Z},$ (4)
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 $x_{j} \geq l_{j}$ or $x_{j} \leq u_{j}$ 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
$B x_{B} + S x_{S} + N x_{N} = 0,$ (5)
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 Monitoring Information]). Given values of $x_{N}$ and $x_{S}$ , the basic variables $x_{B}$ are adjusted so that $(x, s)$ satisfies (5).
If $P$ is a permutation matrix such that $(\begin{matrix} A & - I \end{matrix}) P = (\begin{matrix} B & S & N \end{matrix})$ , then $W$ satisfies
$W P = (\begin{matrix} B & S & N \\ 0 & 0 & I_{N} \end{matrix}),$ (6)
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 $B$ that alters as the working set $W$ changes. Given the permutation $P$ , the null space basis is given by
$Z = P (\begin{matrix} - B^{- 1} S \\ I \\ 0 \end{matrix}) .$ (7)
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 $B$ or $B^{T}$ . 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:
$g_{Z} = Z^{T} g and H_{Z} = Z^{T} H Z,$ (8)
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 Cond Hz in the monitoring file output; see [Description of Monitoring Information].)
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
$W^{T} λ = g (x) .$ (9)
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 $ifail = 3$ (see [Error Indicators and Warnings]).
The special form (6) of the working set allows the multiplier vector $λ$ , the solution of (9), to be written in terms of the vector
$d = (\begin{matrix} g \\ 0 \end{matrix}) - {(\begin{matrix} A & - I \end{matrix})}^{T} π = (\begin{matrix} g - A^{T} π \\ π \end{matrix}),$ (10)
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 (8). 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 Norm rg, 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 (7) and (11)). 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 (4) is computed from the equations
$R^{T} R p_{Z} = - g_{Z},$ (11)
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 $α_{m}$ , the step to the ‘nearest’ constraint along $p$ . This constraint is then added to the working set at the next iteration.
If $H_{Z}$ 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
$p_{Z}^{T} H_{Z} p_{Z} = 0 and g_{Z}^{T} p_{Z} \leq 0,$ (12)
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 $ifail = 2$ (see [Error Indicators and Warnings]). 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.

Miscellaneous

If the basis matrix is not chosen carefully, the condition of the null space matrix $Z$ in (7) 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
${(\begin{matrix} B & S \end{matrix})}^{T} = L U,$ (13)
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 (6). 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 $0.5 δ$ to $δ$ (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 $α \leq α_{m}$ ) 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. 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 e04nk. (See also the description of the optional parameters Monitoring File and Print Level.) You can control the level of printed output.
When $Print Level = 5$ or $\geq 10$ and $Monitoring File \geq 0$ , the following line of intermediate printout ( $< 120$ characters) is produced at every iteration on the unit number specified by optional parameter Monitoring File. Unless stated otherwise, the values of the quantities printed are those in effect on completion of the given iteration.
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.
+S is the variable selected by the pricing operation to be added to the superbasic set.
-S is the variable chosen to leave the superbasic set.
-BS is the variable removed from the basis (if any) 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., +S 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). 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 the sum of the magnitudes of constraint violations. If $x$ is feasible, Objective is the value of the objective function. The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point.
During the optimality phase, the value of the objective function will be 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.

L is the number of nonzeros in the basis factor $L$ . Immediately after a basis factorization $B = L U$ , this entry contains lenL. 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 $B = L U$ , this entry contains lenU. 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. If it does not, increase _leniz and _lenz by at least $L + U$ and rerun e04nk (possibly using $start = "W"$ ; see [Parameters]).
Norm rg is $‖d_{S}‖$ , the Euclidean norm of the reduced gradient (see [Main Iteration]). During the optimality phase, this norm will be approximately zero after a unit step. For FP and LP problems, Norm rg is not printed.
Ns is the current number of superbasic variables. For FP and LP problems, Ns is not printed.
Cond Hz 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. For FP and LP
problems, Cond Hz is not printed.

When $Print Level \geq 20$ and $Monitoring File \geq 0$ , the following lines of intermediate printout ( $< 120$ characters) are produced on the unit number specified by optional parameter Monitoring File whenever the matrix $B$ or $B_{S} = {(\begin{matrix} B & S \end{matrix})}^{T}$ is factorized. Gaussian elimination is used to compute an $L U$ factorization of $B$ or $B_{S}$ , 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 ( $default value = 100.0$ ).
Factorize is the factorization count.
Demand is 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 value of the optional parameter Factorization Frequency.
$2$ The number of nonzeros in the updated factors has increased significantly.
$7$ Not enough storage to update factors.
$10$ Row residuals too large (see the description for the optional parameter Check Frequency).
$11$ Ill-conditioning has caused inconsistent results.

Iteration is the iteration count.
Nonlinear is the number of nonlinear variables in the current basis $B$ (not printed if $B_{S}$ is factorized).
Linear is the number of linear variables in $B$ (not printed if $B_{S}$ is factorized).
Slacks is the number of slack variables in $B$ (not printed if $B_{S}$ is factorized).
Elems is the number of nonzeros in $B$ (not printed if $B_{S}$ is factorized).
Density is the percentage nonzero density of $B$ (not printed if $B_{S}$ is factorized). More precisely, $Density = 100 \times Elems / {(Nonlinear + Linear + Slacks)}^{2}$ .
Compressns is the number of times the data structure holding the partially factorized matrix needed to be compressed, in order to recover unused workspace. Ideally, it should be zero. If it is more than $3$ or $4$ , increase _leniz and _lenz and rerun e04nk (possibly using $start = "W"$ ; see [Parameters]).
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 m such quantities. It gives an indication of how much work was required to preserve sparsity during the factorization.
lenL is the number of nonzeros in $L$ .
lenU is the number of nonzeros in $U$ .
Increase is the percentage increase in the number of nonzeros in $L$ and $U$ relative to the number of nonzeros in $B$ . More precisely, $Increase = 100 \times (lenL + lenU - Elems) / Elems$ .
m is the number of rows in the problem. Note that $m = Ut + Lt + bp$ .
Ut is the number of triangular rows of $B$ at the top of $U$ .
d1 is the number of columns remaining when the density of the basis matrix being factorized reached $0.3$ .
Lmax is the maximum subdiagonal element in the columns of $L$ . This will not exceed the value of the optional parameter LU Factor Tolerance.
Bmax is the maximum nonzero element in $B$ (not printed if $B_{S}$ is factorized).
BSmax is the maximum nonzero element in $B_{S}$ (not printed if $B$ is factorized).
Umax is the maximum nonzero element in $U$ , excluding elements of $B$ that remain in $U$ unchanged. (For example, if a slack variable is in the basis, the corresponding row of $B$ will become a row of $U$ without modification. Elements in such rows will not contribute to Umax. If the basis is strictly triangular then none of the elements of $B$ will contribute and Umax will be zero.)
Ideally, Umax should not be significantly larger than Bmax. If it is several orders of magnitude larger, it may be advisable to reset the optional parameter LU Factor Tolerance to some value nearer unity.
Umax is not printed if $B_{S}$ is factorized.

Umin is the magnitude of the smallest diagonal element of $P U Q$ (not printed if $B_{S}$ is factorized).
Growth is the value of the ratio Umax/Bmax, which should not be too large.
Providing Lmax is not large (say, $< 10.0$ ), the ratio $\max (Bmax, Umax) / Umin$ is an estimate of the condition number of $B$ . If this number is extremely large, the basis is nearly singular and some numerical difficulties might occur. (However, an effort is made to avoid near-singularity by using slacks to replace columns of $B$ that would have made Umin extremely small and the modified basis is refactorized.)

Growth is not printed if $B_{S}$ is factorized.

Lt is the number of triangular columns of $B$ at the left of $L$ .
bp is the size of the ‘bump’ or block to be factorized nontrivially after the triangular rows and columns of $B$ have been removed.
d2 is the number of columns remaining when the density of the basis matrix being factorized has reached $0.6$ .

When $Print Level \geq 20$ and $Monitoring File \geq 0$ , the following lines of intermediate printout ( $< 80$ characters) are produced on the unit number specified by optional parameter Monitoring 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.
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., $istate [j - 1] = 3$ for some $j \leq n$ ). It will be a subset of the columns for which $istate [j - 1] = 3$ 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).

When $Print Level \geq 20$ and $Monitoring File \geq 0$ , the following lines of intermediate printout ( $< 80$ characters) are produced on the unit number specified by optional parameter Monitoring File. They refer to the elements of the names array (see [Parameters]).
Name gives the name for the problem (blank if problem unnamed).
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)).
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).

When $Print Level = 1$ or $\geq 10$ and $Monitoring File \geq 0$ , the following lines of final printout ( $< 120$ characters) are produced on the unit number specified by optional parameter Monitoring File.
Let $a_{j}$ denote the $j$ th column of $A$ , for $j = 1, 2, \dots, n$ . The following describes the printout for each column (or variable). A full stop (.) is printed for any numerical value that is zero.
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 gives the state of the variable (LL if nonbasic on its lower bound, UL if nonbasic on its upper bound, EQ if nonbasic and fixed, FR if nonbasic and strictly between its bounds, BS if basic and SBS if 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. The variable is nonbasic or superbasic. If the value of the reduced gradient for the variable exceeds the value of the optional parameter Optimality Tolerance, the solution would not be declared optimal because the reduced gradient for the variable would not be considered negligible.

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 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 $bu [j - 1] \geq bigbnd$ .
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 $m + j$ .

Let $v_{i}$ denote the $i$ th row of $A$ , for $i = 1, 2, \dots, m$ . The following describes the printout for each row (or constraint). A full stop (.) is printed for any numerical value that is zero.
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 gives the state of $v_{i}$ (LL if active on its lower bound, UL if active on its upper bound, EQ if active and fixed, BS if inactive when $s_{i}$ is basic and SBS if inactive when $s_{i}$ 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. The variable is nonbasic or superbasic. If the value of the reduced gradient for the variable exceeds the value of the optional parameter Optimality Tolerance, the solution would not be declared optimal because the reduced gradient for the variable would not be considered negligible.

Activity is the value of $v_{i}$ 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 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 $bu [j - 1] \geq bigbnd$ .
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.
i gives the index $i$ of the $i$ th row.

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

Itn	is the iteration count.
Step	is the step taken along the computed search direction.
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 the sum of the magnitudes of constraint violations. If $x$ is feasible, Objective is the value of the objective function. The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point. During the optimality phase, the value of the objective function will be 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.
Norm rg	is $‖d_{S}‖$ , the Euclidean norm of the reduced gradient (see [Main Iteration]). During the optimality phase, this norm will be approximately zero after a unit step. For FP and LP problems, Norm rg is not printed.

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.
+S	is the variable selected by the pricing operation to be added to the superbasic set.
-S	is the variable chosen to leave the superbasic set.
-BS	is the variable removed from the basis (if any) 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., +S 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). 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 the sum of the magnitudes of constraint violations. If $x$ is feasible, Objective is the value of the objective function. The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point. During the optimality phase, the value of the objective function will be 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.
L	is the number of nonzeros in the basis factor $L$ . Immediately after a basis factorization $B = L U$ , this entry contains lenL. 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 $B = L U$ , this entry contains lenU. 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. If it does not, increase _leniz and _lenz by at least $L + U$ and rerun e04nk (possibly using $start = "W"$ ; see [Parameters]).
Norm rg	is $‖d_{S}‖$ , the Euclidean norm of the reduced gradient (see [Main Iteration]). During the optimality phase, this norm will be approximately zero after a unit step. For FP and LP problems, Norm rg is not printed.
Ns	is the current number of superbasic variables. For FP and LP problems, Ns is not printed.
Cond Hz	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. For FP and LP problems, Cond Hz is not printed.

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., $istate [j - 1] = 3$ for some $j \leq n$ ). It will be a subset of the columns for which $istate [j - 1] = 3$ 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).

Name	gives the name for the problem (blank if problem unnamed).
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)).
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).

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

See Also