e04nq Method
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e04nq solves sparse linear programming or convex quadratic programming problems.

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
)
Visual Basic (Declaration)
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
)
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 

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.
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" or "W".
qphx
Type: NagLibrary..::.E04..::.E04NQ_QPHX
For QP problems, you must supply a version of qphx to compute the matrix product Hx 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,zT and only y enters the objective quadratically then In this case, ncolh should be the dimension of y, and qphx should compute H1y. 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: m1.
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: n1.
ne
Type: System..::.Int32
On entry: the number of nonzero elements in A.
Constraint: 1nen×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 or 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: 0lencn.
ncolh
Type: System..::.Int32
On entry: nH, 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: 0ncolhn.
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 cTx.
If iobj=0, there is no free row, i.e., the problem is either an FP problem, or a QP problem with c=0.
Constraint: 0iobjm.
objadd
Type: System..::.Double
On entry: the constant q, to be added to the objective for printing purposes. Typically objadd=0.0D0.
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,,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: 1inda[i-1]m, for i=1,2,,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 jth column, for j=1,2,,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 jth 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]=-infbnd and bu[n]=infbnd, where infbnd is the value of the optional parameter Infinite Bound Size.
Constraints:
  • loca[0]=1;
  • loca[j]1, for j=1,2,,n-1;
  • loca[n]=ne+1;
  • 0loca[j+1]-loca[j]m, for j=0,1,,n-1.
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 Ax (which, equivalently, are the bounds for the slacks, s) and the free row (if any). To fix the jth variable, set bl[j-1]=bu[j-1]=β, say, where β<infbnd . To specify a nonexistent lower bound (i.e., lj=-), set bl[j-1]-infbnd. Here, infbnd is the value of the optional parameter Infinite Bound Size. To specify the jth constraint as an equality, set bl[n+j-1]=bu[n+j-1]=β, say, where β<infbnd. Note that the lower bound corresponding to the free row must be set to - and stored in bl[n+iobj-1].
Constraint: if iobj>0, bl[n+iobj-1]-infbnd
(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 Ax (which, equivalently, are the bounds for the slacks, s) and the free row (if any). To specify a nonexistent upper bound (i.e., uj=+), set bu[j-1]infbnd. Note that the upper bound corresponding to the free row must be set to + and stored in bu[n+iobj-1].
Constraints:
  • if iobj>0, bu[n+iobj-1]infbnd;
  • otherwise bl[i-1]bu[i-1].
c
Type: array< System..::.Double >[]()[]
An array of size [max1,lenc]
Note: the dimension of the array c must be at least max1,lenc if iobj0, 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 Mode0, helast[j-1]=0,1,2,3 for j=1,2,,n+m.
hs
Type: array< System..::.Int32 >[]()[]
An array of size [n+m]
On entry:
If start="C" or "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 A -I . 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,,n. All variables will then be eligible for the initial basis. Less trivially, to say that the jth variable will probably be equal to one of its bounds, set hs[j-1]=4 and x[j-1]=bl[j-1] or 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] if bl[j-1]x[j-1]bu[j-1], or at the value bl[j-1] or bu[j-1] closest to x[j-1].
If 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", 0hs[j-1]5, for j=1,2,,n;
  • if start="W", 0hs[j-1]3, for j=1,2,,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]
If 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.
If 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-A-ITπ. 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, gi=0, for i=n+1:m, hence rcn+1:m=π.
ns
Type: System..::.Int32 %
On entry: nS, 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 12xTHx (if any).
If ninf>0, obj is just the linear objective term cTx (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 (see [Error Indicators and Warnings]).
e04nq returns with ifail=0 if the reduced gradient (rgNorm; see [Description of the Printed Output]) is negligible, the Lagrange multipliers (Lagr Mult; see [Description of the Printed Output]) are optimal, x satisfies the constraints to the accuracy requested by the value of the optional parameter Feasibility Tolerance and the reduced Hessian factor R (see [Definition of the Working Set and Search Direction]) is nonsingular.

Description

The possible forms for the function fx are summarized in Table 1. The most general form for fx is
fx = q + cTx + 12 xTHx = q + j=1 n cj xj + 12 i=1 n j=1 n xi Hij xj
where q is a constant, c is a constant n-vector and H is a constant symmetric n by n matrix with elements Hij . 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 xT H x 0  for all x. If not, fx  is nonconvex and e04nq will terminate with the error indicator ifail=11. If fx  is nonconvex it may be more appropriate to call e04vh instead.
If 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 Hx for any given vector x. In many cases, the product Hx 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 or e04nc.
There is considerable flexibility allowed in the definition of fx  in Table 1. The vector c defining the linear term cTx 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, cTx 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 -  and + . 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 Ax 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=s1,s2,,smT. For example, the linear constraint 52x1+3x2+ is replaced by 2x1+3x2-s1=0, together with the bounded slack 5s1+. The problem defined by (1) can therefore be re-written in the following equivalent form:
minimize x Rn , s Rm fx  subject to ​ Ax-s=0 ,   l x s u.
Since the slack variables s are subject to the same upper and lower bounds as the elements of Ax, the bounds on x and Ax 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 fx 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  
  minimize x, s, v, w j=1 n+m vj + wj  
  subject to ​ Ax-s=0 ,  l x s -v + w u ,  v 0 ,  w 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 fx .
If the constraints are infeasible (i.e., v0  or w0  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 fx  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 fx  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)  
  minimize x, s, v, w fx + γ j=1 n+m vj + wj  
  subject to ​ Ax-s=0 ,  l x s -v + w u ,  v 0 ,  w 0 ,
where γ is a non-negative parameter known as the elastic weight (see the description of the optional parameter Elastic Weight), and fx + γ j vj + wj  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 fx , 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 xj  is allowed to violate its lower bound (say) and an explicit value of v can be recovered as vj = lj - xj .
A constraint is said to be active or binding at x if the associated element of either x or Ax is equal to one of its upper or lower bounds. Since an active constraint in Ax 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 Ax-s=0 are (conceptually) partitioned into the form
BxB+SxS+NxN=0,
where xN consists of the nonbasic elements of x,s and the basis matrix B is square and nonsingular. The elements of xB and xS are called the basic and superbasic variables respectively; with xN 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, xS 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 Ax-s=0. The number of superbasic variables (nS say) therefore indicates the number of degrees of freedom remaining after the constraints have been satisfied. In broad terms, nS is a measure of how nonlinear the problem is. In particular, nS 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 nS 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 nS is decreased by one.
Associated with each of the m equality constraints Ax-s=0 is a dual variable πi. Similarly, each variable in x,s has an associated reduced gradient dj (also known as a reduced cost). The reduced gradients for the variables x are the quantities g-ATπ, 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 dj0 for all nonbasic variables at their lower bounds, dj0 for all nonbasic variables at their upper bounds and dj=0 for all superbasic variables. In practice, an approximate QP solution is found by slightly relaxing these conditions on dj (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 xj means that the reduced gradient dj associated with the relevant active upper or lower bound on xj is computed via the formula dj=gj-ajTπ, where aj is the jth column of A -I . (The variable selected by such a process and the corresponding value of dj (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., nm), 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 djs.
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 LU 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

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:
Some error messages may refer to parameters that are dropped from this interface (cuser, lencu, iuser, leniu, ruser, lenru, cw, lencw, iw, leniw, rw, lenrw) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
ifail=1
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., fx  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 Ax-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 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 12s2 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 LU  factorization of the basis has just been obtained and used to recompute the basic variables xB , 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 Ax-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.
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 Hx 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
ifail=13
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=-8000
ifail=-6000

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.

Further Comments

This section contains a description of the printed output.

Description of the Printed Output

If 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 App  or Ipp , the ppth segments of the constraint matrix A -I .
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 dj = gj - πT aj , for j = jq , where gj  is the gradient of the current objective function, π  is the vector of dual variables, and aj  is the j th column of the constraint matrix A -I .
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+αp. 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 rth element of a vector y satisfying By=aq whenever aq (the qth column of the constraint matrix A -I  replaces the rth 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=LU, 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=LU, 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

Algorithmic Details

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 summarize 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 x-,s- is defined by 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 mW denote the number of constraints in the working set (including bounds), and let W denote the associated mW by n+m working set matrix consisting of the mW gradients of the working set constraints.
If P is a permutation matrix such that A -I P= B S N , then W satisfies where IN is the identity matrix with the same number of columns as N.
At each iteration, an upper triangular factor R is available such that HZ=RTR. Normally, R is computed from RTR=ZTHZ 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, nS1000). 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 not zero, Lagrange multipliers need not be computed and the search direction is given by p=ZpZ (see (8) and (12)). The step length is chosen to maintain feasibility with respect to the satisfied constraints.

Miscellaneous

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 0.5δ to δ (in steps of 0.5δ/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 0.5δ.
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 ααN) 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 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 jn). 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 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 or BS= B S T is factorized. Gaussian elimination is used to compute an LU factorization of B or BS, where PLPT is a lower triangular matrix and PUQ 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 LU  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 LU  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 BS  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.
m is the number of rows in B  or BS .
n is the number of columns in B  or BS . Preceded by ‘=’ or ‘>’ respectively.
Elems is the number of nonzero elements in B  or BS .
Amax is the largest nonzero in B  or BS .
Density is the percentage nonzero density of B  or BS .
Merit/MerRP/MerCP Merit is the average Markowitz merit count for the elements chosen to be the diagonals of PUQ . 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=LU , 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 or 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  or BS .
Utri is the number of triangular rows of B  or BS  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 or 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 or 2.0, say (but bigger than 1.0).
As long as Lmax is not large (say 5.0 or less), maxAmax,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  or BS  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 LU  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  or BS  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 PUQ .
DUmin is the smallest diagonal of PUQ .
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 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 Ax u . The i th constraint is therefore of the form
α νi x β ,
where νi  is the i th row of A .
Internally, the constraints take the form Ax - s = 0 , where s  is the set of slack variables (which happen to satisfy the bounds l s u ). For the i th constraint, the slack variable si  is directly available, and it is sometimes convenient to refer to its state. It should satisfy α si 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 si in the intermediate output.)
Row gives the name of νi.
State the state of νi  (the state of si  relative to the bounds α  and β ). The various states possible are as follows:
LL si  is nonbasic at its lower limit, α .
UL si  is nonbasic at its upper limit, β .
EQ si  is nonbasic and fixed at the value α=β .
FR si  is nonbasic and currently zero, even though it is free to take any value between its bounds α  and β .
BS si  is basic.
SBS si  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 πi is not sufficiently small, as measured by the Optimality Tolerance. If the slack is nonbasic, πi 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 νix 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 is α, the lower bound specified for the variable si. None indicates that bl[j-1]-infbnd.
Upper Limit is β, the upper bound specified for the variable si. None indicates that bu[j-1]infbnd.
Dual Activity is the value of the dual variable πi (the Lagrange multiplier for νi; see [Main Iteration]). For FP problems, πi is set to zero.
i gives the index i of the ith row.

The COLUMNS Section

Let the j th component of x  be the variable xj  and assume that it satisfies the bounds α xj 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 xj in the intermediate output.)
Column gives the name of xj.
State the state of xj  relative to the bounds α  and β . The various states possible are as follows:
LL xj  is nonbasic at its lower limit, α .
UL xj  is nonbasic at its upper limit, β .
EQ xj  is nonbasic and fixed at the value α = β .
FR xj  is nonbasic and currently zero, even though it is free to take any value between its bounds α  and β .
BS xj  is basic.
SBS xj  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 πi is not sufficiently small, as measured by the Optimality Tolerance. If the slack is nonbasic, πi 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 xj at the final iterate.
Obj Gradient is the value of gj at the final iterate. For FP problems, gj is set to zero.
Lower Limit is the lower bound specified for the variable. None indicates that bl[j-1]-infbnd.
Upper Limit is the upper bound specified for the variable. None indicates that bu[j-1]infbnd.
Reduced Gradnt is the value of dj at the final iterate (see [Main Iteration]). For FP problems, dj is set to zero.
m + j is the value of m+j.
Note:  if two problems are the same except that one minimizes fx  and the other maximizes -fx , their solutions will be the same but the signs of the dual variables πi  and the reduced gradients dj will be reversed.

The Solution File

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.
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.

See Also