nag_opt_lin_lsq (e04ncc) (PDF version)
e04 Chapter Contents
e04 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_opt_lin_lsq (e04ncc)

+ Contents

    1  Purpose
    7  Accuracy

1  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.

2  Specification

#include <nag.h>
#include <nage04.h>
void  nag_opt_lin_lsq (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)

3  Description

nag_opt_lin_lsq (e04ncc) is designed to solve a class of quadratic programming problems stated in the following general form:
minimize ​ x R n F x   subject to   l x Ax u , (1)
where A  is an n L  by 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 cT x Not applicable
QP1 cT x + 1 2 xT Hx n  by n  symmetric positive semidefinite
QP2 cT x + 1 2 xT Hx n  by n  symmetric positive semidefinite
QP3 cT x + 1 2 xT HT Hx m  by n  upper trapezoidal
QP4 cT x + 1 2 xT HT Hx m  by n  upper trapezoidal
LS1 cT x + 1 2 b - H x 2 m  by n
LS2 cT x + 1 2 b - H x 2 m  by n
LS3 cT x + 1 2 b - H x 2 m  by n  upper trapezoidal
LS4 cT x + 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 argument options.prob (see Section 11.2).
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 QR  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  or u  can be set to special values that will be treated as -  or + . (See the description of the optional argument options.inf_bound in Section 11.2.
The function F x  is a quadratic function, whose defining feature is that its second-derivative matrix 2 F x  (the Hessian matrix) is constant. For the LP case, 2 F x = 0 ; for QP1 and QP2, 2 F x = H ; and for QP3, QP4 and LS problems, 2 F x = HT 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, nag_opt_lin_lsq (e04ncc) will solve the resulting linear programming problem; however, this can be accomplished more efficiently by using nag_opt_lp (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.
If H  is of full rank then nag_opt_lin_lsq (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 nag_opt_lin_lsq (e04ncc) is described in detail in Section 10.

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:     mIntegerInput
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 argument options.prob, see Section 11.2.
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:     nIntegerInput
On entry: n , the number of variables.
Constraint: n>0 .
3:     nclinIntegerInput
On entry: n L , the number of general linear constraints.
Constraint: nclin0 .
4:     a[nclin×tda]const doubleInput
Note: the i,jth element of the matrix A is stored in a[i-1×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,, n L . If nclin=0  then the array a is not referenced.
5:     tdaIntegerInput
On entry: the stride separating matrix column elements in the array a.
Constraint: tdan  if nclin>0 .
6:     bl[n+nclin]const doubleInput
7:     bu[n+nclin]const doubleInput
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 = - ), set bl[j-1] -options.inf_bound , and to specify a nonexistent upper bound (i.e., u j = + ), set bu[j-1] options.inf_bound, where options.inf_bound is one of the optional arguments (default value 10 20  (see Section 11.2). 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] bu[j] , for j=0,1,, n + nclin - 1 ;
  • if bl[j] = bu[j] = β , β < options.inf_bound .
8:     cvec[n]const doubleInput
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.
9:     b[m]doubleInput/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 may be set to the null pointer.
10:   h[m×tdh]doubleInput/Output
Note: the i,jth element of the matrix H is stored in h[i-1×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  by m  rows and columns of the symmetric Hessian matrix. Only the diagonal and upper triangular elements of the leading m  rows and columns of 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  by 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 non-singular 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 nag_opt_lin_lsq (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 argument options.hessian=Nag_TRUE  (see Section 11.2), and the problem is one of the LS or QP types, h contains the upper triangular Cholesky factor of the Hessian matrix 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 set to the null pointer.
11:   tdhIntegerInput
On entry: the stride separating matrix column elements in the array h.
Constraint: tdhn .
12:   kx[n]IntegerInput/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  of 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 kx[i] n , for i=0,1,,n-1;
  • if ij , kx[i] 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]doubleInput/Output
On entry: an initial estimate of the solution.
On exit: the point at which nag_opt_lin_lsq (e04ncc) terminated. If fail.code=NE_NOERROR , NW_SOLN_NOT_UNIQUE or NW_NOT_FEASIBLE, x contains an estimate of the solution.
14:   objfdouble *Output
On exit: the value of the objective function at x  if 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:   optionsNag_E04_Opt *Input/Output
On entry/exit: a pointer to a structure of type Nag_E04_Opt whose members are optional arguments for nag_opt_lin_lsq (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 11. Some of the results returned in options can be used by nag_opt_lin_lsq (e04ncc) to perform a ‘warm start’ (see the member options.start in Section 11.2).
If any of these optional arguments are required then the structure options should be declared and initialized by a call to nag_opt_init (e04xxc) and supplied as an argument to nag_opt_lin_lsq (e04ncc). However, if the optional arguments are not required the NAG defined null pointer, E04_DEFAULT, can be used in the function call.
16:   commNag_Comm *Input/Output
Note: comm is a NAG defined type (see Section 3.2.1.1 in the Essential Introduction).
On entry/exit: structure containing pointers for communication with an optional user-defined printing function; see Section 11.3.1 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 nag_opt_lin_lsq (e04ncc); comm will then be declared internally for use in calls to user-supplied functions.
17:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

5.1  Description of Printed Output

Intermediate and final results are printed out by default. The level of printed output can be controlled with the structure member options.print_level (see Section 11.2). The default, options.print_level=Nag_Soln_Iter provides a single line of output at each iteration and the final result. This section describes the default printout produced by nag_opt_lin_lsq (e04ncc).
The convention for numbering the constraints in the iteration results is that indices 1 to n  refer to the bounds on the variables, and indices n+1  to n + n L  refer to the general constraints.
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 10.3).
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 Z1T g FR , the Euclidean norm of the reduced gradient with respect to Z 1  (see Section 10.3). During the optimality phase, this norm will be approximately zero after a unit step.
The printout of the final result consists of:
Varbl gives the name (V) and index j , for j=1,2,,n of the variable.
State gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the optional argument options.ftol (default value ε , where ε  is the machine precision; see Section 11.2), 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] -options.inf_bound , where options.inf_bound is the optional argument.)
Upper bound is the upper bound specified for variable j . (None indicates that bu[j-1] options.inf_bound, where options.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 bl[j-1] -options.inf_bound  and bu[j-1] 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] -options.inf_bound  and bu[j-1] 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,, 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.

6  Error Indicators and Warnings

NE_2_INT_ARG_LT
On entry, tda=value  while n=value . These arguments must satisfy tdan .
On entry, tdh=value  while n=value . These arguments must satisfy tdhn .
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 , , 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. (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, 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 argument options.ftol (default value = ε , where ε  is the machine precision; see Section 11.2). 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 argument 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: m1.
On entry, n=value.
Constraint: n1.
On entry, nclin=value.
Constraint: nclin0.
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_iter0 .
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 .
Value value given to options.max_iter is not valid. Correct range is options.max_iter0 .
Value value given to options.rank_tol is not valid. Correct range is 0.0 < options.rank_tol < 1.0 .
NE_INVALID_REAL_RANGE_F
Value value given to options.ftol is not valid. Correct range is options.ftol>0.0 .
NE_INVALID_REAL_RANGE_FF
Value value given to options.crash_tol is not valid. Correct range is 0.0 options.crash_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 argument options.inf_step (default value 10 20 ; see Section 11.2) 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 5.1) at the final point. The modified problem (with an altered value of the optional feasibility tolerance, options.ftol) may then be solved using optional argument options.start=Nag_Warm (see Section 11.2). 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 argument 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 10.3) 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 arguments options.max_iter and options.fmax_iter, see Section 11.2) 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 nag_opt_lin_lsq (e04ncc) using the optional argument 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 11.3) 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 10.2), which will be printed if optional argument options.print_level=Nag_Soln_Iter_Full (default value options.print_level=Nag_Soln_Iter; see Section 11.2.) In this case, the algorithm could be cycling (see the comments below for fail.code=NE_CYCLING ).

7  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.

8  Further Comments

8.1  Termination Criteria

nag_opt_lin_lsq (e04ncc) exits with fail.code=NE_NOERROR  if x  is a strong local minimizer, i.e., the reduced gradient is negligible, the Lagrange multipliers are optimal (see Section 5.1) and R z  (see Section 10.3) is non-singular.

8.2  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.

9  Example

To minimize the quadratic function cT x + 1 2 xT Hx , where
c = -4.0,-1.0,-1.0,-1.0,-1.0,-1.0,-1.0,-1.0,-0.3T ,
H = 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
subject to the bounds
-2 x 1 2 -2 x 2 2 -2 x 3 2 -2 x 4 2 -2 x 5 2 -2 x 6 2 -2 x 7 2 -2 x 8 2 -2 x 9 2
and to the general constraints
-2.0 x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + 4x9 1.5 -2.0 x1 + 2x2 + 3x3 + 4x4 - 2x5 + x6 + x7 + x8 + x9 1.5 -2.0 x1 - x2 + x3 - x4 + x5 + x6 + x7 + x8 + x9 4.0
The initial point, which is feasible, is
x 0 = 0,0,0,0,0,0,0,0,0T ,
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,-1.7777,-0.45555T ,
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 argument 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 nag_opt_init (e04xxc). Then options.prob is assigned directly, before calling nag_opt_lin_lsq (e04ncc). On return from nag_opt_lin_lsq (e04ncc), nag_opt_free (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.

9.1  Program Text

Program Text (e04ncce.c)

9.2  Program Data

Program Data (e04ncce.d)

9.3  Program Results

Program Results (e04ncce.r)

10  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 11, may be omitted if the more sophisticated features of the algorithm and software are not currently of interest.

10.1  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. (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).) 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 n L n . 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., nag_opt_nlp (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 argument options.start in Section 11.2.
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 x -  is defined by
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 argument options.ftol; see Section 11.2). 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 nag_opt_lin_lsq (e04ncc); see Section 11.3). 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).

10.2  Definition of the Search Direction

Let A FR  denote the n W  by 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  if
A FR p FR = 0 . (3)
In order to compute p FR , the TQ  factorization of A FR  is used:
A FR Q FR = 0 T (4)
where T  is a non-singular n W  by n W  reverse-triangular matrix (i.e., t ij = 0  if i + j < n W ), and the non-singular n FR  by 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 = Z Y , (5)
where Y  is n FR  by 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 n R 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 nag_opt_lin_lsq (e04ncc); see Section 11.3. 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.

10.3  The Main Iteration

Let Q  denote the n  by n  matrix
Q = Q FR I FX (7)
where I FX  is the identity matrix of order n FX . Let R  denote an n  by n  upper triangular matrix (the Cholesky factor) such that
QT 2 F ~ Q H Q = RT R , (8)
and let the matrix of the first n Z  rows and columns of R  be denoted by R Z . (The matrix 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 non-singular case, p Z  satisfies the equations
RZT R Z p Z = - g Z (9)
where g Z  denotes the vector ZT g FR  and g  denotes the objective gradient. (The norm of g FR  is the printed quantity Norm Gf; see Section 11.3.) 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 RZT 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   gZT 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 ZT 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
AFRT λ A = g FR   and   λ B = g FX - AFXT 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 δ  when the associated constraint is at its upper bound, or if λ j -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 11.3) 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 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 λ j  corresponding to an inequality constraint in the working set will be such that - 1+δ λ j δ  when the associated constraint is at its upper bound, and -δ λ j 1+δ  when the associated constraint is at its lower bound. Lagrange multipliers for equality constraints will satisfy λ j 1 + δ .
The choice of step length is based on remaining feasible with respect to the satisfied constraints. If R Z  is non-singular and x+p  is feasible, α  will be taken as unity. In this case, the projected gradient at x -  will be zero, and Lagrange multipliers are computed. Otherwise, α  is set to α M , the step to the ‘nearest’ constraint (with index Jadd; see Section 11.3), which is added to the working set at the next iteration.
If 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 QR  factorization in the LS case. Column interchanges are used in both cases, and an estimate is made of the rank of the triangular factor. Thereafter, the dependent rows of R  are eliminated from the problem.
Each change in the working set leads to a simple change to 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 QT g , QT c  and f , which are related by the formulae
f = Pb - R 0 QT x , b 0 ​ for the QP case ​ ,
and
QT g = QT c - RT 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 nag_opt_lin_lsq (e04ncc): if a given factor R Z  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 R - Z p Z = e Z , where 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 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 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 A FR  contains n FR  constraints.) The idea is to include as many general constraints as necessary to ensure a non-singular R Z .
At the beginning of each phase, an upper triangular matrix R 1  is determined that is the largest non-singular 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 argument options.rank_tol; see Section 11.2.) Let Z 1  denote the columns of Z  corresponding to R 1 , and let Z  be partitioned as Z = Z 1 Z 2 . A working set for which Z 1  defines the null space can be obtained by including the rows of Z2T  as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by Z 1 .
The artificially augmented working set is given by
A - FR = A FR Z2T , (13)
so that p FR  will satisfy A FR p FR = 0  and Z2T p FR = 0 . By definition of the TQ  factorization, A - FR  automatically satisfies the following:
A - FR Q FR = A FR Z2T Q FR = A FR Z2T Z 1 Z 2 Y = 0 T- ,
where
T - = 0 T I 0 ,
and hence the TQ  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 TQ  factorization can therefore be updated in the normal fashion as the iterations proceed. No work is required to ‘delete’ the artificial constraints associated with Z 2  when Z1T 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 Z2T  is equal to Z2T 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 Z1T 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 11.3).
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., n L 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 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 TQ  factor T  (the printed value Cond T; see Section 11.3). 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 options.ftol, Section 11.2) 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 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.

11  Optional Arguments

A number of optional input and output arguments to nag_opt_lin_lsq (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 arguments you should use the NAG defined null pointer, E04_DEFAULT, in place of options when calling nag_opt_lin_lsq (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 nag_opt_init (e04xxc). Values may then be assigned to the structure members in the normal C manner.
Option settings may also be read from a text file using the function nag_opt_read (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 nag_opt_read (e04xyc).

11.1  Optional Argument Checklist and Default Values

For easy reference, the following list shows the members of options which are valid for nag_opt_lin_lsq (e04ncc) together with their default values where relevant. The number ε  is a generic notation for machine precision (see nag_machine_precision (X02AJC)).
Nag_ProblemType prob Nag_LS1
Nag_Start start Nag_Cold
Boolean list Nag_TRUE
Nag_PrintType print_level Nag_Soln_Iter  
char outfile[80] stdout
void (*print_fun)() NULL  
Integer fmax_iter max50, 5 n+nclin
Integer max_iter max50, 5 n+nclin
double crash_tol 0.01  
double ftol ε
double inf_bound 10 20
double inf_step maxoptions.inf_bound, 10 20
double rank_tol 100 ε  or 10 ε
Integer *state size n+nclin
double *ax size nclin
double *lambda size n+nclin
Boolean hessian Nag_FALSE
Integer iter  

11.2  Description of the Optional Arguments

prob – Nag_ProblemTypeDefault =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×tdh, symmetric, b and cvec not referenced;
Nag_QP2 h of size m×tdh, symmetric, b not referenced, cvec of size n;
Nag_QP3 h of size m×tdh, upper trapezoidal, b and cvec not referenced;
Nag_QP4 h of size m×tdh, upper trapezoidal, b not referenced, cvec of size n.
Nag_LS1 h of size m×tdh, b of size m, cvec not referenced;
Nag_LS2 h of size m×tdh, b of size m, cvec of size n;
Nag_LS3 h of size m×tdh, upper trapezoidal, b of size m, cvec not referenced;
Nag_LS4 h of size m×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 nag_opt_lin_lsq (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 or Nag_LS4.
start – Nag_StartDefault =Nag_Cold
On entry: specifies how the initial working set is chosen. With options.start=Nag_Cold, 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 options.crash_tol; see below).
With options.start=Nag_Warm, you must provide a valid definition of every array element of the optional argument options.state (see below). nag_opt_lin_lsq (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 nag_opt_lin_lsq (e04ncc) is called repeatedly to solve related problems.
Constraint: options.start=Nag_Cold or Nag_Warm.
list – Nag_BooleanDefault =Nag_TRUE
On entry: if options.list=Nag_TRUE  the argument settings in the call to nag_opt_lin_lsq (e04ncc) will be printed.
print_level – Nag_PrintTypeDefault =Nag_Soln_Iter
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 x , the constraint values Ax  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 TQ  factorization (see (4) in Section 10.2) 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 11.3.
Constraint: options.print_level=Nag_NoPrint, Nag_Soln, Nag_Iter, Nag_Soln_Iter, Nag_Iter_Long, Nag_Soln_Iter_Long, Nag_Soln_Iter_Const or Nag_Soln_Iter_Full.
outfile – const char[80]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 functionDefault =  NULL
On entry: printing function defined by you; the prototype of options.print_fun is
void(*print_fun)(const Nag_Search_State *st, Nag_Comm *comm);
See Section 11.3.1 below for further details.
fmax_iter – IntegerDefault = max50, 5 n+nclin
max_iter – IntegerDefault = max50, 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 nag_opt_lin_lsq (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 argument options.list=Nag_TRUE  (the default setting).
Constraints:
crash_tol – doubleDefault =0.01
On entry: options.crash_tol is used when optional argument options.start=Nag_Cold (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 options.crash_tol of their bounds. In particular, a constraint of the form ajT x l  will be included in the initial working set if ajT x-l options.crash_tol × 1 + l .
Constraint: 0.0 options.crash_tol 1.0 .
ftol – doubleDefault = 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 as 10 -6 .
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 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 – doubleDefault = 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 + (and similarly any lower bound less than or equal to -options.inf_bound  will be regarded as -).
Constraint: options.inf_bound>0.0 .
inf_step – doubleDefault = maxoptions.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 – doubleDefault = 100 ε  or 10 ε
The default value is 100 ε  for problem types QP1, LS1 and LS3 but is 10 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 10.3). If ρ i  denotes the function ρ i = max R 11 , R 22 , , R ii , the rank of R  is defined to be smallest index i  such that R i + 1 , i + 1 options.rank_tol × 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 nag_opt_lin_lsq (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 nag_opt_lin_lsq (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 nag_opt_lin_lsq (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 argument 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 options.state[j-1] 4 , for j=1,2,,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 nag_opt_lin_lsq (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 Ax .
lambda – double *Default memory = n + nclin
On entry: n+nclin  values of memory will be automatically allocated by nag_opt_lin_lsq (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_BooleanDefault =Nag_FALSE
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 H Q  (8), and hence extra computation is required to form the Hessian itself. If the optional argument 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 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.

11.3  Description of Printed Output

The level of printed output can be controlled with the structure members options.list and options.print_level (see Section 11.2). If options.list=Nag_TRUE  then the argument values to nag_opt_lin_lsq (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 nag_opt_lin_lsq (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  to n + n L  refer to the general constraints.
When options.print_level=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 1.0  only if the factor R Z  is singular (see Section 10.3).
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 Z1T g FR , the Euclidean norm of the reduced gradient with respect to Z 1  (see Section 10.3). During the optimality phase, this norm will be approximately zero after a unit step.
If options.print_level=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 Z 2  (see Section 10.3).
Zr is the number of columns of Z 1  (see Section 10.2). 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 10) 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 or 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 Ax  currently held in options.ax.
State is the current value of options.state associated with Ax .
Also printed are the Lagrange Multipliers for the bound constraints, linear constraints and artificial constraints.
If 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 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 j , for j=1,2,,n of the variable.
State gives the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the optional argument options.ftol (default value ε , where ε  is the machine precision; see Section 11.2), 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] -options.inf_bound , where options.inf_bound is the optional argument.)
Upper bound is the upper bound specified for variable j . (None indicates that bu[j-1] options.inf_bound, where options.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 bl[j-1] -options.inf_bound  and bu[j-1] 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] -options.inf_bound  and bu[j-1] 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,, 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.
If options.print_level=Nag_NoPrint then printout will be suppressed; you can print the final solution when nag_opt_lin_lsq (e04ncc) returns to the calling program.

11.3.1  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 nag_opt_lin_lsq (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.
If commit_prt = Nag_TRUE then the results from the last iteration of nag_opt_lin_lsq (e04ncc) are provided through st. Note that options.print_fun will be called with commit_prt = Nag_TRUE only if options.print_level=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:
nInteger
The number of variables.
nclinInteger
The number of linear constraints.
iterInteger
The iteration count.
jdelInteger
Index of constraint deleted from the working set.
jaddInteger
Index of constraint added to the working set.
stepdouble
The step taken along the computed search direction.
ninfInteger
The number of violated constraints (infeasibilities).
fdouble
The current value of the objective function if stninf = 0 ; otherwise, stf is a weighted sum of the magnitudes of constraint violations.
bndInteger
Number of bound constraints in the working set.
linInteger
Number of general linear constraints in the working set.
nartInteger
Number of artificial constraints in the working set (see Section 10.3).
nrankInteger
The rank of the upper triangular matrix R  (see Section 10.3).
nrzInteger
Number of columns of Z 1  (see Section 10.2).
norm_gzdouble
Euclidean norm of the reduced gradient, Z1T g FR  (see Section 10.3).
norm_gfdouble
Euclidean norm of the gradient function with respect to the free variables.
cond_tdouble
A lower bound on the condition number of the working set.
cond_rdouble
A lower bound on the condition number of the triangular factor R 1  (see Section 10.3).
xdouble *
The components stx[j-1]  of the current point x , for j=1,2,,stn.
axdouble *
If stnclin > 0 , the stnclin components of the linear constraints Ax .
stateInteger *
options.state contains the status of the stn variables and stnclin general linear constraints. See Section 11.2 for a description of the possible status values.
diagtdouble *
If stlin > 0 , the stlin elements in the diagonal of the matrix T .
diagrdouble *
If stnrank > 0 , the first stnrank elements of the diagonal of the upper triangular matrix R .
If commnew_lm = Nag_TRUE then the Lagrange multipliers have been updated and the following members of st are set:
bndInteger
The number of bound constraints in the working set.
kxInteger *
bclambdadouble *
Indices of the bound constraints in the working set, with associated multipliers. stkx[i]  is the index of the constraint with multiplier stbclambda[i] , for i=0,1,,stbnd - 1.
linInteger
The number of linear constraints in the working set.
kactiveInteger *
lambdadouble *
Indices of the linear constraints in the working set, with associated multipliers. stkactive[i]  is the index of the constraint with multiplier stlambda[ stbnd + i ] , for i=0,1,,stlin - 1.
nartInteger
The number of artificial constraints in the working set (see Section 10.3).
gqdouble *
stgq[i] , for i=0,1,,stnart - 1, hold the multipliers for the artificial constraints.
If commsol_prt = Nag_TRUE then the final result from nag_opt_lin_lsq (e04ncc) is available and the following members of st are set:
nInteger
The number of variables.
nclinInteger
The number of linear constraints.
iterInteger
The iteration count.
xdouble *
The components stx[j-1]  of the final point x , for j=1,2,,stn.
feasibleNag_Boolean
Will be Nag_TRUE if the final point is feasible.
fdouble
The final value of the objective function if stfeasible is Nag_TRUE; otherwise, the sum of infeasibilities. If the problem is of type FP and x  is feasible then stf is set to zero.
axdouble *
If stnclin > 0 , the stnclin components of the final linear constraint activities, Ax .
stateInteger *
Contains the final status of the stn variables and stnclin general linear constraints. See Section 11.2 for a description of the possible status values.
lambdadouble *
Contains the stn + stnclin final values of the Lagrange multipliers.
bldouble *
Contains the stn + stnclin lower bounds.
budouble *
Contains the stn + stnclin upper bounds.
endstateNag_EndState 
The state of termination of nag_opt_lin_lsq (e04ncc). Possible values of stendstate and their correspondence to the exit value of fail are:
Value of stendstate 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_prtNag_Boolean
Will be Nag_TRUE when the print function is called with the result of the current iteration.
sol_prtNag_Boolean
Will be Nag_TRUE when the print function is called with the final result.
new_lmNag_Boolean
Will be Nag_TRUE when the Lagrange multipliers have been updated.
userdouble *
iuserInteger *
pPointer 
Pointers for communication of user information. If used they must be allocated memory either before entry to nag_opt_lin_lsq (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.

nag_opt_lin_lsq (e04ncc) (PDF version)
e04 Chapter Contents
e04 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012