e04ncc solves linearly constrained linear least squares problems and convex quadratic programming problems. It is not intended for large sparse problems.

2 Specification

#include <nag.h>

void	e04ncc (Integer m, Integer n, Integer nclin, const double a[], Integer tda, const double bl[], const double bu[], const double cvec[], double b[], double h[], Integer tdh, Integer kx[], double x[], double objf, Nag_E04_Opt options, Nag_Comm comm, NagError fail)

The function may be called by the names: e04ncc, nag_opt_lsq_lincon_solve or nag_opt_lin_lsq.

3 Description

e04ncc is designed to solve a class of quadratic programming problems stated in the following general form:

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

(1)

where

A

is an

n_{L}

n

matrix and the objective function

F (x)

may be specified in a variety of ways depending upon the particular problem to be solved. The available forms for

F (x)

are listed in Table 1 below, 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

‖x‖

denotes the Euclidean length of

x

Problem Type	$f (x)$	Matrix $H$
FP	Not applicable	Not applicable
LP	$c^{T} x$	Not applicable
QP1	$\frac{1}{2} x^{T} H x$	$n$ by $n$ symmetric positive semidefinite
QP2	$c^{T} x + \frac{1}{2} x^{T} H x$	$n$ by $n$ symmetric positive semidefinite
QP3	$\frac{1}{2} x^{T} H^{T} H x$	$m$ by $n$ upper trapezoidal
QP4	$c^{T} x + \frac{1}{2} x^{T} H^{T} H x$	$m$ by $n$ upper trapezoidal
LS1	$\frac{1}{2} {‖b - H x‖}^{2}$	$m$ by $n$
LS2	$c^{T} x + \frac{1}{2} {‖b - H x‖}^{2}$	$m$ by $n$
LS3	$\frac{1}{2} {‖b - H x‖}^{2}$	$m$ by $n$ upper trapezoidal
LS4	$c^{T} x + \frac{1}{2} {‖b - H x‖}^{2}$	$m$ by $n$ upper trapezoidal

Table 1

For problems of type LS,

H

is referred to as the least squares matrix, or the matrix of observations, and

b

as the vector of observations. The default problem type is LS1, and other objective functions are selected by using the optional parameter

options . prob

(see Section 12.2).

When

H

is upper trapezoidal it will usually be the case that

m = n

, so that

H

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

Q R

factorization, has been performed.

The constraints involving

A

are called the general constraints. Note that upper and lower bounds are specified for all the variables and for all the general constraints. An equality constraint can be specified by setting

l_{i} = u_{i}

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

l

u

can be set to special values that will be treated as

- \infty

+ \infty

. (See the description of the optional parameter

options . inf_bound

in Section 12.2.

The function

F (x)

is a quadratic function, whose defining feature is that its second-derivative matrix

\nabla^{2} F (x)

(the Hessian matrix) is constant. For the LP case,

\nabla^{2} F (x) = 0

; for QP1 and QP2,

\nabla^{2} F (x) = H

; and for QP3, QP4 and LS problems,

\nabla^{2} F (x) = H^{T} H

and the Hessian matrix is positive semidefinite (positive definite if

H

is full rank), so that

F (x)

is convex. If

H

is defined as the zero matrix, e04ncc will solve the resulting linear programming problem; however, this can be accomplished more efficiently by using e04mfc.

Problems of type QP3 and QP4 for which

H

is not in upper trapezoidal form should be solved as problems of type LS1 and LS2 respectively, with

b = 0

You must supply an initial estimate of the solution.

H

is of full rank then e04ncc will obtain the unique (global) minimum. If

H

is not of full rank 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

F (x)

contains an explicit linear term (as in problems LP, QP2, QP4, LS2 and LS4).

The method used by e04ncc is described in detail in Section 11.

4 References

Gill P E, Hammarling S, Murray W, Saunders M A and Wright M H (1986) Users' guide for LSSOL (Version 1.0) Report SOL 86-1 Department of Operations Research, Stanford University

Gill P E, Murray W, Saunders M A and Wright M H (1984) Procedures for optimization problems with a mixture of bounds and general linear constraints ACM Trans. Math. Software 10 282–298

Gill P E, Murray W and Wright M H (1981) Practical Optimization Academic Press

Stoer J (1971) On the numerical solution of constrained least squares problems SIAM J. Numer. Anal. 8 382–411

5 Arguments

1: $m$ – Integer Input

On entry:

m

, the number of rows in the matrix

H

. If the problem is of type FP or LP, m is not referenced and is assumed to be zero. The default type is LS1; other problem types can be specified using the optional parameter

options . prob

, see Section 12.2.

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 problem type QP3 or QP4 if

H

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 QP cases,

m

should not be greater than

m

; if it is, the last

(m - n)

rows of

H

are ignored.

If the problem is a least squares problem (in particular, the default type LS1), 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 problem is not FP or LP.

2: $n$ – Integer Input

On entry:

n

, the number of variables.

Constraint:

n > 0

3: $nclin$ – Integer Input

On entry:

n_{L}

, the number of general linear constraints.

Constraint:

nclin \geq 0

4: $a [nclin \times tda]$ – const double Input

Note: the

(i, j)

th element of the matrix

A

is stored in

a [(i - 1) \times tda + j - 1]

On entry: the

i

th row of a must contain the coefficients of the

i

th general linear constraint (the

i

th row of

A

), for

i = 1, 2, \dots, n_{L}

. If

nclin = 0

then the array a is not referenced.

5: $tda$ – Integer Input

On entry: the stride separating matrix column elements in the array a.

Constraint:

tda \geq n

nclin > 0

6: $bl [n + nclin]$ – const double Input

7: $bu [n + nclin]$ – const double Input

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

n_{L}

elements the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e.,

l_{j} = - \infty

), set

bl [j - 1] \leq - options . inf_bound

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

u_{j} = + \infty

), set

bu [j - 1] \geq options . inf_bound

, where

options . inf_bound

is one of the optional parameters (default value

10^{20}

(see Section 12.2). To specify the

j

th constraint as an equality, set

bl [j - 1] = bu [j - 1] = β

, say, where

|β| < options . inf_bound

Constraints:

$bl [j] \leq bu [j]$ , for $j = 0, 1, \dots, n + nclin - 1$ ;
if $bl [j] = bu [j] = β$ , $|β| < options . inf_bound$ .

8: $cvec [n]$ – const double Input

On entry: the coefficients of the explicit linear term of the objective function when the problem is of type LP, QP2, QP4, LS2 or LS4.

If the problem is of type FP, QP1, QP3, LS1 (the default) or LS3, cvec is not referenced and may be NULL.

9: $b [m]$ – double Input/Output

On entry: the

m

elements of the vector of observations.

On exit: the transformed residual vector of equation (10).

b is referenced only in the case of least squares problem types (in particular, the default type LS1. For other problem types, b is not referenced and may be NULL.

10: $h [m \times tdh]$ – double Input/Output

Note: the

(i, j)

th element of the matrix

H

is stored in

h [(i - 1) \times tdh + j - 1]

On entry: the array h must contain the matrix

H

as specified in Table 1 (see Section 3).

For problems QP1 and QP2, the first

m

rows and columns of h must contain the leading

m

m

rows and columns of the symmetric Hessian matrix. Only the diagonal and upper triangular elements of the leading

m

rows and columns of h are referenced. The remaining elements are assumed to be zero and need not be assigned.

For problems QP3, QP4, LS3 and LS4, the first

m

rows of h must contain an

m

n

upper trapezoidal factor of either the Hessian or the least squares matrix, ordered according to the array kx (see below). 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

H

, the fewer iterations will be required. Elements outside the upper trapezoidal part of the first

m

rows of

H

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

H

into a triangular matrix before the call to e04ncc and solve as type LS3 or LS4.

On exit: by default, h contains the upper triangular Cholesky factor

R

of equation (8), with columns ordered as indicated by kx (see below). If the optional parameter

options . hessian = Nag_TRUE

(see Section 12.2), and the problem is one of the LS or QP types, h contains the upper triangular Cholesky factor of the Hessian matrix

\nabla^{2} F

, with columns ordered as indicated by kx (see below). In either case, this matrix 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, h is not referenced and may be NULL.

11: $tdh$ – Integer Input

On entry: the stride separating matrix column elements in the array h.

Constraint:

tdh \geq n

12: $kx [n]$ – Integer Input/Output

On entry: for problems of type QP3, QP4, LS3 or LS4 the array kx must specify the order of the columns of the matrix

H

with respect to the ordering of x. Thus if column

j

H

is the column associated with the variable

x_{i}

then

kx [j - 1] = i

If the problem is of any other type then the array kx need not be initialized.

Constraints:

$1 \leq kx [i] \leq n$ , for $i = 0, 1, \dots, n - 1$ ;
if $i \neq j$ , $kx [i] \neq kx [j]$ .

On exit: defines the order of the columns of

H

with respect to the ordering of x, as described above.

13: $x [n]$ – double Input/Output

On entry: an initial estimate of the solution.

On exit: the point at which e04ncc terminated. If

fail . code = NE_NOERROR

, NW_SOLN_NOT_UNIQUE or NW_NOT_FEASIBLE, x contains an estimate of the solution.

14: $objf$ – double * Output

On exit: the value of the objective function at

x

x

is feasible, or the sum of infeasibilities at

x

otherwise. If the problem is of type FP and

x

is feasible, objf is set to zero.

15: $options$ – Nag_E04_Opt * Input/Output

On entry/exit: a pointer to a structure of type Nag_E04_Opt whose members are optional parameters for e04ncc. These structure members offer the means of adjusting some of the argument values of the algorithm and on output will supply further details of the results. A description of the members of options is given below in Section 12. Some of the results returned in options can be used by e04ncc to perform a ‘warm start’ (see the member

options . start

in Section 12.2).

If any of these optional parameters are required then the structure options should be declared and initialized by a call to e04xxc and supplied as an argument to e04ncc. However, if the optional parameters are not required the NAG defined null pointer, E04_DEFAULT, can be used in the function call.

16: $comm$ – Nag_Comm * Input/Output

Note: comm is a NAG defined type (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).

On entry/exit: structure containing pointers for communication with an optional user-defined printing function; see Section 12.3.1 for details. If you do not need to make use of this communication feature the null pointer NAGCOMM_NULL may be used in the call to e04ncc; comm will then be declared internally for use in calls to user-supplied functions.

17: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $tda = 〈value〉$ while $n = 〈value〉$ . These arguments must satisfy $tda \geq n$ .

On entry, $tdh = 〈value〉$ while $n = 〈value〉$ . These arguments must satisfy $tdh \geq n$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_ARRAY_CONS: The contents of array kx are not valid. Constraint: must contain a permutation of integers $1, 2, \dots, n$ .
NE_B_NULL: $options . prob = 〈value〉$ but argument $b =$ NULL.
NE_BAD_PARAM: On entry, argument $options . print_level$ had an illegal value.

On entry, argument $options . prob$ had an illegal value.

On entry, argument $options . start$ had an illegal value.
NE_BOUND: The lower bound for variable $〈value〉$ (array element $bl [〈value〉]$ ) is greater than the upper bound.
NE_BOUND_LCON: The lower bound for linear constraint $〈value〉$ (array element $bl [〈value〉]$ ) is greater than the upper bound.
NE_CVEC_NULL: $options . prob = 〈value〉$ but argument $cvec =$ NULL.
NE_CYCLING: The algorithm could be cycling, since a total of 50 changes were made to the working set without altering $x$ . Check the detailed iteration printout for a repeated pattern of constraint deletions and additions.
If a sequence of constraint changes is being repeated, the iterates are probably cycling. (e04ncc 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 $options . ftol$ (default value $= \sqrt{ε}$ , where $ε$ is the machine precision; see Section 12.2). If this error exit occurs but no suspicious pattern of constraint changes can be observed, it may be worthwhile to restart with the final $x$ (with optional parameter $options . start = Nag_Cold$ or $Nag_Warm$ ).
NE_H_NULL_QP: $options . prob = 〈value〉$ but argument $h =$ NULL. This problem type requires an array to be supplied in argument h.
NE_INT_ARG_LT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .

On entry, $nclin = 〈value〉$ .
Constraint: $nclin \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_INVALID_INT_RANGE_1: Value $〈value〉$ given to $options . fmax_iter$ is not valid. Correct range is $options . fmax_iter \geq 0$ .

Value $〈value〉$ given to $options . max_iter$ is not valid. Correct range is $options . max_iter \geq 0$ .
NE_INVALID_REAL_RANGE_F: Value $〈value〉$ given to $options . ftol$ is not valid. Correct range is $options . ftol > 0.0$ .

Value $〈value〉$ given to $options . inf_bound$ is not valid. Correct range is $options . inf_bound > 0.0$ .

Value $〈value〉$ given to $options . inf_step$ is not valid. Correct range is $options . inf_step > 0.0$ .
NE_INVALID_REAL_RANGE_FF: Value $〈value〉$ given to $options . crash_tol$ is not valid. Correct range is $0.0 \leq options . crash_tol \leq 1.0$ .

Value $〈value〉$ given to $options . rank_tol$ is not valid. Correct range is $0.0 < options . rank_tol < 1.0$ .
NE_NOT_APPEND_FILE: Cannot open file $〈string〉$ for appending.
NE_NOT_CLOSE_FILE: Cannot close file $〈string〉$ .
NE_OPT_NOT_INIT: Options structure not initialized.
NE_STATE_VAL: $options . state [〈value〉]$ is out of range. $options . state [〈value〉] = 〈value〉$ .
NE_UNBOUNDED: Solution appears to be unbounded.
This error indicator implies that a step as large as optional parameter $options . inf_step$ (default value $10^{20}$ ; see Section 12.2) would have to be taken in order to continue the algorithm. This situation can occur only when $H$ is singular, there is an explicit linear term, and at least one variable has no upper or lower bound.
NE_WARM_START: $options . start = Nag_Warm$ but pointer $options . state =$ NULL.
NE_WRITE_ERROR: Error occurred when writing to file $〈string〉$ .
NW_NOT_FEASIBLE: No feasible point was found for the linear constraints.
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, if the feasibility tolerance for each violated constraint exceeds its Residual (see Section 12.3) at the final point. The modified problem (with an altered value of the optional feasibility tolerance, $options . ftol$ ) may then be solved using optional parameter $options . start = Nag_Warm$ (see Section 12.2). 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 $options . ftol$ is greater than $σ$ . For example, if all elements of $A$ are of order unity and are accurate only to three decimal places, $options . ftol$ should be at least $10^{- 3}$ .
NW_OVERFLOW_WARN: Serious ill conditioning in the working set after adding constraint $〈value〉$ . Overflow may occur in subsequent iterations.
If overflow occurs preceded by this warning then serious ill conditioning has probably occurred in the working set when adding a constraint. It may be possible to avoid the difficulty by increasing the magnitude of the optional parameter $options . ftol$ and re-running the program. If the message recurs even after this change, the offending linearly dependent constraint $j$ must be removed from the problem.
NW_SOLN_NOT_UNIQUE: Optimal solution is not unique.
The point in x is a weak local minimum, i.e., the projected gradient is negligible, the Lagrange multipliers are optimal, but either $R_{z}$ (see Section 11.3) is singular or there is a small multiplier. This means that $x$ is not unique.
NW_TOO_MANY_ITER: The maximum number of iterations, $〈value〉$ , have been performed.
The limiting number of iterations (determined by the optional parameters $options . max_iter$ and $options . fmax_iter$ , see Section 12.2) 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 iteration limits or, alternatively, rerun e04ncc using the optional parameter $options . start = Nag_Warm$ 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 extended iteration printout (see Section 12.3) 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 Section 11.2), which will be printed if optional parameter $options . print_level = Nag_Soln_Iter_Full$ (default value $options . print_level = Nag_Soln_Iter$ ; see Section 12.2.) In this case, the algorithm could be cycling (see the comments below for $fail . code = NE_CYCLING$ ).

7 Accuracy

e04ncc implements a numerically stable active set strategy and returns solutions that are as accurate as the condition of the problem warrants on the machine.

8 Parallelism and Performance

e04ncc is not threaded in any implementation.

9 Further Comments

9.1 Termination Criteria

e04ncc exits with

fail . code = NE_NOERROR

x

is a strong local minimizer, i.e., the reduced gradient is negligible, the Lagrange multipliers are optimal (see Section 12.3) and

R_{z}

(see Section 11.3) is nonsingular.

9.2 Scaling

Sensible scaling of the problem is likely to reduce the number of iterations required and make the problem less sensitive to perturbations in the data, thus improving the condition of the problem. In the absence of better information it is usually sensible to make the Euclidean lengths of each constraint of comparable magnitude. See the E04 Chapter Introduction and Gill et al. (1981) for further information and advice.

10 Example

To minimize the quadratic function

c^{T} x + \frac{1}{2} x^{T} H x

, where

c = {(- 4.0, - 1.0, - 1.0, - 1.0, - 1.0, - 1.0, - 1.0, - 0.1, - 0.3)}^{T},

H = (\begin{matrix} 2 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 2 & 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 2 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix})

subject to the bounds

\begin{matrix} - 2 \leq x_{1} \leq 2 \\ - 2 \leq x_{2} \leq 2 \\ - 2 \leq x_{3} \leq 2 \\ - 2 \leq x_{4} \leq 2 \\ - 2 \leq x_{5} \leq 2 \\ - 2 \leq x_{6} \leq 2 \\ - 2 \leq x_{7} \leq 2 \\ - 2 \leq x_{8} \leq 2 \\ - 2 \leq x_{9} \leq 2 \end{matrix}

and to the general constraints

\begin{matrix} - 2.0 \leq x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + 4 x_{9} \leq 1.5 \\ - 2.0 \leq x_{1} + 2 x_{2} + 3 x_{3} + 4 x_{4} - 2 x_{5} + x_{6} + x_{7} + x_{8} + x_{9} \leq 1.5 \\ - 2.0 \leq x_{1} - x_{2} + x_{3} - x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} \leq 4.0 \end{matrix}

The initial point, which is feasible, is

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

and

F (x_{0}) = 0

The optimal solution (to five figures) is

x^{*} = {(2.0, - 0.23333, - 0.26667, - 0.3, - 0.1, 2.0, 2.0, - 1.7777, - 0.45555)}^{T},

and

F (x^{*}) = - 8.0678

. Three bound constraints and two general constraints are active at the solution. Note that, although the Hessian matrix is positive semidefinite, the point

x^{*}

is unique.

This example illustrates the use of the options structure. Since the problem is of type QP2 (as described in Section 3) and the default value of the optional parameter

options . prob = Nag_LS1

, it is necessary to reset this argument to

options . prob = Nag_QP2

in order to solve the problem. This is achieved by declaring the options structure and initializing it by calling e04xxc. Then

options . prob

is assigned directly, before calling e04ncc. On return from e04ncc, e04xzc is used to free the memory assigned to the pointers in the options structure. You must not use the standard C function free() for this purpose.

11 Further Description

This section gives a detailed description of the algorithm used in e04ncc. This, and possibly the next section, Section 12, may be omitted if the more sophisticated features of the algorithm and software are not currently of interest.

11.1 Overview

e04ncc is based on an inertia-controlling method that maintains a Cholesky factorization of the reduced Hessian (see below). The method is based on that of Gill and Murray (1978) and is described in detail by Gill et al. (1981). Here we briefly summarise the main features of the method.

e04ncc uses essentially the same algorithm as the subroutine 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, see Stoer (1971).) e04ncc 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

n_{L} \leq n

. Once any iterate is feasible, all subsequent iterates remain feasible.

e04ncc 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., e04ucc). 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

options . start

in Section 12.2.

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

\bar{x}

is defined by

\bar{x} = x + α p,

(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

options . ftol

; see Section 12.2). 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

A

corresponding to fixed variables may be ignored.

Let

n_{W}

denote the number of general constraints in the working set and let

n_{FX}

denote the number of variables fixed at one of their bounds (

n_{W}

and

n_{FX}

are the quantities Lin and Bnd in the extended iteration printout from e04ncc; see Section 12.3). Similarly, let

n_{FR}

(n_{FR} = n - n_{FX})

denote the number of free variables. At every iteration, the variables are re-ordered so that the last

n_{FX}

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 Section 5).

11.2 Definition of the Search Direction

Let

A_{FR}

denote the

n_{W}

n_{FR}

sub-matrix of general constraints in the working set corresponding to the free variables, and let

p_{FR}

denote the search direction with respect to the free variables only. The general constraints in the working set will be unaltered by any move along

p

A_{FR} p_{FR} = 0 .

(3)

In order to compute

p_{FR}

, the

T Q

factorization of

A_{FR}

is used:

A_{FR} Q_{FR} = (\begin{matrix} 0 & T \end{matrix})

(4)

where

T

is a nonsingular

n_{W}

n_{W}

reverse-triangular matrix (i.e.,

t_{i j} = 0

i + j < n_{W}

), and the nonsingular

n_{FR}

n_{FR}

matrix

Q_{FR}

is the product of orthogonal transformations (see Gill et al. (1984)). If the columns of

Q_{FR}

are partitioned so that

Q_{FR} = (\begin{matrix} Z & Y \end{matrix}),

(5)

where

Y

n_{FR}

n_{W}

, then the

n_{Z}

(n_{Z} = n_{FR} - n_{W})

columns of

Z

form a basis for the null space of

A_{FR}

. Let

n_{R}

be an integer such that

0 \leq n_{R} \leq n_{Z}

, and let

Z_{1}

denote a matrix whose

n_{R}

columns are a subset of the columns of

Z

. (The integer

n_{R}

is the quantity Zr in the extended iteration printout from e04ncc; see Section 12.3. In many cases,

Z_{1}

will include all the columns of

Z

.) The direction

p_{FR}

will satisfy (3) if

p_{FR} = Z_{1} p_{Z}

(6)

where

p_{Z}

is any

n_{R}

-vector.

11.3 The Main Iteration

Let

Q

denote the

n

n

matrix

Q = (\begin{matrix} Q_{FR} \\ I_{FX} \end{matrix})

(7)

where

I_{FX}

is the identity matrix of order

n_{FX}

. Let

R

denote an

n

n

upper triangular matrix (the Cholesky factor) such that

Q^{T} \tilde{\nabla^{2} F} Q \equiv H_{Q} = R^{T} R,

(8)

and let the matrix of the first

n_{Z}

rows and columns of

R

be denoted by

R_{Z}

. (The matrix

\tilde{\nabla^{2} F}

in (8) is the Hessian with its rows and columns permuted so that the free variables come first.)

The definition of

p_{Z}

in (6) depends on whether or not the matrix

R_{Z}

is singular at

x

. In the nonsingular case,

p_{Z}

satisfies the equations

R_{Z}^{T} R_{Z} p_{Z} = - g_{Z}

(9)

where

g_{Z}

denotes the vector

Z^{T} g_{FR}

and

g

denotes the objective gradient. (The norm of

g_{FR}

is the printed quantity Norm Gf; see Section 12.3.) When

p_{Z}

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

f_{Z}

is available such that

R_{Z}^{T} f_{Z} = - g_{Z}

, which allows

p_{Z}

to be computed from a single back-substitution

R_{Z} p_{Z} = f_{Z}

. For example, when solving problem LS1,

f_{Z}

comprises the first

n_{Z}

elements of the transformed residual vector

f = P (b - H x)

(10)

which is recurred from one iteration to the next, where

P

is an orthogonal matrix.

In the singular case,

p_{Z}

is defined such that

R_{Z} p_{Z} = 0 and g_{Z}^{T} p_{Z} < 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 + α p

, where

α > 0

The vector

Z^{T} g_{FR}

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

λ_{A}

and

λ_{B}

for the general and bound constraints are defined from the equations

A_{FR}^{T} λ_{A} = g_{FR} and λ_{B} = g_{FX} - A_{FX}^{T} λ_{A} .

(12)

Given a positive constant

δ

of the order of the machine precision, the Lagrange multiplier

λ_{j}

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

λ_{j} \leq δ

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

λ_{j} \geq - δ

when the associated constraint is at its lower bound. If a multiplier is non-optimal, the objective function (either the true objective or the sum of infeasibilities) can be reduced by deleting the corresponding constraint (with index Jdel; see Section 12.3) 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 e04ncc will continue until the minimum value of the sum of infeasibilities has been found. At this point, the Lagrange multiplier

λ_{j}

corresponding to an inequality constraint in the working set will be such that

- (1 + δ) \leq λ_{j} \leq δ

when the associated constraint is at its upper bound, and

- δ \leq λ_{j} \leq (1 + δ)

when the associated constraint is at its lower bound. Lagrange multipliers for equality constraints will satisfy

|λ_{j}| \leq 1 + δ

The choice of step length is based on remaining feasible with respect to the satisfied constraints. If

R_{Z}

is nonsingular and

x + p

is feasible,

α

will be taken as unity. In this case, the projected gradient at

\bar{x}

will be zero, and Lagrange multipliers are computed. Otherwise,

α

is set to

α_{M}

, the step to the ‘nearest’ constraint (with index Jadd; see Section 12.3), which is added to the working set at the next iteration.

H

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

Q R

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

A_{FR}

: if the status of a general constraint changes, a row of

A_{FR}

is altered; if a bound constraint enters or leaves the working set, a column of

A_{FR}

changes. Explicit representations are recurred of the matrices

T, Q_{FR}

and

R

; and of vectors

Q^{T} g

Q^{T} c

and

f

, which are related by the formulae

f = P b - (\begin{matrix} R \\ 0 \end{matrix}) Q^{T} x, (b \equiv 0 ​ for the QP case ​),

and

Q^{T} g = Q^{T} c - R^{T} f .

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 e04ncc: if a given factor

R_{Z}

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

{\bar{R}}_{Z} p_{Z} = e_{Z}

, where

{\bar{R}}_{Z}

is the matrix

R_{Z}

with a unit last diagonal, and

e_{Z}

is a vector of all zeros except in the last position. If the Hessian matrix

\nabla^{2} F

is singular, the matrix

R

(and hence

R_{Z}

) may be singular at the start of the optimality phase. However,

R_{Z}

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

A_{FR}

contains

n_{FR}

constraints.) The idea is to include as many general constraints as necessary to ensure a nonsingular

R_{Z}

At the beginning of each phase, an upper triangular matrix

R_{1}

is determined that is the largest nonsingular leading sub-matrix of

R_{Z}

. The use of interchanges during the factorization of

H

tends to maximize the dimension of

R_{1}

. (The rank of

R_{1}

is estimated using the optional parameter

options . rank_tol

; see Section 12.2.) Let

Z_{1}

denote the columns of

Z

corresponding to

R_{1}

, and let

Z

be partitioned as

Z = (\begin{matrix} Z_{1} & Z_{2} \end{matrix})

. A working set for which

Z_{1}

defines the null space can be obtained by including the rows of

Z_{2}^{T}

as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by

Z_{1}

The artificially augmented working set is given by

{\bar{A}}_{FR} = (\begin{matrix} A_{FR} \\ Z_{2}^{T} \end{matrix}),

(13)

so that

p_{FR}

will satisfy

A_{FR} p_{FR} = 0

and

Z_{2}^{T} p_{FR} = 0

. By definition of the

T Q

factorization,

{\bar{A}}_{FR}

automatically satisfies the following:

{\bar{A}}_{FR} Q_{FR} = (\begin{matrix} A_{FR} \\ Z_{2}^{T} \end{matrix}) Q_{FR} = (\begin{matrix} A_{FR} \\ Z_{2}^{T} \end{matrix}) (\begin{matrix} Z_{1} & Z_{2} & Y \end{matrix}) = (\begin{matrix} 0 & \bar{T} \end{matrix}),

where

\bar{T} = (\begin{matrix} 0 & T \\ I & 0 \end{matrix}),

and hence the

T Q

factorization of (13) requires no additional work.

The matrix

Z_{2}

need not be kept fixed, since its role is purely to define an appropriate null space; the

T Q

factorization can therefore be updated in the normal fashion as the iterations proceed. No work is required to ‘delete’ the artificial constraints associated with

Z_{2}

when

Z_{1}^{T} g_{FR} = 0

, since this simply involves repartitioning

Q_{FR}

. When deciding which constraint to delete, the ‘artificial’ multiplier vector associated with the rows of

Z_{2}^{T}

is equal to

Z_{2}^{T} g_{FR}

, 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

Z_{2}

and

Z_{1}

, the Euclidean norm of

Z_{1}^{T} g_{FR}

, and the condition estimator of

R_{1}

appear in the extended iteration printout as Art, Zr, Norm Gz and Cond Rz respectively (see Section 12.3).

Although the algorithm of e04ncc does not perform simplex steps in general, there is one exception: a linear program with fewer general constraints than variables (i.e.,

n_{L} \leq n

). (Use of the simplex method in this situation leads to savings in storage.) At the starting point, the ‘natural’ working set (the set of constraints exactly or nearly satisfied at the starting point) is augmented with a suitable number of ‘temporary’ bounds, each of which has the effect of temporarily fixing a variable at its current value. In subsequent iterations, a temporary bound is treated as a standard constraint until it is deleted from the working set, in which case it is never added again.

One of the most important features of e04ncc 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

T Q

factor

T

(the printed value Cond T; see Section 12.3). In constructing the initial working set, constraints are excluded that would result in a large value of Cond T. Thereafter, e04ncc allows constraints to be violated by as much as a user-specified feasibility tolerance (see

options . ftol

, Section 12.2) in order to provide, whenever possible, a choice of constraints to be added to the working set at a given iteration. Let

α_{M}

denote the maximum step at which

x + α_{M} p

does not violate any constraint by more than its feasibility tolerance. All constraints at distance

α (α \leq α_{M})

along

p

from the current point are then viewed as acceptable candidates for inclusion in the working set. The constraint whose normal makes the largest angle with the search direction is added to the working set. 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.

12 Optional Parameters

A number of optional input and output arguments to e04ncc are available through the structure argument options, type Nag_E04_Opt. An argument may be selected by assigning an appropriate value to the relevant structure member; those arguments not selected will be assigned default values. If no use is to be made of any of the optional parameters you should use the NAG defined null pointer, E04_DEFAULT, in place of options when calling e04ncc; the default settings will then be used for all arguments.

Before assigning values to options directly the structure must be initialized by a call to the function e04xxc. Values may then be assigned to the structure members in the normal C manner.

After return from e04ncc, the options structure may only be re-used for future calls of e04ncc if the dimensions of the new problem are the same. Otherwise, the structure must be cleared by a call of e04xzc) and re-initialized by a call of e04xxc before future calls. Failure to do this will result in unpredictable behaviour.

Option settings may also be read from a text file using the function e04xyc in which case initialization of the options structure will be performed automatically if not already done. Any subsequent direct assignment to the options structure must not be preceded by initialization.

If assignment of functions and memory to pointers in the options structure is required, then this must be done directly in the calling program; they cannot be assigned using e04xyc.

12.1 Optional Parameter Checklist and Default Values

For easy reference, the following list shows the members of options which are valid for e04ncc together with their default values where relevant. The number

ε

is a generic notation for machine precision (see X02AJC).

Nag_ProblemType prob	Nag_LS1
Nag_Start start	$Nag_Cold$
Boolean list	Nag_TRUE
Nag_PrintType print_level	Nag_Soln_Iter
char outfile[512]	stdout
void (*print_fun)()	NULL
Integer fmax_iter	$\max (50, 5 (n + nclin))$
Integer max_iter	$\max (50, 5 (n + nclin))$
double crash_tol	0.01
double ftol	$\sqrt{ε}$
double inf_bound	$10^{20}$
double inf_step	$\max (options . inf_bound, 10^{20})$
double rank_tol	$100 ε$ or $10 \sqrt{ε}$
Integer *state	size $n + nclin$
double *ax	size nclin
double *lambda	size $n + nclin$
Boolean hessian	Nag_FALSE
Integer iter

12.2 Description of the Optional Parameters

prob – Nag_ProblemType

Default

= Nag_LS1

On entry: specifies the type of objective function to be minimized during the optimality phase. The following are the ten possible values of

options . prob

and the size of the arrays h, kx, b and cvec that are required to define the objective function:

$Nag_FP$	h, b and cvec not referenced;
$Nag_LP$	h and b not referenced, cvec of size n;
$Nag_QP1$	h of size $m \times tdh$ , symmetric, b and cvec not referenced;
$Nag_QP2$	h of size $m \times tdh$ , symmetric, b not referenced, cvec of size n;
$Nag_QP3$	h of size $m \times tdh$ , upper trapezoidal, b and cvec not referenced;
$Nag_QP4$	h of size $m \times tdh$ , upper trapezoidal, b not referenced, cvec of size n.
$Nag_LS1$	h of size $m \times tdh$ , b of size m, cvec not referenced;
$Nag_LS2$	h of size $m \times tdh$ , b of size m, cvec of size n;
$Nag_LS3$	h of size $m \times tdh$ , upper trapezoidal, b of size m, cvec not referenced;
$Nag_LS4$	h of size $m \times tdh$ , upper trapezoidal, b of size m, cvec of size n.

The array kx of size n must be supplied for all problem types but need only be initialized for types

Nag_QP3

Nag_QP4

Nag_LS3

and

Nag_LS4

. If

H = 0

, i.e., the objective function is purely linear, the efficiency of e04ncc may be increased by specifying

options . prob = Nag_LP

Constraint:

options . prob = Nag_FP

Nag_LP

Nag_QP1

Nag_QP2

Nag_QP3

Nag_QP4

Nag_LS1

Nag_LS2

Nag_LS3

Nag_LS4

start – Nag_Start

Default

= Nag_Cold

On entry: specifies how the initial working set is chosen. With

options . start = Nag_Cold

, e04ncc chooses the initial working set based on the values of the variables and constraints at the initial point. Broadly speaking, the initial working set will include equality constraints and bounds or inequality constraints that violate or ‘nearly’ satisfy their bounds (to within the value of the optional parameter

options . crash_tol

; see below).

With

options . start = Nag_Warm

, you must provide a valid definition of every array element of the optional parameter

options . state

(see below). e04ncc will override your specification of

options . state

if necessary, so that a poor choice of the working set will not cause a fatal error. For instance, any elements of

options . state

which are set to

- 2

- 1

or 4 will be reset to zero, as will any elements which are set to 3 when the corresponding elements of bl and bu are not equal. A warm start will be advantageous if a good estimate of the initial working set is available – for example, when e04ncc is called repeatedly to solve related problems.

Constraint:

options . start = Nag_Cold

Nag_Warm

list – Nag_Boolean

Default

= Nag_TRUE

On entry: if

options . list = Nag_TRUE

the argument settings in the call to e04ncc will be printed.

print_level – Nag_PrintType

Default

= Nag_Soln_Iter

On entry: the level of results printout produced by e04ncc. The following values are available:

$Nag_NoPrint$	No output.
$Nag_Soln$	The final solution.
$Nag_Iter$	One line of output for each iteration.
$Nag_Iter_Long$	A longer line of output for each iteration with more information (line exceeds 80 characters).
$Nag_Soln_Iter$	The final solution and one line of output for each iteration.
$Nag_Soln_Iter_Long$	The final solution and one long line of output for each iteration (line exceeds 80 characters).
$Nag_Soln_Iter_Const$	As $Nag_Soln_Iter_Long$ with the Lagrange multipliers, the variables $x$ , the constraint values $A x$ and the constraint status also printed at each iteration.
$Nag_Soln_Iter_Full$	As $Nag_Soln_Iter_Const$ with the diagonal elements of the matrix $T$ associated with the $T Q$ factorization (see (4) in Section 11.2) of the working set, and the diagonal elements of the upper triangular matrix $R$ printed at each iteration.

Details of each level of results printout are described in Section 12.3.

Constraint:

options . print_level = Nag_NoPrint

Nag_Soln

Nag_Iter

Nag_Soln_Iter

Nag_Iter_Long

Nag_Soln_Iter_Long

Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

outfile – const char[512]

Default

= stdout

On entry: the name of the file to which results should be printed. If

options . outfile [0] =' \0'

then the stdout stream is used.

print_fun – pointer to function

Default

=

NULL

On entry: printing function defined by you; the prototype of

options . print_fun

void(*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);

See Section 12.3.1 below for further details.

fmax_iter – Integer

Default

= \max (50, 5 (n + nclin))

max_iter – Integer

Default

= \max (50, 5 (n + nclin))

On entry:

options . fmax_iter

and

options . max_iter

specify the maximum number of iterations allowed in the feasibility and optimality phase, respectively.

If you wish to check that a call to e04ncc is correct before attempting to solve the problem in full then

options . fmax_iter

may be set to

0

. No iterations will then be performed but all initialization prior to the first iteration will be done and a listing of argument settings will be output, if optional parameter

options . list = Nag_TRUE

(the default setting).

Constraints:

$options . fmax_iter \geq 0$ ;
$options . max_iter \geq 0$ .

crash_tol – double

Default

= 0.01

On entry:

options . crash_tol

is used when optional parameter

options . start = Nag_Cold

(the default) and e04ncc selects an initial working set. The initial working set will include (if possible) bounds or general inequality constraints that lie within

options . crash_tol

of their bounds. In particular, a constraint of the form

a_{j}^{T} x \geq l

will be included in the initial working set if

|a_{j}^{T} x - l| \leq options . crash_tol \times (1 + |l|)

Constraint:

0.0 \leq options . crash_tol \leq 1.0

ftol – double

Default

= \sqrt{ε}

On entry: defines the maximum acceptable absolute violation in each constraint at a ‘feasible’ point. For example, if the variables and the coefficients in the general constraints are of order unity, and the latter are correct to about 6 decimal digits, it would be appropriate to specify

options . ftol

10^{- 6}

e04ncc attempts to find a feasible solution before optimizing the objective function. If the sum of infeasibilities cannot be reduced to zero, e04ncc finds the minimum value of the sum. Let Sinf be the corresponding sum of infeasibilities. If Sinf is quite small, it may be appropriate to raise

options . ftol

by a factor of 10 or

100

. Otherwise, some error in the data should be suspected.

Note that a ‘feasible solution’ is a solution that satisfies the current constraints to within the feasibility tolerance

options . ftol

Constraint:

options . ftol > 0.0

inf_bound – double

Default

= 10^{20}

On entry:

options . inf_bound

defines the ‘infinite’ bound in the definition of the problem constraints. Any upper bound greater than or equal to

options . inf_bound

will be regarded as

+ \infty

(and similarly any lower bound less than or equal to

- options . inf_bound

will be regarded as

- \infty

Constraint:

options . inf_bound > 0.0

inf_step – double

Default

= \max (options . inf_bound, 10^{20})

On entry: specifies the magnitude of the change in variables that will be considered a step to an unbounded solution. (Note that an unbounded solution can occur only when the Hessian is singular and the objective contains an explicit linear term.) If the change in

x

during an iteration would exceed the value of

options . inf_step

, the objective function is considered to be unbounded below in the feasible region.

Constraint:

options . inf_step > 0.0

rank_tol – double

Default

= 100 ε

10 \sqrt{ε}

The default value is

100 ε

for problem types QP1, LS1 and LS3 but is

10 \sqrt{ε}

for other QP and LS problem types. This option does not apply to FP or LP problem types.

On entry:

options . rank_tol

enables you to control the estimate of the triangular factor

R_{1}

(see Section 11.3). If

ρ_{i}

denotes the function

ρ_{i} = \max \{|R_{11}|, |R_{22}|, \dots, |R_{i i}|\}

, the rank of

R

is defined to be smallest index

i

such that

|R_{i + 1, i + 1}| \leq options . rank_tol \times |ρ_{i + 1}|

Constraint:

0.0 < options . rank_tol < 1.0

state – Integer *

Default memory

= n + nclin

On entry:

options . state

need not be set if the default option of

options . start = Nag_Cold

is used as

n + nclin

values of memory will be automatically allocated by e04ncc.

If the option

options . start = Nag_Warm

has been chosen,

options . state

must point to a minimum of

n + nclin

elements of memory. This memory will already be available if the options structure has been used in a previous call to e04ncc from the calling program, with

options . start = Nag_Cold

and the same values of n and nclin. If a previous call has not been made sufficient memory must be allocated to

options . state

by you.

When a warm start is chosen

options . state

should specify the status of the constraints at the start of the feasibility phase. More precisely, the first

n

elements of

options . state

refer to the upper and lower bounds on the variables, and the next

n_{L}

elements refer to the general linear constraints (if any). Possible values for

options . state [j]

are as follows:

$options . state [j]$	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 should only be specified if $bl [j] = bu [j]$ .

The values

- 2

- 1

and 4 are also acceptable but will be reset to zero by the function, as will any elements which are set to 3 when the corresponding elements of bu and bl are not equal. If e04ncc has been called previously with the same values of n and nclin,

options . state

already contains satisfactory information. (See also the description of the optional parameter

options . start

.) The function also adjusts (if necessary) the values supplied in x to be consistent with the values supplied in

options . state

Constraint:

- 2 \leq options . state [j - 1] \leq 4

, for

j = 1, 2, \dots, 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

options . state [j]

is as follows:

$options . state [j]$	Meaning
$- 2$	The constraint violates its lower bound by more than the feasibility tolerance.
$- 1$	The constraint violates its upper bound by more than the feasibility tolerance.
$0$	The constraint is satisfied to within the feasibility tolerance, but is not in the working set.
$1$	This inequality constraint is included in the working set at its lower bound.
$2$	This inequality constraint is included in the working set at its upper bound.
$3$	This constraint is included in the working set as an equality. This value of $options . state$ can occur only when $bl [j] = bu [j]$ .
$4$	This corresponds to optimality being declared with $x [j]$ being temporarily fixed at its current value. This value of $options . state$ can only occur when $fail . code = NW_SOLN_NOT_UNIQUE$ .

ax – double *

Default memory

= nclin

On entry: nclin values of memory will be automatically allocated by e04ncc and this is the recommended method of use of

options . ax

. However you may supply memory from the calling program.

On exit: if

nclin > 0

options . ax

points to the final values of the linear constraints

A x

lambda – double *

Default memory

= n + nclin

On entry:

n + nclin

values of memory will be automatically allocated by e04ncc and this is the recommended method of use of

options . lambda

. However you may supply memory from the calling program.

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

n_{L}

elements contain the multipliers for the general linear constraints (if any). If

options . state [j - 1] = 0

(i.e., constraint

j

is not in the working set),

options . lambda [j - 1]

is zero. If

x

is optimal,

options . lambda [j - 1]

should be non-negative if

options . state [j - 1] = 1

, non-positive if

options . state [j - 1] = 2

and zero if

options . state [j - 1] = 4

hessian – Nag_Boolean

Default

= Nag_FALSE

On entry: controls the contents of the argument h on return from e04ncc. e04ncc works exclusively with the transformed and reordered matrix

H_{Q}

(8), and hence extra computation is required to form the Hessian itself. If the optional parameter

options . hessian = Nag_FALSE

, h contains the Cholesky factor of the matrix

H_{Q}

with columns ordered as indicated by kx (see Section 5). If

options . hessian = Nag_TRUE

, h contains the Cholesky factor of the Hessian matrix

\nabla^{2} F

, with columns ordered as indicated by kx.

iter – Integer

On exit: the total number of iterations performed in the feasibility phase and (if appropriate) the optimality phase.

12.3 Description of Printed Output

The level of printed output can be controlled with the structure members

options . list

and

options . print_level

(see Section 12.2). If

options . list = Nag_TRUE

then the argument values to e04ncc are listed, whereas the printout of results is governed by the value of

options . print_level

. The default of

options . print_level = Nag_Soln_Iter

provides a single line of output at each iteration and the final result. This section describes all of the possible levels of results printout available from e04ncc.

To aid interpretation of the printed results, the following convention is used for numbering the constraints: indices 1 to

n

refer to the bounds on the variables, and indices

n + 1

n + n_{L}

refer to the general constraints.

When

options . print_level = Nag_Iter

Nag_Soln_Iter

the following line of output is produced at every iteration. In all cases, the values of the quantities printed are those in effect on completion of the given iteration.

Itn	is the iteration count.
Step	is the step taken along the computed search direction. If a constraint is added during the current iteration, Step will be the step to the nearest constraint. During the optimality phase, the step can be greater than $1.0$ only if the factor $R_{Z}$ is singular (see Section 11.3).
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. 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 non-increasing. 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	$‖Z_{1}^{T} g_{FR}‖$ , the Euclidean norm of the reduced gradient with respect to $Z_{1}$ (see Section 11.3). During the optimality phase, this norm will be approximately zero after a unit step.

options . print_level = Nag_Iter_Long

Nag_Soln_Iter_Long

Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

the line of printout is extended to give the following additional information. (Note that this longer line extends over more than 80 characters.)

Jdel	is the index of the constraint deleted from the working set, along with the designation L (lower bound), U (upper bound), E (equality), F (temporarily fixed variable) or A (artificial constraint). If Jdel is zero, no constraint was deleted.
Jadd	is the index of the constraint added to the working set, along with a designation as for Jdel. If Jadd is zero, no constraint was added.
Bnd	is the number of simple bound constraints in the current working set.
Lin	is the number of general linear constraints in the current working set.
Art	is the number of artificial constraints in the working set, i.e., the number of columns of $Z_{2}$ (see Section 11.3).
Zr	is the number of columns of $Z_{1}$ (see Section 11.2). Zr is the dimension of the subspace in which the objective function is currently being minimized. The value of Zr is the number of variables minus the number of constraints in the working set; i.e., $Zr = n - (Bnd + Lin + Art)$ .
	The value of $n_{Z}$ , the number of columns of $Z$ (see Section 11) can be calculated as $n_{Z} = n - (Bnd + Lin)$ . A zero value of $n_{Z}$ implies that $x$ lies at a vertex of the feasible region.
Norm 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 $R_{1}$ (the first Zr rows and columns of the factor $R_{Z}$ ).

When

options . print_level = Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

more detailed results are given at each iteration. For the setting

options . print_level = Nag_Soln_Iter_Const

additional values output are:

Value of x	is the value of $x$ currently held in x.
State	is the current value of $options . state$ associated with $x$ .
Value of Ax	is the value of $A x$ currently held in $options . ax$ .
State	is the current value of $options . state$ associated with $A x$ .

Also printed are the Lagrange Multipliers for the bound constraints, linear constraints and artificial constraints.

options . print_level = Nag_Soln_Iter_Full

then the diagonals of

T

and

R

are also output at each iteration.

When

options . print_level = Nag_Soln

Nag_Soln_Iter

Nag_Soln_Iter_Long

Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

the final printout from e04ncc includes a listing of the status of every variable and constraint. The following describes the printout for each variable.

Varbl

gives the name (V) and index

j

, for

j = 1, 2, \dots, n

of the variable.

State

gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the optional parameter

options . ftol

(default value

\sqrt{ε}

, where

ε

is the machine precision; see Section 12.2), State will be ++ or -- respectively.

A key is sometimes printed before State to give some additional information about the state of a variable.

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, 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 $options . ftol$ .

Value

is the value of the variable at the final iteration.

Lower bound

is the lower bound specified for variable

j

. (None indicates that

bl [j - 1] \leq - options . inf_bound

, where

options . inf_bound

is the optional parameter.)

Upper bound

is the upper bound specified for variable

j

. (None indicates that

bu [j - 1] \geq options . inf_bound

, where

options . inf_bound

is the optional parameter.)

Lagr mult

is the value of the Lagrange multiplier for the associated bound. This will be zero if State is FR unless

bl [j - 1] \leq - options . inf_bound

and

bu [j - 1] \geq options . inf_bound

, 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.

Residual

is the difference between the variable Value and the nearer of its (finite) bounds

bl [j - 1]

and

bu [j - 1]

. A blank entry indicates that the associated variable is not bounded (i.e.,

bl [j - 1] \leq - options . inf_bound

and

bu [j - 1] \geq options . inf_bound

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]

replaced by

bl [n + j - 1]

and

bu [n + j - 1]

respectively, and with the following change in the heading:

L Con

the name (L) and index

j

, for

j = 1, 2, \dots, n_{L}

of the linear constraint.

Note that movement off a constraint (as opposed to a variable moving away from its bound) can be interpreted as allowing the entry in the 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.

options . print_level = Nag_NoPrint

then printout will be suppressed; you can print the final solution when e04ncc returns to the calling program.

12.3.1 Output of results via a user-defined printing function

You may also specify your own print function for output of iteration results and the final solution by use of the

options . print_fun

function pointer, which has prototype

void (*print_fun)(const Nag_Search_State *st, Nag_Comm *comm)

The rest of this section can be skipped if you wish to use the default printing facilities.

When a user-defined function is assigned to

options . print_fun

this will be called in preference to the internal print function of e04ncc. Calls to the user-defined function are again controlled by means of the

options . print_level

member. Information is provided through st and comm, the two structure arguments to

options . print_fun

comm \to it_prt = Nag_TRUE

then the results from the last iteration of e04ncc are provided through st. Note that

options . print_fun

will be called with

comm \to it_prt = Nag_TRUE

only if

options . print_level = Nag_Iter

Nag_Iter_Long

Nag_Soln_Iter

Nag_Soln_Iter_Long

Nag_Soln_Iter_Const

Nag_Soln_Iter_Full

. The following members of st are set:

n – Integer: The number of variables.

nclin – Integer: The number of linear constraints.

iter – Integer: The iteration count.

jdel – Integer: Index of constraint deleted from the working set.

jadd – Integer: Index of constraint added to the working set.

step – double: The step taken along the computed search direction.

ninf – Integer: The number of violated constraints (infeasibilities).

f – double: The current value of the objective function if $st \to ninf = 0$ ; otherwise, $st \to f$ is a weighted sum of the magnitudes of constraint violations.

bnd – Integer: Number of bound constraints in the working set.

lin – Integer: Number of general linear constraints in the working set.

nart – Integer: Number of artificial constraints in the working set (see Section 11.3).

nrank – Integer: The rank of the upper triangular matrix $R$ (see Section 11.3).

nrz – Integer: Number of columns of $Z_{1}$ (see Section 11.2).

norm_gz – double: Euclidean norm of the reduced gradient, $‖Z_{1}^{T} g_{FR}‖$ (see Section 11.3).

norm_gf – double: Euclidean norm of the gradient function with respect to the free variables.

cond_t – double: A lower bound on the condition number of the working set.

cond_r – double: A lower bound on the condition number of the triangular factor $R_{1}$ (see Section 11.3).

x – double *: The components $st \to x [j - 1]$ of the current point $x$ , for $j = 1, 2, \dots, st \to n$ .

ax – double *: If $st \to nclin > 0$ , the $st \to nclin$ components of the linear constraints $A x$ .

state – Integer *: $options . state$ contains the status of the $st \to n$ variables and $st \to nclin$ general linear constraints. See Section 12.2 for a description of the possible status values.

diagt – double *: If $st \to lin > 0$ , the $st \to lin$ elements in the diagonal of the matrix $T$ .

diagr – double *: If $st \to nrank > 0$ , the first $st \to nrank$ elements of the diagonal of the upper triangular matrix $R$ .

comm \to new_lm = Nag_TRUE

then the Lagrange multipliers have been updated and the following members of st are set:

bnd – Integer: The number of bound constraints in the working set.

kx – Integer *
bclambda – double *: Indices of the bound constraints in the working set, with associated multipliers. $st \to kx [i]$ is the index of the constraint with multiplier $st \to bclambda [i]$ , for $i = 0, 1, \dots, st \to bnd - 1$ .

lin – Integer: The number of linear constraints in the working set.

kactive – Integer *
lambda – double *: Indices of the linear constraints in the working set, with associated multipliers. $st \to kactive [i]$ is the index of the constraint with multiplier $st \to lambda [st \to bnd + i]$ , for $i = 0, 1, \dots, st \to lin - 1$ .

nart – Integer: The number of artificial constraints in the working set (see Section 11.3).

gq – double *: $st \to gq [i]$ , for $i = 0, 1, \dots, st \to nart - 1$ , hold the multipliers for the artificial constraints.

comm \to sol_prt = Nag_TRUE

then the final result from e04ncc is available and the following members of st are set:

n – Integer: The number of variables.

nclin – Integer: The number of linear constraints.

iter – Integer: The iteration count.

x – double *: The components $st \to x [j - 1]$ of the final point $x$ , for $j = 1, 2, \dots, st \to n$ .

feasible – Nag_Boolean: Will be Nag_TRUE if the final point is feasible.

f – double: The final value of the objective function if $st \to feasible$ is Nag_TRUE; otherwise, the sum of infeasibilities. If the problem is of type FP and $x$ is feasible then $st \to f$ is set to zero.

ax – double *: If $st \to nclin > 0$ , the $st \to nclin$ components of the final linear constraint activities, $A x$ .

state – Integer *: Contains the final status of the $st \to n$ variables and $st \to nclin$ general linear constraints. See Section 12.2 for a description of the possible status values.

lambda – double *: Contains the $st \to n + st \to nclin$ final values of the Lagrange multipliers.

bl – double *: Contains the $st \to n + st \to nclin$ lower bounds.

bu – double *: Contains the $st \to n + st \to nclin$ upper bounds.

endstate – Nag_EndState

The state of termination of e04ncc. Possible values of

st \to endstate

and their correspondence to the exit value of fail are:

Value of $st \to endstate$	Value of fail
$Nag_Feasible$ or $Nag_Optimal$	NE_NOERROR
$Nag_Weakmin$	NW_SOLN_NOT_UNIQUE
$Nag_Unbounded$	NE_UNBOUNDED
$Nag_Infeasible$	NW_NOT_FEASIBLE
Nag_Too_Many_Iter	NW_TOO_MANY_ITER
$Nag_Cycling$	NE_CYCLING

The relevant members of the structure comm are:

it_prt – Nag_Boolean: Will be Nag_TRUE when the print function is called with the result of the current iteration.

sol_prt – Nag_Boolean: Will be Nag_TRUE when the print function is called with the final result.

new_lm – Nag_Boolean: Will be Nag_TRUE when the Lagrange multipliers have been updated.

user – double *
iuser – Integer *
p – Pointer: Pointers for communication of user information. If used they must be allocated memory either before entry to e04ncc or during a call to $options . print_fun$ . The type Pointer will be void * with a C compiler that defines void * and char * otherwise.