e04nc Method
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e04nc solves linearly constrained linear least-squares problems and convex quadratic programming problems. It is not intended for large sparse problems.

Syntax

C#
public static void e04nc(
	int m,
	int n,
	int nclin,
	double[,] c,
	double[] bl,
	double[] bu,
	double[] cvec,
	int[] istate,
	int[] kx,
	double[] x,
	double[,] a,
	double[] b,
	out int iter,
	out double obj,
	double[] clamda,
	E04..::.e04ncOptions options,
	out int ifail
)
Visual Basic (Declaration)
Public Shared Sub e04nc ( _
	m As Integer, _
	n As Integer, _
	nclin As Integer, _
	c As Double(,), _
	bl As Double(), _
	bu As Double(), _
	cvec As Double(), _
	istate As Integer(), _
	kx As Integer(), _
	x As Double(), _
	a As Double(,), _
	b As Double(), _
	<OutAttribute> ByRef iter As Integer, _
	<OutAttribute> ByRef obj As Double, _
	clamda As Double(), _
	options As E04..::.e04ncOptions, _
	<OutAttribute> ByRef ifail As Integer _
)
Visual C++
public:
static void e04nc(
	int m, 
	int n, 
	int nclin, 
	array<double,2>^ c, 
	array<double>^ bl, 
	array<double>^ bu, 
	array<double>^ cvec, 
	array<int>^ istate, 
	array<int>^ kx, 
	array<double>^ x, 
	array<double,2>^ a, 
	array<double>^ b, 
	[OutAttribute] int% iter, 
	[OutAttribute] double% obj, 
	array<double>^ clamda, 
	E04..::.e04ncOptions^ options, 
	[OutAttribute] int% ifail
)
F#
static member e04nc : 
        m:int * 
        n:int * 
        nclin:int * 
        c:float[,] * 
        bl:float[] * 
        bu:float[] * 
        cvec:float[] * 
        istate:int[] * 
        kx:int[] * 
        x:float[] * 
        a:float[,] * 
        b:float[] * 
        iter:int byref * 
        obj:float byref * 
        clamda:float[] * 
        options:E04..::.e04ncOptions * 
        ifail:int byref -> unit

Parameters

m
Type: System..::.Int32
On entry: m, the number of rows in the matrix A. If the problem is specified as type FP or LP, m is not referenced and is assumed to be zero.
If the problem is of type QP, m will usually be n, the number of variables. However, a value of m less than n is appropriate for QP3 or QP4 if A is an upper trapezoidal matrix with m rows. Similarly, m may be used to define the dimension of a leading block of nonzeros in the Hessian matrices of QP1 or QP2, in which case the last n-m rows and columns of a are assumed to be zero. In the QP case, m should not be greater than n; if it is, the last m-n rows of A are ignored.
If the problem is of type LS1 (the default) or specified as type LS2, LS3 or LS4, m is also the dimension of the array b. Note that all possibilities (m<n, m=n and m>n) are allowed in this case.
Constraint: m>0 if the problem is not of type FP or LP.
n
Type: System..::.Int32
On entry: n, the number of variables.
Constraint: n>0.
nclin
Type: System..::.Int32
On entry: nL, the number of general linear constraints.
Constraint: nclin≥0.
c
Type: array< System..::.Double ,2>[,](,)[,]
An array of size [ldc, dim2]
Note: ldc must satisfy the constraint: ldc≥max1,nclin
Note: the second dimension of the array c must be at least n if nclin>0, and at least 1 otherwise.
On entry: the ith row of c must contain the coefficients of the ith general constraint, for i=1,2,…,nclin.
If nclin=0, c is not referenced.
bl
Type: array< System..::.Double >[]()[]
An array of size [n+nclin]
On entry: bl must contain the lower bounds and bu the upper bounds, for all the constraints, in the following order. The first n elements of each array must contain the bounds on the variables, and the next nL elements must contain the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e., lj=-∞), set bl[j-1]≤-bigbnd, and to specify a nonexistent upper bound (i.e., uj=+∞), set bu[j-1]≥bigbnd; the default value of bigbnd is 1020, but this may be changed by the optional parameter Infinite Bound Size. To specify the jth constraint as an equality, set bu[j-1]=bl[j-1]=β, say, where β<bigbnd.
Constraints:
  • bl[j]≤bu[j], for j=0,1,…,n+nclin-1;
  • if bl[j]=bu[j]=β, β<bigbnd.
bu
Type: array< System..::.Double >[]()[]
An array of size [n+nclin]
On entry: bl must contain the lower bounds and bu the upper bounds, for all the constraints, in the following order. The first n elements of each array must contain the bounds on the variables, and the next nL elements must contain the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e., lj=-∞), set bl[j-1]≤-bigbnd, and to specify a nonexistent upper bound (i.e., uj=+∞), set bu[j-1]≥bigbnd; the default value of bigbnd is 1020, but this may be changed by the optional parameter Infinite Bound Size. To specify the jth constraint as an equality, set bu[j-1]=bl[j-1]=β, say, where β<bigbnd.
Constraints:
  • bl[j]≤bu[j], for j=0,1,…,n+nclin-1;
  • if bl[j]=bu[j]=β, β<bigbnd.
cvec
Type: array< System..::.Double >[]()[]
An array of size [dim1]
Note: the dimension of the array cvec must be at least n if the problem type is LP, QP2, QP4, LS2, LS4, and at least 1 otherwise.
On entry: the coefficients of the explicit linear term of the objective function.
If the problem is of type FP, QP1, QP3, LS1 (the default) or LS3, cvec is not referenced.
istate
Type: array< System..::.Int32 >[]()[]
An array of size [n+nclin]
On entry: need not be set if the (default) optional parameter Cold Start is used.
If the optional parameter Warm Start has been chosen, istate specifies the desired status of the constraints at the start of the feasibility phase. More precisely, the first n elements of istate refer to the upper and lower bounds on the variables, and the next nL elements refer to the general linear constraints (if any). Possible values for istate[j-1] are as follows:
istate[j-1] Meaning
0 The constraint should not be in the initial working set.
1 The constraint should be in the initial working set at its lower bound.
2 The constraint should be in the initial working set at its upper bound.
3 The constraint should be in the initial working set as an equality. This value must not be specified unless bl[j-1]=bu[j-1].
The values -2, -1 and 4 are also acceptable but will be reset to zero by the method. If e04nc has been called previously with the same values of n and nclin, istate already contains satisfactory information. (See also the description of the optional parameter Warm Start.) The method also adjusts (if necessary) the values supplied in x to be consistent with istate.
Constraint: -2≤istate[j]≤4, for j=0,1,…,n+nclin-1.
On exit: the status of the constraints in the working set at the point returned in x. The significance of each possible value of istate[j-1] is as follows:
istate[j-1] Meaning
-2 The constraint violates its lower bound by more than the feasibility tolerance.
-1 The constraint violates its upper bound by more than the feasibility tolerance.
-0 The constraint is satisfied to within the feasibility tolerance, but is not in the working set.
-1 This inequality constraint is included in the working set at its lower bound.
-2 This inequality constraint is included in the working set at its upper bound.
-3 The constraint is included in the working set as an equality. This value of istate can occur only when bl[j-1]=bu[j-1].
-4 This corresponds to optimality being declared with x[j-1] being temporarily fixed at its current value.
kx
Type: array< System..::.Int32 >[]()[]
An array of size [n]
On entry: need not be initialized for problems of type FP, LP, QP1, QP2, LS1 (the default) or LS2.
For problems QP3, QP4, LS3 or LS4, kx must specify the order of the columns of the matrix A with respect to the ordering of x. Thus if column j of A is the column associated with the variable xi then kx[j-1]=i.
Constraints:
  • 1≤kx[i]≤n, for i=0,1,…,n-1;
  • if i≠j, kx[i]≠kx[j].
On exit: defines the order of the columns of a with respect to the ordering of x, as described above.
x
Type: array< System..::.Double >[]()[]
An array of size [n]
On entry: an initial estimate of the solution.
On exit: the point at which e04nc terminated. If ifail=0, 1 or 4, x contains an estimate of the solution.
a
Type: array< System..::.Double ,2>[,](,)[,]
An array of size [lda, dim2]
Note: lda must satisfy the constraint: lda≥max1,m
Note: the second dimension of the array a must be at least 1 if the problem type is FP, LP, and at least n otherwise.
On entry: the array a must contain the matrix A as specified in Table 1 (see [Description]).
If the problem is of type QP1 or QP2, the first m rows and columns of a must contain the leading m by m rows and columns of the symmetric Hessian matrix. Only the diagonal and upper triangular elements of the leading m rows and columns of a are referenced. The remaining elements are assumed to be zero and need not be assigned.
For problems QP3, QP4, LS3 or LS4, the first m rows of a must contain an m by n upper trapezoidal factor of either the Hessian matrix or the least-squares matrix, ordered according to the kx array. The factor need not be of full rank, i.e., some of the diagonals may be zero. However, as a general rule, the larger the dimension of the leading nonsingular sub-matrix of A, the fewer iterations will be required. Elements outside the upper triangular part of the first m rows of a are assumed to be zero and need not be assigned.
If a constrained least-squares problem contains a very large number of observations, storage limitations may prevent storage of the entire least-squares matrix. In such cases, you should transform the original A into a triangular matrix before the call to e04nc and solve the problem as type LS3 or LS4.
On exit: if Hessian=No and the problem is of type LS or QP, a contains the upper triangular Cholesky factor R of (8) (see [Main Iteration]), with columns ordered as indicated by kx. If Hessian=Yes and the problem is of type LS or QP, a contains the upper triangular Cholesky factor R of the Hessian matrix H, with columns ordered as indicated by kx. In either case R may be used to obtain the variance-covariance matrix or to recover the upper triangular factor of the original least-squares matrix.
If the problem is of type FP or LP, a is not referenced.
b
Type: array< System..::.Double >[]()[]
An array of size [dim1]
Note: the dimension of the array b must be at least m if the problem is of type LS1 (the default), LS2, LS3 or LS4, and at least 1 otherwise.
On entry: the m elements of the vector of observations.
On exit: the transformed residual vector of equation (10) (see [Main Iteration]).
If the problem is of type FP, LP, QP1, QP2, QP3 or QP4, b is not referenced.
iter
Type: System..::.Int32 %
On exit: the total number of iterations performed.
obj
Type: System..::.Double %
On exit: the value of the objective function at x if x is feasible, or the sum of infeasibiliites at x otherwise. If the problem is of type FP and x is feasible, obj is set to zero.
clamda
Type: array< System..::.Double >[]()[]
An array of size [n+nclin]
On exit: the values of the Lagrange multipliers for each constraint with respect to the current working set. The first n elements contain the multipliers for the bound constraints on the variables, and the next nL elements contain the multipliers for the general linear constraints (if any). If istate[j-1]=0 (i.e., constraint j is not in the working set), clamda[j-1] is zero. If x is optimal, clamda[j-1] should be non-negative if istate[j-1]=1, non-positive if istate[j-1]=2 and zero if istate[j-1]=4.
options
Type: NagLibrary..::.E04..::.e04ncOptions
An object of type E04.e04ncOptions. 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]).
e04nc returns with ifail=0 if x is a strong local minimizer, i.e., the projected gradient (Norm Gz; see [Description of the Printed Output]) is negligible, the Lagrange multipliers (Lagr Mult; see [Definition of Search Direction]) are optimal and RZ (see [Main Iteration]) is nonsingular.

Description

e04nc is designed to solve a class of quadratic programming problems of the following general form:
minimize x∈Rn Fx  subject to  l≤ x Cx ≤u (1)
where c is an nL by n matrix and the objective function Fx may be specified in a variety of ways depending upon the particular problem to be solved. The available forms for Fx are listed in Table 1, in which the prefixes FP, LP, QP and LS stand for ‘feasible point’, ‘linear programming’, ‘quadratic programming’ and ‘least-squares’ respectively, c is an n element vector, b is an m element vector and z denotes the Euclidean length of z.
Problem type Fx Matrix A
FP None Not applicable
LP cTx Not applicable
QP1 cTx+12xTAx n by n symmetric positive semi-definite
QP2 cTx+12xTAx n by n symmetric positive semi-definite
QP3 cTx+12xTATAx m by n upper trapezoidal
QP4 cTx+12xTATAx m by n upper trapezoidal
LS1 cTx+12b-Ax2 m by n
LS2 cTx+12b-Ax2 m by n
LS3 cTx+12b-Ax2 m by n upper trapezoidal
LS4 cTx+12b-Ax2 m by n upper trapezoidal
Table 1
In the standard LS problem Fx will usually have the form LS1, and in the standard convex QP problem Fx will usually have the form QP2. The default problem type is LS1 and other objective functions are selected by using the optional parameter Problem Type.
When A is upper trapezoidal it will usually be the case that m=n, so that A is upper triangular, but full generality has been allowed for in the specification of the problem. The upper trapezoidal form is intended for cases where a previous factorization, such as a QR factorization, has been performed.
The constraints involving c are called the general constraints. Note that upper and lower bounds are specified for all the variables and for all the general constraints. An equality constraint can be specified by setting li=ui. If certain bounds are not present, the associated elements of l or u can be set to special values that will be treated as -∞ or +∞. (See the description of the optional parameter Infinite Bound Size.)
The defining feature of a quadratic function Fx is that the second-derivative matrix H (the Hessian matrix) is constant. For the LP case H=0; for QP1 and QP2, H=A; for QP3 and QP4, H=ATA and for LS1 (the default), LS2, LS3 and LS4, H=ATA.
Problems of type QP3 and QP4 for which A is not in upper trapezoidal form should be solved as types LS1 and LS2 respectively, with b=0.
For problems of type LS, we refer to A as the least-squares matrix, or the matrix of observations and to b as the vector of observations.
You must supply an initial estimate of the solution.
If H is nonsingular then e04nc will obtain the unique (global) minimum. If H is singular then the solution may still be a global minimum if all active constraints have nonzero Lagrange multipliers. Otherwise the solution obtained will be either a weak minimum (i.e., with a unique optimal objective value, but an infinite set of optimal x), or else the objective function is unbounded below in the feasible region. The last case can only occur when Fx contains an explicit linear term (as in problems LP, QP2, QP4, LS2 and LS4).
The method used by e04nc is described in detail in [Algorithmic Details].

References

Error Indicators and Warnings

Note: e04nc may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
ifail=1
x is a weak local minimum, (i.e., the projected gradient is negligible, the Lagrange multipliers are optimal, but either RZ (see [Main Iteration]) is singular, or there is a small multiplier). This means that x is not unique.
ifail=2
The solution appears to be unbounded. This value of ifail implies that a step as large as Infinite Bound Size (default value=1020) would have to be taken in order to continue the algorithm. This situation can occur only when A is singular, there is an explicit linear term, and at least one variable has no upper or lower bound.
ifail=3
No feasible point was found, i.e., it was not possible to satisfy all the constraints to within the feasibility tolerance. In this case, the constraint violations at the final x will reveal a value of the tolerance for which a feasible point will exist – for example, when the feasibility tolerance for each violated constraint exceeds its Slack (see [Description of the Printed Output]) at the final point. The modified problem (with an altered feasibility tolerance) may then be solved using a Warm Start. You should check that there are no constraint redundancies. If the data for the constraints are accurate only to the absolute precision σ, you should ensure that the value of the optional parameter Feasibility Tolerance (default value=ε, where ε is the machine precision) is greater than σ. For example, if all elements of c are of order unity and are accurate only to three decimal places, the Feasibility Tolerance should be at least 10-3.
ifail=4
The limiting number of iterations (determined by the optional parameters Feasibility Phase Iteration Limit (default value=max50,5n+nL) and Optimality Phase Iteration Limit (default value=max50,5n+nL)) was reached before normal termination occurred. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), either increase the iterations limit and rerun e04nc or, alternatively, rerun e04nc using the Warm Start facility to specify the initial working set. If the iteration limit is already large, but some of the constraints could be nearly linearly dependent, check the monitoring information (see [Description of Monitoring Information]) for a repeated pattern of constraints entering and leaving the working set. (Near-dependencies are often indicated by wide variations in size in the diagonal elements of the matrix T (see [Definition of Search Direction]), which will be printed if Print Level≥30 (default value=10). In this case, the algorithm could be cycling (see the comments for ifail=5).
ifail=5
The algorithm could be cycling, since a total of 50 changes were made to the working set without altering x. You should check the monitoring information (see [Description of Monitoring Information]) for a repeated pattern of constraint deletions and additions.
If a sequence of constraint changes is being repeated, the iterates are probably cycling. (e04nc does not contain a method that is guaranteed to avoid cycling; such a method would be combinatorial in nature.) Cycling may occur in two circumstances: at a constrained stationary point where there are some small or zero Lagrange multipliers; or at a point (usually a vertex) where the constraints that are satisfied exactly are nearly linearly dependent. In the latter case, you have the option of identifying the offending dependent constraints and removing them from the problem, or restarting the run with a larger value of the optional parameter Feasibility Tolerance (default value=ε, where ε is the machine precision). If e04nc terminates with ifail=5, but no suspicious pattern of constraint changes can be observed, it may be worthwhile to restart with the final x (with or without the Warm Start option).
ifail=6
An input parameter is invalid.
Overflow
If the printed output before the overflow error contains a warning about serious ill-conditioning in the working set when adding the jth constraint, it may be possible to avoid the difficulty by increasing the magnitude of the optional parameter Feasibility Tolerance and rerunning the program. If the message recurs even after this change, the offending linearly dependent constraint (with index ‘j’) must be removed from the problem.
ifail=-4000
Invalid dimension for array value
ifail=-8000
Negative dimension for array value
ifail=-6000
Invalid Parameters value

Accuracy

e04nc 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 some comments on scaling and a description of the printed output.

Scaling

Description of the Printed Output

This section describes the intermediate printout and final printout produced by e04nc. 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≥10 (the default for e04nc, by default no output is produced by e04nc).
The following line of summary output (<80 characters) is produced at every iteration. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.
Itn is the iteration count.
Step is the step taken along the computed search direction. If a constraint is added during the current iteration (i.e., Jadd is positive), Step will be the step to the nearest constraint. During the optimality phase, the step can be greater than one only if the factor RZ is singular. (See [Main Iteration].)
Ninf is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Objective is the value of the current objective function. If x is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If x is feasible, Objective is the value of the objective function of (1). The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point.
During the optimality phase the value of the objective function will be nonincreasing. During the feasibility phase the number of constraint infeasibilities will not increase until either a feasible point is found or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.
Norm Gz is Z1T gFR , the Euclidean norm of the reduced gradient with respect to Z1. During the optimality phase, this norm will be approximately zero after a unit step. (See [Definition of Search Direction] and [Main Iteration].)
The final printout includes a listing of the status of every variable and constraint.
The following describes the printout for each variable. A full stop (.) is printed for any numerical value that is zero.
Varbl gives the name (V) and index j, for j=1,2,…,n, of the variable.
State gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the Feasibility Tolerance, State will be ++ or -- respectively.
A key is sometimes printed before State.
A Alternative optimum possible. The variable is active at one of its bounds, but its Lagrange multiplier is essentially zero. This means that if the variable were allowed to start moving away from its bound then there would be no change to the objective function. The values of the other free variables might change, giving a genuine alternative solution. However, if there are any degenerate variables (labelled D), the actual change might prove to be zero, since one of them could encounter a bound immediately. In either case the values of the Lagrange multipliers might also change.
D Degenerate. The variable is free, but it is equal to (or very close to) one of its bounds.
I Infeasible. The variable is currently violating one of its bounds by more than the Feasibility Tolerance.
Value is the value of the variable at the final iteration.
Lower Bound is the lower bound specified for the variable. None indicates that bl[j-1]≤-bigbnd.
Upper Bound is the upper bound specified for the variable. None indicates that bu[j-1]≥bigbnd.
Lagr Mult is the Lagrange multiplier for the associated bound. This will be zero if State is FR unless bl[j-1]≤-bigbnd and bu[j-1]≥bigbnd, in which case the entry will be blank. If x is optimal, the multiplier should be non-negative if State is LL and non-positive if State is UL.
Slack is the difference between the variable Value and the nearer of its (finite) bounds bl[j-1] and bu[j-1]. A blank entry indicates that the associated variable is not bounded (i.e., bl[j-1]≤-bigbnd and bu[j-1]≥bigbnd).
The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, bl[j-1] and bu[j-1] are replaced by bl[n+j-1] and bu[n+j-1] respectively, and with the following change in the heading:
L Con gives the name (L) and index j, for j=1,2,…,nL, of the linear constraint.
Note that movement off a constraint (as opposed to a variable moving away from its bound) can be interpreted as allowing the entry in the Slack column to become positive.
Numerical values are output with a fixed number of digits; they are not guaranteed to be accurate to this precision.

Example

Algorithmic Details

This section contains a detailed description of the method used by e04nc.

Overview

e04nc is essentially identical to the method LSSOL described in Gill et al. (1986). It is based on a two-phase (primal) quadratic programming method with features to exploit the convexity of the objective function due to Gill et al. (1984). (In the full-rank case, the method is related to that of Stoer (1971).) e04nc has two 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 two-phase nature of the algorithm is reflected by changing the function being minimized from the sum of infeasibilities to the quadratic objective function. The feasibility phase does not perform the standard simplex method (i.e., it does not necessarily find a vertex), except in the LP case when nL≤n. Once any iterate is feasible, all subsequent iterates remain feasible.
e04nc has been designed to be efficient when used to solve a sequence of related problems – for example, within a sequential quadratic programming method for nonlinearly constrained optimization (e.g., e04uf or e04wd). In particular, you may specify an initial working set (the indices of the constraints believed to be satisfied exactly at the solution); see the discussion of the optional parameter Warm Start.
In general, an iterative process is required to solve a quadratic program. (For simplicity, we shall always consider a typical iteration and avoid reference to the index of the iteration.) Each new iterate x- is defined by
x-=x+αp, (2)
where the step length α is a non-negative scalar, and p is called the search direction.
At each point x, a working set of constraints is defined to be a linearly independent subset of the constraints that are satisfied ‘exactly’ (to within the tolerance defined by the optional parameter Feasibility Tolerance). The working set is the current prediction of the constraints that hold with equality at a solution of (1). The search direction is constructed so that the constraints in the working set remain unaltered for any value of the step length. For a bound constraint in the working set, this property is achieved by setting the corresponding element of the search direction to zero. Thus, the associated variable is fixed, and specification of the working set induces a partition of x into fixed and free variables. During a given iteration, the fixed variables are effectively removed from the problem; since the relevant elements of the search direction are zero, the columns of c corresponding to fixed variables may be ignored.
Let nW denote the number of general constraints in the working set and let nFX denote the number of variables fixed at one of their bounds (nW and nFX are the quantities Lin and Bnd in the monitoring file output from e04nc; see [Description of Monitoring Information]). Similarly, let nFRnFR=n-nFX denote the number of free variables. At every iteration, the variables are reordered so that the last nFX variables are fixed, with all other relevant vectors and matrices ordered accordingly. The order of the variables is indicated by the contents of the array kx on exit (see [Parameters]).

Definition of Search Direction

Main Iteration

Let Q denote the n by n matrix
Q= QFR IFX , (7)
where IFX is the identity matrix of order nFX. Let R denote an n by n upper triangular matrix (the Cholesky factor) such that
RTR=HQ≡QTH~Q, (8)
where H~ is the Hessian H with rows and columns permuted so that the free variables are first.
Let the matrix of the first nZ rows and columns of R be denoted by RZ. The definition of pZ in (6) depends on whether or not the matrix RZ is singular at x. In the nonsingular case, pZ satisfies the equations
RZT RZ pZ = - gZ (9)
where gZ denotes the vector ZTgFR and g denotes the objective gradient. (The norm of gFR is the printed quantity Norm Gf; see [Description of Monitoring Information].) When pZ is defined by (9), x+p is the minimizer of the objective function subject to the constraints (bounds and general) in the working set treated as equalities. In general, a vector fZ is available such that RZT fZ = - gZ , which allows pZ to be computed from a single back-substitution RZpZ=fZ. For example, when solving problem LS1, fZ comprises the first nZ elements of the transformed residual vector
f=Pb-Ax, (10)
which is recurred from one iteration to the next, where P is an orthogonal matrix.
In the singular case, pZ is defined such that
RZ pZ = 0   and   gZT pZ < 0 . (11)
This vector has the property that the objective function is linear along p and may be reduced by any step of the form x+αp, where α>0.
The vector ZTgFR is known as the projected gradient at x. If the projected gradient is zero, x is a constrained stationary point in the subspace defined by Z. During the feasibility phase, the projected gradient will usually be zero only at a vertex (although it may be zero at non-vertices in the presence of constraint dependencies). During the optimality phase, a zero projected gradient implies that x minimizes the quadratic objective when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers λc and λb for the general and bound constraints are defined from the equations
CFRTλC=gFRand ​λB=gFX-CFXTλC. (12)
Given a positive constant δ of the order of the machine precision, the Lagrange multiplier λj corresponding to an inequality constraint in the working set is said to be optimal if λj≤δ when the associated constraint is at its upper bound, or if λj≥-δ when the associated constraint is at its lower bound. If a multiplier is nonoptimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint (with index Jdel; see [Description of Monitoring Information]) from the working set.
If optimal multipliers occur during the feasibility phase and the sum of infeasibilities is nonzero, there is no feasible point, and e04nc will continue until the minimum value of the sum of infeasibilities has been found. At this point, the Lagrange multiplier λj corresponding to an inequality constraint in the working set will be such that -1+δ≤λj≤δ when the associated constraint is at its upper bound, and -δ≤λj≤1+δ when the associated constraint is at its lower bound. Lagrange multipliers for equality constraints will satisfy λj≤1+δ.
The choice of step length is based on remaining feasible with respect to the satisfied constraints. If RZ is nonsingular and x+p is feasible, α will be taken as unity. In this case, the projected gradient at x- will be zero, and Lagrange multipliers are computed. Otherwise, α is set to αm, the step to the ‘nearest’ constraint (with index Jadd; see [Description of Monitoring Information]), which is added to the working set at the next iteration.
If A is not input as a triangular matrix, it is overwritten by a triangular matrix R satisfying (8) obtained using the Cholesky factorization in the QP case, or the QR factorization in the LS case. Column interchanges are used in both cases, and an estimate is made of the rank of the triangular factor. Thereafter, the dependent rows of R are eliminated from the problem.
Each change in the working set leads to a simple change to CFR: if the status of a general constraint changes, a row of CFR is altered; if a bound constraint enters or leaves the working set, a column of CFR changes. Explicit representations are recurred of the matrices T,QFR and R; and of vectors QTg, QTc and f, which are related by the formulae
f=Pb- R 0 QTx ,   b≡0 for the ​ QP ​ case ,
and
QTg=QTc-RTf.
Note that the triangular factor R associated with the Hessian of the original problem is updated during both the optimality and the feasibility phases.
The treatment of the singular case depends critically on the following feature of the matrix updating schemes used in e04nc: if a given factor RZ is nonsingular, it can become singular during subsequent iterations only when a constraint leaves the working set, in which case only its last diagonal element can become zero. This property implies that a vector satisfying (11) may be found using the single back-substitution R-ZpZ=eZ, where R-Z is the matrix RZ with a unit last diagonal, and eZ is a vector of all zeros except in the last position. If H is singular, the matrix R (and hence RZ) may be singular at the start of the optimality phase. However, RZ will be nonsingular if enough constraints are included in the initial working set. (The matrix with no rows and columns is positive-definite by definition, corresponding to the case when CFR contains nFR constraints.) The idea is to include as many general constraints as necessary to ensure a nonsingular RZ.
At the beginning of each phase, an upper triangular matrix R1 is determined that is the largest nonsingular leading sub-matrix of RZ. The use of interchanges during the factorization of A tends to maximize the dimension of R1. (The rank of R1 is estimated using the optional parameter Rank Tolerance.) Let Z1 denote the columns of Z corresponding to R1, and let Z be partitioned as Z=Z1 Z2. A working set for which Z1 defines the null space can be obtained by including the rows of Z2T  as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by Z1.
The artificially augmented working set is given by
C-FR= CFR Z2T , (13)
so that pFR will satisfy CFRpFR=0 and Z2T pFR = 0 . By definition of the TQ factorization, c-FR automatically satisfies the following:
C-FRQFR= CFR Z2T QFR= CFR Z2T Z1 Z2 Y = 0 T- ,
where
T-= 0 T I 0 ,
and hence the TQ factorization of (13) requires no additional work.
The matrix Z2 need not be kept fixed, since its role is purely to define an appropriate null space; the TQ factorization can therefore be updated in the normal fashion as the iterations proceed. No work is required to ‘delete’ the artificial constraints associated with Z2 when Z1T gFR = 0 , since this simply involves repartitioning QFR. When deciding which constraint to delete, the ‘artificial’ multiplier vector associated with the rows of Z2T  is equal to Z2T gFR , and the multipliers corresponding to the rows of the ‘true’ working set are the multipliers that would be obtained if the temporary constraints were not present.
The number of columns in Z2 and Z1 , the Euclidean norm of Z1T gFR , and the condition estimator of R1 appear in the monitoring file output as Art, Zr, Norm Gz and Cond Rz respectively (see [Description of Monitoring Information]).
Although the algorithm of e04nc does not perform simplex steps in general, there is one exception: a linear program with fewer general constraints than variables (i.e., nL≤n). Use of the simplex method in this situation leads to savings in storage. At the starting point, the ‘natural’ working set (the set of constraints exactly or nearly satisfied at the starting point) is augmented with a suitable number of ‘temporary’ bounds, each of which has the effect of temporarily fixing a variable at its current value. In subsequent iterations, a temporary bound is treated as a standard constraint until it is deleted from the working set, in which case it is never added again.
One of the most important features of e04nc is its control of the conditioning of the working set, whose nearness to linear dependence is estimated by the ratio of the largest to smallest diagonals of the TQ factor T (the printed value Cond T; see [Description of Monitoring Information]). In constructing the initial working set, constraints are excluded that would result in a large value of Cond T. Thereafter, e04nc allows constraints to be violated by as much as a user-specified optional parameter Feasibility Tolerance in order to provide, whenever possible, a choice of constraints to be added to the working set at a given iteration. Let αm denote the maximum step at which x+αmp does not violate any constraint by more than its feasibility tolerance. All constraints at distance αα≤αm 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. In order to ensure that the new iterate satisfies the constraints in the working set as accurately as possible, the step taken is the exact distance to the newly added constraint. As a consequence, negative steps are occasionally permitted, since the current iterate may violate the constraint to be added by as much as the feasibility tolerance.

Description of Monitoring Information

This section describes the long line of output (>80 characters) which forms part of the monitoring information produced by e04nc. (See also the description of the optional parameters Monitoring File and Print Level.) You can control the level of printed output.
To aid interpretation of the printed results, the following convention is used for numbering the constraints: indices 1 through n refer to the bounds on the variables, and indices n+1 through n+nL refer to the general constraints. When the status of a constraint changes, the index of the constraint is printed, along with the designation L (lower bound), U (upper bound), E (equality), F (temporarily fixed variable) or A (artificial constraint).
When Print Level≥5 and Monitoring File≥0, the following line of output is produced at every iteration on the unit number specified by optional parameter Monitoring File. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.
Itn is the iteration count.
Jdel is the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted.
Jadd is the index of the constraint added to the working set. If Jadd is zero, no constraint was added.
Step is the step taken along the computed search direction. If a constraint is added during the current iteration (i.e., Jadd is positive), Step will be the step to the nearest constraint. During the optimality phase, the step can be greater than one only if the factor RZ is singular.
Ninf is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Objective is the value of the current objective function. If x is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If x is feasible, Objective is the value of the objective function of (1). The output line for the final iteration of the feasibility phase (i.e., the first iteration for which Ninf is zero) will give the value of the true objective at the first feasible point.
During the optimality phase the value of the objective function will be nonincreasing. During the feasibility phase the number of constraint infeasibilities will not increase until either a feasible point is found or the optimality of the multipliers implies that no feasible point exists. Once optimal multipliers are obtained the number of infeasibilities can increase, but the sum of infeasibilities will either remain constant or be reduced until the minimum sum of infeasibilities is found.
Bnd is the number of simple bound constraints in the current working set.
Lin is the number of general linear constraints in the current working set.
Art is the number of artificial constraints in the working set, i.e., the number of columns of Z2 (see [Main Iteration]).
Zr is the number of columns of Z1 (see [Definition of Search Direction]). Zr is the dimension of the subspace in which the objective function is currently being minimized. The value of Zr is the number of variables minus the number of constraints in the working set; i.e., Zr=n-Bnd+Lin+Art.
The value of nZ, the number of columns of Z (see [Definition of Search Direction]) can be calculated as nZ=n-Bnd+Lin. A zero value of nZ implies that x lies at a vertex of the feasible region.
Norm Gz is Z1T gFR , the Euclidean norm of the reduced gradient with respect to Z1. During the optimality phase, this norm will be approximately zero after a unit step.
Norm Gf is the Euclidean norm of the gradient function with respect to the free variables, i.e., variables not currently held at a bound.
Cond T is a lower bound on the condition number of the working set.
Cond Rz is a lower bound on the condition number of the triangular factor R1 (the first Zr rows and columns of the factor RZ). If the problem is specified to be of type LP or the estimated rank of the data matrix A is zero then Cond Rz is not printed.

See Also