Solves linear least-squares and convex quadratic programming problems (non-sparse)

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NAG[e04ncc] NAG[nag_opt_lin_lsq] - Solves linear least-squares and convex quadratic programming problems (non-sparse)

Calling Sequence

e04ncc(a, bl, bu, kx, x, objf, 'm'=m, 'n'=n, 'nclin'=nclin, 'tda'=tda, 'cvec'=cvec, 'b'=b, 'h'=h, 'tdh'=tdh, 'optional_settings'=optional_settings, 'comm'=comm, 'fail'=fail)

nag_opt_lin_lsq(. . .)

Parameters

a - Matrix(1..nclin, 1..tda, datatype=float[8], order=C_order);

On entry: the th row of a must contain the coefficients of the th general linear constraint (the th row of ), for . If then the array a is not referenced.

bl - Vector(1.., datatype=float[8]);
bu - Vector(1.., datatype=float[8]);

On entry: bl must contain the lower bounds and bu the upper bounds, for all the constraints in the following order. The first elements of each array must contain the bounds on the variables, and the next elements the bounds for the general linear constraints (if any). To specify a non-existent lower bound (i.e., ), set , and to specify a non-existent upper bound (i.e., ), set , where optional_settings[inf_bound] is one of the optional arguments (default value (see Section [The optional-settings Parameter]). To specify the th constraint as an equality, set , say, where .

Constraints:

, for ;

if , .

kx - Vector(1..n, datatype=integer[kernelopts('wordsize')/8]);

On entry: for problems of type QP3, QP4, LS3 or LS4 the array kx must specify the order of the columns of the matrix with respect to the ordering of x. Thus if column of is the column associated with the variable then .

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

Constraints:

, for ;

if , .

On exit: defines the order of the columns of with respect to the ordering of x, as described above.

x - Vector(1..n, datatype=float[8]);

On entry: an initial estimate of the solution.

On exit: the point at which nag_opt_lin_lsq (e04ncc) terminated. If no error is raised, NW_SOLN_NOT_UNIQUE or NW_NOT_FEASIBLE, x contains an estimate of the solution.

objf - assignable;

Note: On exit the variable objf will have a value of type float.

On exit: the value of the objective function at if is feasible, or the sum of infeasibilities at otherwise. If the problem is of type FP and is feasible, objf is set to zero.

'm'=m - integer; (optional)

Default value: the first dimension of the arrays b, h.

On entry: , the number of rows in the matrix . 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 argument optional_settings[prob], see Section [The optional-settings Parameter].

If the problem is of type QP, m will usually be , the number of variables. However, a value of m less than is appropriate for problem type QP3 or QP4 if is an upper trapezoidal matrix with rows. Similarly, m may be used to define the dimension of a leading block of non-zeros in the Hessian matrices of QP1 or QP2. In QP cases, should not be greater than ; if it is, the last rows of 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 (, and ) are allowed in this case.

Constraint: if problem is not FP or LP. .

'n'=n - integer; (optional)

Default value: the first dimension of the arrays cvec, kx, x and the second dimension of the arrays cvec, kx, xthe arrays a, h.

On entry: , the number of variables.

Constraint: . .

'nclin'=nclin - integer; (optional)

Default value: the first dimension of the array a.

On entry: , the number of general linear constraints.

Constraint: . .

'tda'=tda - integer; (optional)

On entry: the second dimension of the array a as declared in the function from which nag_opt_lin_lsq (e04ncc) is called.

Constraint: if . .

'cvec'=cvec - Vector(1..n, datatype=float[8]); (optional)

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 set to the null pointer.

'b'=b - Vector(1..m, datatype=float[8]); (optional)

On entry: the 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 may be set to the null pointer.

'h'=h - Matrix(1..m, 1..tdh, datatype=float[8], order=C_order); (optional)

On entry: the array h must contain the matrix as specified in Table 1 (see Section [Description]).

For problems QP1 and QP2, the first rows and columns of h must contain the leading by rows and columns of the symmetric Hessian matrix. Only the diagonal and upper triangular elements of the leading 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 rows of h must contain an by 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 non-singular sub-matrix of , the fewer iterations will be required. Elements outside the upper trapezoidal part of the first rows of 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, the user should transform the original into a triangular matrix before the call to nag_opt_lin_lsq (e04ncc) and solve as type LS3 or LS4.

On exit: by default, h contains the upper triangular Cholesky factor of equation (8), with columns ordered as indicated by kx (see below). If the optional argument (see Section [The optional-settings Parameter]), and the problem is one of the LS or QP types, h contains the upper triangular Cholesky factor of the Hessian matrix , 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 set to the null pointer.

'tdh'=tdh - integer; (optional)

On entry: the second dimension of the array h as declared in the function from which nag_opt_lin_lsq (e04ncc) is called.

Constraint: . .

'optional_settings'=optional_settings - Vector; (optional)

On entry/exit: A Maple Vector, which should be generated using NAG[Nag_E04_Opt], corresponding to the Nag_E04_Opt structure. This offers the means of adjusting some of the parameter values of the algorithm. A description of the members of optional_settings is given below in Section [The optional_settings Parameter]. Further information and examples on setting and using vectors of this type are available, see the documentation for NAG[SetOptions], NAG[GetOptions], NAG[FreeOptions] and NAG[Nag_E04_Opt].

'comm'=comm - table; (optional)

A Maple table, which should be generated using NAG[Nag_Comm], corresponding to the Nag_Comm structure.

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

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

Description

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

(1)

( ) F(x) l<={[[x],[Ax]]}<=u

where is an by matrix and the objective function may be specified in a variety of ways depending upon the particular problem to be solved. The available forms for 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, is an element vector, is an element vector, and denotes the Euclidean length of .

Problem Type		Matrix
FP	Not applicable	Not applicable
LP		Not applicable
QP1		by symmetric positive semi-definite
QP2		by symmetric positive semi-definite
QP3		by upper trapezoidal
QP4		by upper trapezoidal
LS1		by
LS2		by
LS3		by upper trapezoidal
LS4		by upper trapezoidal

Table 1

For problems of type LS, is referred to as the least-squares matrix, or the matrix of observations, and as the vector of observations. The default problem type is LS1, and other objective functions are selected by using the optional argument optional_settings[prob] (see Section [The optional-settings Parameter]).

When is upper trapezoidal it will usually be the case that , so that 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 factorization, has been performed.

The constraints involving 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 . If certain bounds are not present, the associated elements of or can be set to special values that will be treated as or . (See the description of the optional argument optional_settings[inf_bound] in Section [The optional-settings Parameter].

The function is a quadratic function, whose defining feature is that its second-derivative matrix (the Hessian matrix) is constant. For the LP case, ; for QP1 and QP2, ; and for QP3, QP4 and LS problems, and the Hessian matrix is positive semi-definite (positive definite if is full rank), so that is convex. If is defined as the zero matrix, nag_opt_lin_lsq (e04ncc) will solve the resulting linear programming problem; however, this can be accomplished more efficiently by using e04mfc (nag_opt_lp).

Problems of type QP3 and QP4 for which is not in upper trapezoidal form should be solved as problems of type LS1 and LS2 respectively, with .

The user must supply an initial estimate of the solution.

If is of full rank then nag_opt_lin_lsq (e04ncc) will obtain the unique (global) minimum. If is not of full rank then the solution may still be a global minimum if all active constraints have non-zero 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 ), or else the objective function is unbounded below in the feasible region. The last case can only occur when contains an explicit linear term (as in problems LP, QP2, QP4, LS2 and LS4).

The method used by nag_opt_lin_lsq (e04ncc) is described in detail in Section [Further Description].

Error Indicators and Warnings

"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 .

"NE_BAD_PARAM"

On entry, argument optional_settings[print_level] had an illegal value.

"NE_B_NULL"

but argument NULL.

"NE_BOUND"

The lower bound for variable (array element ) is greater than the upper bound.

"NE_BOUND_LCON"

The lower bound for linear constraint (array element ) is greater than the upper bound.

"NE_CVEC_NULL"

but argument NULL.

"NE_CYCLING"

The algorithm could be cycling, since a total of 50 changes were made to the working set without altering . 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. (nag_opt_lin_lsq (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, the user has 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 argument optional_settings[ftol] (default value , where is the machine precision; see Section [The optional-settings Parameter]). If this error exit occurs but no suspicious pattern of constraint changes can be observed, it may be worthwhile to restart with the final (with optional argument "Nag_Cold" or "Nag_Warm").

"NE_H_NULL_QP"

but argument NULL. This problem type requires an array to be supplied in argument h.

"NE_INT_ARG_LT"

On entry, m must not be less than 1: .

"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 consult NAG for assistance.

"NE_INVALID_INT_RANGE_1"

Value given to optional_settings[max_iter] is not valid. Correct range is .

"NE_INVALID_REAL_RANGE_F"

Value given to optional_settings[ftol] is not valid. Correct range is .

"NE_INVALID_REAL_RANGE_FF"

Value given to optional_settings[crash_tol] is not valid. Correct range is .

"NE_NOT_APPEND_FILE"

Cannot open file for appending.

"NE_NOT_CLOSE_FILE"

Cannot close file .

"NE_OPT_NOT_INIT"

Options structure not initialized.

"NE_STATE_VAL"

is out of range. .

"NE_UNBOUNDED"

Solution appears to be unbounded.

This error indicator implies that a step as large as optional argument optional_settings[inf_step] (default value ; see Section [The optional-settings Parameter]) would have to be taken in order to continue the algorithm. This situation can occur only when is singular, there is an explicit linear term, and at least one variable has no upper or lower bound.

"NE_WARM_START"

but pointer NULL.

"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 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 [Description of Printed Output]) at the final point. The modified problem (with an altered value of the optional feasibility tolerance, optional_settings[ftol]) may then be solved using optional argument (see Section [The optional-settings Parameter]). The user should check that there are no constraint redundancies. If the data for the constraints are accurate only to the absolute precision , the user should ensure that the value of optional_settings[ftol] is greater than . For example, if all elements of are of order unity and are accurate only to three decimal places, optional_settings[ftol] should be at least .

"NW_OVERFLOW_WARN"

Serious ill conditioning in the working set after adding constraint . 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 argument optional_settings[ftol] and re-running the program. If the message recurs even after this change, the offending linearly dependent constraint 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 (see Section [The Main Iteration]) is singular or there is a small multiplier. This means that is not unique.

	Accuracy
	nag_opt_lin_lsq (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.

Further Comments

	Termination Criteria
	nag_opt_lin_lsq (e04ncc) exits with no error is raised if is a strong local minimizer, i.e., the reduced gradient is negligible, the Lagrange multipliers are optimal (see Section [Description of Printed Output]) and (see Section [The Main Iteration]) is non-singular.

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.

Further Description

This section gives a detailed description of the algorithm used in nag_opt_lin_lsq (e04ncc). This, and possibly the next section, Section [Optional Arguments], may be omitted if the more sophisticated features of the algorithm and software are not currently of interest.

Overview

nag_opt_lin_lsq (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 summarize the main features of the method.

nag_opt_lin_lsq (e04ncc) uses essentially the same algorithm as the subroutine LSSOL described in Gill et al. (1986a). 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. (1984b). (In the full-rank case, the method is related to that of Stoer, see Stoer (1971).) nag_opt_lin_lsq (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 . Once any iterate is feasible, all subsequent iterates remain feasible.

nag_opt_lin_lsq (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 (nag_opt_nlp)). In particular, the user 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 argument optional_settings[start] in Section [The optional-settings Parameter].

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 is defined by

(2)

where the step length is a non-negative scalar, and is called the search direction.

At each point , 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 argument optional_settings[ftol]; see Section [The optional-settings Parameter]). 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 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 corresponding to fixed variables may be ignored.

Let denote the number of general constraints in the working set and let denote the number of variables fixed at one of their bounds ( and are the quantities Lin and Bnd in the extended iteration printout from nag_opt_lin_lsq (e04ncc); see Section [Description of Printed Output]). Similarly, let denote the number of free variables. At every iteration, the variables are re-ordered so that the last 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 [Parameters]).

Definition of the Search Direction

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

(3)

In order to compute , the factorization of is used:

(4)

where is a non-singular by reverse-triangular matrix (i.e., if ), and the non-singular by matrix is the product of orthogonal transformations (see Gill et al. (1984b)). If the columns of are partitioned so that

(5)

where is by , then the columns of form a basis for the null space of . Let be an integer such that , and let denote a matrix whose columns are a subset of the columns of . (The integer is the quantity Zr in the extended iteration printout from nag_opt_lin_lsq (e04ncc); see Section [Description of Printed Output]. In many cases, will include all the columns of .) The direction will satisfy (3) if

(6)

where is any -vector.

The Main Iteration

Let denote the by matrix

(7)

Q = binomial(Q[FR], I[FX])

where is the identity matrix of order . Let denote an by upper triangular matrix (the Cholesky factor) such that

(8)

and let the matrix of the first rows and columns of be denoted by . (The matrix in (8) is the Hessian with its rows and columns permuted so that the free variables come first.)

The definition of in (6) depends on whether or not the matrix is singular at . In the non-singular case, satisfies the equations

(9)

where denotes the vector and denotes the objective gradient. (The norm of is the printed quantity Norm Gf; see Section [Description of Printed Output].) When is defined by (9), 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 is available such that , which allows to be computed from a single back-substitution . For example, when solving problem LS1, comprises the first elements of the transformed residual vector

(10)

which is recurred from one iteration to the next, where is an orthogonal matrix.

In the singular case, is defined such that

(11)

This vector has the property that the objective function is linear along and may be reduced by any step of the form , where .

The vector is known as the projected gradient at . If the projected gradient is zero, is a constrained stationary point in the subspace defined by . 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 minimizes the quadratic objective when the constraints in the working set are treated as equalities. At a constrained stationary point, Lagrange multipliers and for the general and bound constraints are defined from the equations

(12)

Given a positive constant of the order of the machine precision, the Lagrange multiplier corresponding to an inequality constraint in the working set is said to be optimal if when the associated constraint is at its upper bound, or if 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 [Description of Printed Output]) from the working set.

If optimal multipliers occur during the feasibility phase and the sum of infeasibilities is non-zero, there is no feasible point, and nag_opt_lin_lsq (e04ncc) will continue until the minimum value of the sum of infeasibilities has been found. At this point, the Lagrange multiplier corresponding to an inequality constraint in the working set will be such that when the associated constraint is at its upper bound, and when the associated constraint is at its lower bound. Lagrange multipliers for equality constraints will satisfy .

The choice of step length is based on remaining feasible with respect to the satisfied constraints. If is non-singular and is feasible, will be taken as unity. In this case, the projected gradient at will be zero, and Lagrange multipliers are computed. Otherwise, is set to , the step to the "nearest" constraint (with index Jadd; see Section [Description of Printed Output]), which is added to the working set at the next iteration.

If is not input as a triangular matrix, it is overwritten by a triangular matrix satisfying (8) obtained using the Cholesky factorization in the QP case, or the 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 are eliminated from the problem.

Each change in the working set leads to a simple change to : if the status of a general constraint changes, a row of is altered; if a bound constraint enters or leaves the working set, a column of changes. Explicit representations are recurred of the matrices and ; and of vectors , and , which are related by the formulae

f=Pb-([[R],[0]])Q^Tx,(b,&equiv;,0, )

and

Note that the triangular factor 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 nag_opt_lin_lsq (e04ncc): if a given factor is non-singular, 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 , where is the matrix with a unit last diagonal, and is a vector of all zeros except in the last position. If the Hessian matrix is singular, the matrix (and hence ) may be singular at the start of the optimality phase. However, will be non-singular 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 contains constraints.) The idea is to include as many general constraints as necessary to ensure a non-singular .

At the beginning of each phase, an upper triangular matrix is determined that is the largest non-singular leading sub-matrix of . The use of interchanges during the factorization of tends to maximize the dimension of . (The rank of is estimated using the optional argument optional_settings[rank_tol]; see Section [The optional-settings Parameter].) Let denote the columns of corresponding to , and let be partitioned as . A working set for which defines the null space can be obtained by including the rows of as "artificial constraints". Minimization of the objective function then proceeds within the subspace defined by .

The artificially augmented working set is given by

(13)

so that will satisfy and . By definition of the factorization, automatically satisfies the following:

where

and hence the factorization of (13) requires no additional work.

The matrix need not be kept fixed, since its role is purely to define an appropriate null space; the factorization can therefore be updated in the normal fashion as the iterations proceed. No work is required to "delete" the artificial constraints associated with when , since this simply involves repartitioning . When deciding which constraint to delete, the "artificial" multiplier vector associated with the rows of is equal to , 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 and , the Euclidean norm of , and the condition estimator of appear in the extended iteration printout as Art, Zr, Norm Gz and Cond Rz respectively (see Section [Description of Printed Output]).

Although the algorithm of nag_opt_lin_lsq (e04ncc) does not perform simplex steps in general, there is one exception: a linear program with fewer general constraints than variables (i.e., ). (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 nag_opt_lin_lsq (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 factor (the printed value Cond T; see Section [Description of Printed Output]). In constructing the initial working set, constraints are excluded that would result in a large value of Cond T. Thereafter, nag_opt_lin_lsq (e04ncc) allows constraints to be violated by as much as a user-specified feasibility tolerance (see optional_settings[ftol], Section [The optional-settings Parameter]) in order to provide, whenever possible, a choice of constraints to be added to the working set at a given iteration. Let denote the maximum step at which does not violate any constraint by more than its feasibility tolerance. All constraints at distance along 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.

The optional_settings Parameter

Further information and examples on setting and using vectors of this type are available, see the documentation for NAG[SetOptions], NAG[GetOptions], NAG[FreeOptions] and NAG[Nag_E04_Opt].

prob - String;

Default

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

"Nag_QP1" symmetric, b and cvec not referenced;

"Nag_QP2" symmetric, b not referenced, ;

"Nag_QP3" upper trapezoidal, b and cvec not referenced;

"Nag_QP4" upper trapezoidal, b not referenced, .

"Nag_LS1" , , cvec not referenced;

"Nag_LS2" , , ;

"Nag_LS3" upper trapezoidal, , cvec not referenced;

"Nag_LS4" upper trapezoidal, , .

The array must be supplied for all problem types but need only be initialized for types "Nag_QP3", "Nag_QP4", "Nag_LS3" and "Nag_LS4". If , i.e., the objective function is purely linear, the efficiency of nag_opt_lin_lsq (e04ncc) may be increased by specifying .

Constraint: "Nag_FP", "Nag_LP", "Nag_QP1", "Nag_QP2", "Nag_QP3", "Nag_QP4", "Nag_LS1", "Nag_LS2", "Nag_LS3" or "Nag_LS4". .

start - String;

Default

On entry: specifies how the initial working set is chosen. With , nag_opt_lin_lsq (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 argument optional_settings[crash_tol]; see below).

With , the user must provide a valid definition of every array element of the optional argument optional_settings[state] (see below). nag_opt_lin_lsq (e04ncc) will override the user's specification of optional_settings[state] if necessary, so that a poor choice of the working set will not cause a fatal error. For instance, any elements of optional_settings[state] which are set to , 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 nag_opt_lin_lsq (e04ncc) is called repeatedly to solve related problems.

Constraint: "Nag_Cold" or "Nag_Warm". .

list - boolean;

Default

On entry: if the argument settings in the call to nag_opt_lin_lsq (e04ncc) will be printed.

print_level - String;

Default

On entry: the level of results printout produced by nag_opt_lin_lsq (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 , the constraint values 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 associated with the factorization (see (4) in Section [Definition of the Search Direction]) of the working set, and the diagonal elements of the upper triangular matrix printed at each iteration.

Details of each level of results printout are described in Section [Description of Printed Output].

Constraint: "Nag_NoPrint", "Nag_Soln", "Nag_Iter", "Nag_Soln_Iter", "Nag_Iter_Long", "Nag_Soln_Iter_Long", "Nag_Soln_Iter_Const" or "Nag_Soln_Iter_Full". .

outfile - Vector(datatype=string);

On entry: The name of a file to which intermediate or diagnostic output should be appended. If a value is not provided for this parameter then the behaviour of this routine is platform dependent. Usually all output will be suppressed, however on some platforms output will be produced and will be displayed in the Maple session.

fmax_iter - integer;
max_iter - integer;

Default

On entry: optional_settings[fmax_iter] and optional_settings[max_iter] specify the maximum number of iterations allowed in the feasibility and optimality phase, respectively.

If the user wishes to check that a call to nag_opt_lin_lsq (e04ncc) is correct before attempting to solve the problem in full then optional_settings[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 argument (the default setting).

Constraints:

;

crash_tol - Vector(datatype=float[8]);

Default

On entry: crash_tol is used when optional argument (the default) and nag_opt_lin_lsq (e04ncc) selects an initial working set. The initial working set will include (if possible) bounds or general inequality constraints that lie within crash_tol of their bounds. In particular, a constraint of the form will be included in the initial working set if .

Constraint: . .

ftol - Vector(datatype=float[8]);

Default

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 ftol as .

nag_opt_lin_lsq (e04ncc) attempts to find a feasible solution before optimizing the objective function. If the sum of infeasibilities cannot be reduced to zero, nag_opt_lin_lsq (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 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 ftol.

Constraint: . .

inf_bound - Vector(datatype=float[8]);

Default

On entry: inf_bound defines the "infinite" bound in the definition of the problem constraints. Any upper bound greater than or equal to inf_bound will be regarded as plus infinity (and similarly any lower bound less than or equal to will be regarded as minus infinity).

Constraint: . .

inf_step - Vector(datatype=float[8]);

Default

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 during an iteration would exceed the value of inf_step, the objective function is considered to be unbounded below in the feasible region.

Constraint: . .

rank_tol - Vector(datatype=float[8]);

Default or

The default value is for problem types QP1, LS1 and LS3 but is for other QP and LS problem types. This option does not apply to FP or LP problem types.

On entry: rank_tol enables the user to control the estimate of the triangular factor (see Section [The Main Iteration]). If denotes the function , the rank of is defined to be smallest index such that .

Constraint: . .

state - assignable;

Default memory

Note: On exit the variable state will have a value of type integer.

On entry: state need not be set if the default option of is used as values of memory will be automatically allocated by nag_opt_lin_lsq (e04ncc).

If the option has been chosen, state must point to a minimum of elements of memory. This memory will already be available if the optional_settings structure has been used in a previous call to nag_opt_lin_lsq (e04ncc) from the calling program, with and the same values of n and nclin. If a previous call has not been made sufficient memory must be allocated to state by the user.

When a warm start is chosen state should specify the status of the constraints at the start of the feasibility phase. More precisely, the first elements of state refer to the upper and lower bounds on the variables, and the next elements refer to the general linear constraints (if any). Possible values for are as follows:

Meaning

The constraint should not be in the initial working set.

The constraint should be in the initial working set at its lower bound.

The constraint should be in the initial working set at its upper bound.

The constraint should be in the initial working set as an equality. This value should only be specified if .

The values , 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 nag_opt_lin_lsq (e04ncc) has been called previously with the same values of n and nclin, state already contains satisfactory information. (See also the description of the optional argument optional_settings[start].) The function also adjusts (if necessary) the values supplied in x to be consistent with the values supplied in state.

Constraint: , for . .

On exit: the status of the constraints in the working set at the point returned in x. The significance of each possible value of is as follows:

Meaning

The constraint violates its lower bound by more than the feasibility tolerance.

The constraint violates its upper bound by more than the feasibility tolerance.

The constraint is satisfied to within the feasibility tolerance, but is not in the working set.

This inequality constraint is included in the working set at its lower bound.

This inequality constraint is included in the working set at its upper bound.

This constraint is included in the working set as an equality. This value of state can occur only when .

This corresponds to optimality being declared with being temporarily fixed at its current value. This value of state can only occur when the error "NW_SOLN_NOT_UNIQUE" is raised.

ax - assignable;

Default memory

Note: On exit the variable ax will have a value of type float.

On entry: nclin values of memory will be automatically allocated by nag_opt_lin_lsq (e04ncc) and this is the recommended method of use of ax. However a user may supply memory from the calling program.

On exit: if , ax points to the final values of the linear constraints .

lambda - assignable;

Default memory

Note: On exit the variable lambda will have a value of type float.

On entry: values of memory will be automatically allocated by nag_opt_lin_lsq (e04ncc) and this is the recommended method of use of lambda. However a user 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 elements contain the multipliers for the bound constraints on the variables, and the next elements contain the multipliers for the general linear constraints (if any). If (i.e., constraint is not in the working set), is zero. If is optimal, should be non-negative if , non-positive if and zero if .

hessian - boolean;

Default

On entry: controls the contents of the argument h on return from nag_opt_lin_lsq (e04ncc). nag_opt_lin_lsq (e04ncc) works exclusively with the transformed and reordered matrix (8), and hence extra computation is required to form the Hessian itself. If the optional argument , h contains the Cholesky factor of the matrix with columns ordered as indicated by kx (see Section [Parameters]). If , h contains the Cholesky factor of the Hessian matrix , with columns ordered as indicated by kx.

iter - assignable;

Note: On exit the variable iter will have a value of type integer.

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

Description of Printed Output

The level of printed output can be controlled by the user with the structure members optional_settings[list] and optional_settings[print_level] (see Section [The optional-settings Parameter]). If then the argument values to nag_opt_lin_lsq (e04ncc) are listed, whereas the printout of results is governed by the value of optional_settings[print_level]. The default of 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 nag_opt_lin_lsq (e04ncc).

To aid interpretation of the printed results, the following convention is used for numbering the constraints: indices 1 to refer to the bounds on the variables, and indices to refer to the general constraints.

When "Nag_Iter" or "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 only if the factor is singular (see Section [The 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 is not feasible, Sinf gives a weighted sum of the magnitudes of constraint violations. If 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 , the Euclidean norm of the reduced gradient with respect to (see Section [The Main Iteration]). During the optimality phase, this norm will be approximately zero after a unit step.

If "Nag_Iter_Long", "Nag_Soln_Iter_Long", "Nag_Soln_Iter_Const" or "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 (see Section [The Main Iteration]).

Zr is the number of columns of (see Section [Definition of the 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., .

The value of , the number of columns of (see Section [Further Description]) can be calculated as . A zero value of implies that 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 (the first Zr rows and columns of the factor ).

When "Nag_Soln_Iter_Const" or "Nag_Soln_Iter_Full" more detailed results are given at each iteration. For the setting additional values output are:

Value of x is the value of currently held in x.

State is the current value of optional_settings[state] associated with .

Value of Ax is the value of currently held in optional_settings[ax].

State is the current value of optional_settings[state] associated with .

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

If then the diagonals of and are also output at each iteration.

When "Nag_Soln", "Nag_Soln_Iter", "Nag_Soln_Iter_Long", "Nag_Soln_Iter_Const" or "Nag_Soln_Iter_Full" the final printout from nag_opt_lin_lsq (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 , for 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 argument optional_settings[ftol] (default value , where is the machine precision; see Section [The optional-settings Parameter]), 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 optional_settings[ftol].

Value is the value of the variable at the final iteration.

Lower bound is the lower bound specified for variable . (None indicates that , where optional_settings[inf_bound] is the optional argument.)

Upper bound is the upper bound specified for variable . (None indicates that , where optional_settings[inf_bound] is the optional argument.)

Lagr mult is the value of the Lagrange multiplier for the associated bound. This will be zero if State is FR unless and , in which case the entry will be blank. If 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 and . A blank entry indicates that the associated variable is not bounded (i.e., and ).

The meaning of the printout for general constraints is the same as that given above for variables, with "variable" replaced by "constraint", and replaced by and respectively, and with the following change in the heading:

L Con the name (L) and index , for 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.

If then printout will be suppressed; the user can print the final solution when nag_opt_lin_lsq (e04ncc) returns to the calling program.

Output of results via a user-defined printing function

The rest of this section can be skipped by a user who only wishes to use the default printing facilities.

When a user-defined function is assigned to optional_settings[print_fun] this will be called in preference to the internal print function of nag_opt_lin_lsq (e04ncc). Calls to the user-defined function are again controlled by means of the optional_settings[print_level] member. Information is provided through st and comm, the two structure arguments to optional_settings[print_fun].

If then the results from the last iteration of nag_opt_lin_lsq (e04ncc) are provided through st. Note that optional_settings[print_fun] will be called with only if "Nag_Iter", "Nag_Iter_Long", "Nag_Soln_Iter", "Nag_Soln_Iter_Long", "Nag_Soln_Iter_Const" or "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 - float

The step taken along the computed search direction.

ninf - integer

The number of violated constraints (infeasibilities).

f - float

The current value of the objective function if ; otherwise, 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 [The Main Iteration]).

nrank - integer

The rank of the upper triangular matrix (see Section [The Main Iteration]).

nrz - integer

Number of columns of (see Section [Definition of the Search Direction]).

norm_gz - float

Euclidean norm of the reduced gradient, (see Section [The Main Iteration]).

norm_gf - float

Euclidean norm of the gradient function with respect to the free variables.

cond_t - float

A lower bound on the condition number of the working set.

cond_r - float

A lower bound on the condition number of the triangular factor (see Section [The Main Iteration]).

x - Vector(datatype=float[8])

The components of the current point , .

ax - Vector(datatype=float[8])

If , the components of the linear constraints .

state - integer

optional_settings[state] contains the status of the variables and general linear constraints. See Section [The optional-settings Parameter] for a description of the possible status values.

diagt - Vector(datatype=float[8])

If , the elements in the diagonal of the matrix .

diagr - Vector(datatype=float[8])

If , the first elements of the diagonal of the upper triangular matrix .

If 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 - Vector(datatype=integer[kernelopts('wordsize')/8])

bclambda - Vector(datatype=float[8])

Indices of the bound constraints in the working set, with associated multipliers. is the index of the constraint with multiplier , for .

lin - integer

The number of linear constraints in the working set.

kactive - Vector(datatype=integer[kernelopts('wordsize')/8])

lambda - Vector(datatype=float[8])

Indices of the linear constraints in the working set, with associated multipliers. is the index of the constraint with multiplier , for .

nart - integer

The number of artificial constraints in the working set (see Section [The Main Iteration]).

gq - Vector(datatype=float[8])

, for , hold the multipliers for the artificial constraints.

If then the final result from nag_opt_lin_lsq (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 - Vector(datatype=float[8])

The components of the final point , for .

feasible - boolean

Will be true if the final point is feasible.

f - float

The final value of the objective function if is true; otherwise, the sum of infeasibilities. If the problem is of type FP and is feasible then f is set to zero.

ax - Vector(datatype=float[8])

If , the components of the final linear constraint activities, .

state - Vector(datatype=integer[kernelopts('wordsize')/8])

Contains the final status of the variables and general linear constraints. See Section [The optional-settings Parameter] for a description of the possible status values.

lambda - Vector(datatype=float[8])

Contains the final values of the Lagrange multipliers.

bl - Vector(datatype=float[8])

Contains the lower bounds.

bu - Vector(datatype=float[8])

Contains the upper bounds.

endstate - table

The state of termination of nag_opt_lin_lsq (e04ncc). Possible values of endstate and their correspondence to the error raised are:

Value of endstate	Error raised
Nag_Feasible or Nag_Optimal	no error raised
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 - boolean

Will be true when the print function is called with the result of the current iteration.

sol_prt - boolean

Will be true when the print function is called with the final result.

new_lm - boolean

Will be true when the Lagrange multipliers have been updated.

p - Maple

Before calling nag_opt_lin_lsq (e04ncc) this field may be initialized for use by when called from nag_opt_lin_lsq (e04ncc).

Examples

m := 10:
n := 9:
nclin := 3:
tda := 9:
tdh := 9:
optional_settings := NAG:-Nag_E04_Opt():
comm := NAG:-Nag_Comm():
a := Matrix([[1, 1, 1, 1, 1, 1, 1, 1, 4], [1, 2, 3, 4, -2, 1, 1, 1, 1], [1, -1, 1, -1, 1, 1, 1, 1, 1]], datatype=float[8], order='C_order'):
bl := Vector([0, 0, -1e+25, 0, 0, 0, 0, 0, 0, 2, -1e+25, 1], datatype=float[8]):
bu := Vector([2, 2, 2, 2, 2, 2, 2, 2, 2, 1e+25, 2, 4], datatype=float[8]):
cvec := Vector(0, datatype=float[8]):
b := Vector([1, 1, 1, 1, 1, 1, 1, 1, 1, 1], datatype=float[8]):
h := Matrix([[1, 1, 1, 1, 1, 1, 1, 1, 1], [1, 2, 1, 1, 1, 1, 2, 0, 0], [1, 1, 3, 1, 1, 1, -1, -1, -3], [1, 1, 1, 4, 1, 1, 1, 1, 1], [1, 1, 1, 3, 1, 1, 1, 1, 1], [1, 1, 2, 1, 1, 0, 0, 0, -1], [1, 1, 1, 1, 0, 1, 1, 1, 1], [1, 1, 1, 0, 1, 1, 1, 1, 1], [1, 1, 0, 1, 1, 1, 2, 2, 3], [1, 0, 1, 1, 1, 1, 0, 2, 2]], datatype=float[8], order='C_order'):
kx := Vector([0, 0, 0, 0, 0, 0, 0, 0, 0], datatype=integer[kernelopts('wordsize')/8]):
x := Vector([1, 0.5, 0.3333, 0.25, 0.2, 0.1667, 0.1428, 0.125, 0.1111], datatype=float[8]):
NAG:-e04ncc(a, bl, bu, kx, x, objf, 'm' = m, 'n' = n, 'nclin' = nclin, 'tda' = tda, 'cvec' = cvec, 'b' = b, 'h' = h, 'tdh' = tdh, 'optional_settings' = optional_settings, 'comm' = comm):