nag_opt_qp (e04nfc) (PDF version)
e04 Chapter Contents
e04 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_opt_qp (e04nfc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_opt_qp (e04nfc) solves general quadratic programming problems. It is not intended for large sparse problems.

2  Specification

#include <nag.h>
#include <nage04.h>
void  nag_opt_qp (Integer n, Integer nclin, const double a[], Integer tda, const double bl[], const double bu[], const double cvec[], const double h[], Integer tdh,
void (*qphess)(Integer n, Integer jthcol, const double h[], Integer tdh, const double x[], double hx[], Nag_Comm *comm),
double x[], double *objf, Nag_E04_Opt *options, Nag_Comm *comm, NagError *fail)

3  Description

nag_opt_qp (e04nfc) 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 A x u ,
where A  is an m lin  by n  matrix and 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 and QP stand for ‘feasible point’, ‘linear programming’ and ‘quadratic programming’ respectively and c  is an n  element vector.
Problem Type fx Matrix H
FP Not applicable Not applicable
LP cT x   Not applicable
QP1 cT x + 1 2 xT Hx   symmetric
QP2 cT x + 1 2 xT Hx   symmetric
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
Table 1
For problems of type FP a feasible point with respect to a set of linear inequality constraints is sought. The default problem type is QP2, other objective functions are selected by using the optional argument options.prob.
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.)
The defining feature of a quadratic function f x  is that the 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 and QP4, 2 f x = HT H . If H  is defined as the zero matrix, nag_opt_qp (e04nfc) will solve the resulting linear programming problem; however, this can be accomplished more efficiently by setting the optional argument options.prob=Nag_LP, or by using nag_opt_lp (e04mfc).
You must supply an initial estimate of the solution.
In the QP case, you may supply H  either explicitly as an m  by n  matrix, or implicitly in a C function that computes the product Hx  for any given vector x . An example of such a function is included in Section 9. There is no restriction on H  apart from symmetry. In general, a successful run of nag_opt_qp (e04nfc) will indicate one of three situations: (i) a minimizer has been found; (ii) the algorithm has terminated at a so-called dead-point; or (iii) the problem has no bounded solution. If a minimizer is found, and H  is positive definite or positive semidefinite, nag_opt_qp (e04nfc) will obtain a global minimizer; otherwise, the solution will be a local minimizer (which may or may not be a global minimizer). A dead-point is a point at which the necessary conditions for optimality are satisfied but the sufficient conditions are not. At such a point, a feasible direction of decrease may or may not exist, so that the point is not necessarily a local solution of the problem. Verification of optimality in such instances requires further information, and is in general an NP-hard problem (see Pardalos and Schnitger (1988)). Termination at a dead-point can occur only if H  is not positive definite. If H  is positive semidefinite, the dead-point will be a weak minimizer (i.e., with a unique optimal objective value, but an infinite set of optimal x ).
Details about the algorithm are described in Section 10, but it is not necessary to read this more advanced section before using nag_opt_qp (e04nfc).

4  References

Bunch J R and Kaufman L C (1980) A computational method for the indefinite quadratic programming problem Linear Algebra and its Applications 34 341–370
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 and Murray W (1978) Numerically stable methods for quadratic programming Math. Programming 14 349–372
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, Saunders M A and Wright M H (1989) A practical anti-cycling procedure for linearly constrained optimization Math. Programming 45 437–474
Gill P E, Murray W, Saunders M A and Wright M H (1991) Inertia-controlling methods for general quadratic programming SIAM Rev. 33 1–36
Pardalos P M and Schnitger G (1988) Checking local optimality in constrained quadratic programming is NP-hard Operations Research Letters 7 33–35

5  Arguments

1:     nIntegerInput
On entry: n , the number of variables.
Constraint: n>0 .
2:     nclinIntegerInput
On entry: m lin , the number of general linear constraints.
Constraint: nclin0 .
3:     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,, m lin . If nclin=0 , the array a is not referenced.
4:     tdaIntegerInput
On entry: the stride separating matrix column elements in the array a.
Constraint: if nclin>0 , tdan
5:     bl[n+nclin]const doubleInput
6:     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 m lin  elements the bounds for the general linear constraints (if any). To specify a nonexistent lower bound (i.e., l j = - ), set bl[j] -options.inf_bound , and to specify a nonexistent upper bound (i.e., u j = + ), set bu[j] options.inf_bound; options.inf_bound is the optional argument, whose default value is 10 20 . To specify the j th constraint as an equality, set bl[j] = bu[j] = 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 .
7:     cvec[n]const doubleInput
On entry: the coefficients of the explicit linear term of the objective function when the problem is of type options.prob=Nag_LP, Nag_QP2 or Nag_QP4. The default problem type is options.prob=Nag_QP2 corresponding to QP2 described in Section 3; other problem types can be specified using the optional argument options.prob.
If the problem is of type options.prob=Nag_FP, Nag_QP1 or Nag_QP3, cvec is not referenced and therefore a NULL pointer may be given.
8:     h[n×tdh]const doubleInput
On entry: h may be used to store the quadratic term H  of the QP objective function if desired. The elements of h are accessed only by the function qphess; thus h is not accessed if the problem is of type options.prob=Nag_FP or Nag_LP. The number of rows of H  is denoted by m , its default value is equal to n . (The optional argument options.hrows may be used to specify a value of m<n .)
If the problem is of type options.prob=Nag_QP1 or Nag_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 need not be assigned.
For problems options.prob=Nag_QP3 or Nag_QP4, the first m  rows of h must contain an m  by n  upper trapezoidal factor of the Hessian matrix. 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.
In some cases, you need not use h to store H  explicitly (see the specification of function qphess).
9:     tdhIntegerInput
On entry: the stride separating matrix column elements in the array h.
Constraint: tdhn  or at least the value of the optional argument options.hrows if it is set.
10:   qphessfunction, supplied by the userExternal Function
In general, you need not provide a version of qphess, because a ‘default’ function is included in the NAG C Library. If the default function is required then the NAG defined null void function pointer, NULLFN, should be supplied in the call to nag_opt_qp (e04nfc). The algorithm of nag_opt_qp (e04nfc) requires only the product of H  and a vector x ; and in some cases you may obtain increased efficiency by providing a version of qphess that avoids the need to define the elements of the matrix H  explicitly.
qphess is not referenced if the problem is of type options.prob=Nag_FP or Nag_LP, in which case qphess should be replaced by NULLFN.
The specification of qphess is:
void  qphess (Integer n, Integer jthcol, const double h[], Integer tdh, const double x[], double hx[], Nag_Comm *comm)
1:     nIntegerInput
On entry: n , the number of variables.
2:     jthcolIntegerInput
On entry: jthcol specifies whether or not the vector x  is a column of the identity matrix.
jthcol = j > 0
The vector x  is the j th column of the identity matrix, and hence Hx  is the j th column of H , which can sometimes be computed very efficiently and qphess may be coded to take advantage of this. However special code is not necessary because x  is always stored explicitly in the array x.
jthcol=0
x  has no special form.
3:     h[n×tdh]const doubleInput
On entry: the matrix H  of the QP objective function. The matrix element H ij  is stored in h[ i-1 × tdh + j - 1 ] , for i=1,2,,n and j=1,2,,n. In some situations, it may be desirable to compute Hx  without accessing h – for example, if H  is sparse or has special structure. (This is illustrated in the function qphess1 in Section 9.) The arguments h and tdh may then refer to any convenient array.
4:     tdhIntegerInput
On entry: the stride separating matrix column elements in the array h.
5:     x[n]const doubleInput
On entry: the vector x .
6:     hx[n]doubleOutput
On exit: the product Hx .
7:     commNag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to qphess.
flagIntegerInput/Output
On entry: commflag  contains a non-negative number.
On exit: if qphess resets commflag  to some negative number nag_opt_qp (e04nfc) will terminate immediately with the error indicator NE_USER_STOP. If fail is supplied to nag_opt_qp (e04nfc), fail.errnum will be set to your setting of commflag .
firstNag_BooleanInput
On entry: will be set to Nag_TRUE on the first call to qphess and Nag_FALSE for all subsequent calls.
nfIntegerInput
On entry: the number of calls made to qphess including the current one.
userdouble *
iuserInteger *
pPointer 
The type Pointer will be void * with a C compiler that defines void * and char * otherwise. Before calling nag_opt_qp (e04nfc) you may allocate memory to these pointers and they may be initialized with various quantities for use by qphess when called from nag_opt_qp (e04nfc).
Note: qphess should be tested separately before being used in conjunction with nag_opt_qp (e04nfc). The input arrays h and x must not be changed within qphess.
11:   x[n]doubleInput/Output
On entry: an initial estimate of the solution.
On exit: the point at which nag_opt_qp (e04nfc) terminated. If fail.code=NE_NOERROR , NW_DEAD_POINT, NW_SOLN_NOT_UNIQUE or NW_NOT_FEASIBLE, x contains an estimate of the solution.
12:   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 options.prob=Nag_FP and x  is feasible, objf is set to zero.
13:   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_qp (e04nfc). 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 in Section 11. Some of the results returned in options can be used by nag_opt_qp (e04nfc) to perform a ‘warm start’ if it is re-entered (see the optional argument options.start).
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_qp (e04nfc). However, if the optional arguments are not required the NAG defined null pointer, E04_DEFAULT, can be used in the function call.
14:   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 user communication with user-supplied functions; see the description of qphess 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_qp (e04nfc); comm will then be declared internally for use in calls to user-supplied functions.
15:   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. You can control the level of printed output with the structure member options.print_level. 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_qp (e04nfc).
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 + m lin  refer to the general constraints. When the status of a constraint changes, the index of the constraint is printed, along with the designation L (lower bound), U (upper bound), E (equality), F (temporarily fixed variable) or A (artificial constraint).
The single line of intermediate results output on completion of each iteration gives:
Itn is the iteration count.
Jdel is the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted.
Jadd is the index of the constraint added to the working set. If Jadd is zero, no constraint was added.
Step is the step taken along the computed search direction. If a constraint is added during the current iteration (i.e., Jadd is positive), Step will be the step to the nearest constraint. During the optimality phase, the step can be greater than 1.0  only if the reduced Hessian is not positive definite.
Ninf is the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Obj 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, Obj 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.
Bnd the number of simple bound constraints in the current working set.
Lin the number of general linear constraints in the current working set.
Nart the number of artificial constraints in the working set. At the start of the optimality phase, Nart provides an estimate of the number of non-positive eigenvalues in the reduced Hessian.
Nrz the dimension of the subspace in which the objective function is currently being minimized. The value of Nrz is the number of variables minus the number of constraints in the working set; i.e., Nrz = n - Bnd + Lin + Nart .
Norm Gz the Euclidean norm of the reduced gradient. During the optimality phase, this norm will be approximately zero after a unit step.
The printout of the final result consists of:
Varbl the name (V) and index j , for j=1,2,,n of the variable.
State the state of the variable (FR if neither bound is in the working set, EQ if a fixed variable, LL if on its lower bound, UL if on its upper bound, TF if temporarily fixed at its current value). If Value lies outside the upper or lower bounds by more than the feasibility tolerance, State will be ++ or -- respectively.
Value the value of the variable at the final iteration.
Lower bound the lower bound specified for the variable. (None indicates that bl[j-1] -options.inf_bound .)
Upper bound the upper bound specified for the variable. (None indicates that bu[j-1] options.inf_bound.)
Lagr mult the value of the Lagrange multiplier for the associated bound constraint. This will be zero if State is FR. If x  is optimal, the multiplier should be non-negative if State is LL, and non-positive if State is UL.
Residual the difference between the variable Value and the nearer of its bounds bl[j-1]  and bu[j-1] .
The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, and with the following change in the heading:
LCon the name (L) and index j , for j=1,2,, m lin  of the constraint.

6  Error Indicators and Warnings

If one of NE_USER_STOP, NE_2_INT_ARG_LT, NE_OPT_NOT_INIT, NE_BAD_PARAM, NE_INVALID_INT_RANGE_1, NE_INVALID_INT_RANGE_2, NE_INVALID_REAL_RANGE_FF, NE_INVALID_REAL_RANGE_F, NE_CVEC_NULL, NE_H_NULL, NE_WARM_START, NE_BOUND, NE_BOUND_LCON, NE_STATE_VAL and NE_ALLOC_FAIL occurs, no values will have been assigned to objf, or to options.ax and options.lambda. x and options.state will be unchanged.
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 .
On entry, tdh=value  while options.hrows=value . These arguments must satisfy tdhoptions.hrows .
NE_ALLOC_FAIL
Dynamic memory allocation failed.
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_H_NULL
options.prob=value , qphess is NULL but argument h is also NULL. If the default function for qphess is to be used for this problem then an array must be supplied in argument h.
NE_HESS_TOO_BIG
Reduced Hessian exceeds assigned dimension. options.max_df=value .
The algorithm needed to expand the reduced Hessian when it was already at its maximum dimension, as specified by the optional argument options.max_df.
The value of the argument options.max_df is too small. Rerun nag_opt_qp (e04nfc) with a larger value (possibly using the options.start=Nag_Warm facility to specify the initial working set).
NE_INT_ARG_LT
On entry, n=value.
Constraint: n1.
On entry, nclin=value.
Constraint: nclin0.
NE_INVALID_INT_RANGE_1
Value value given to options.fcheck not valid. Correct range is options.fcheck1 .
Value value given to options.fmax_iter not valid. Correct range is options.fmax_iter0 .
Value value given to options.hrows not valid. Correct range is n options.hrows 0 .
Value value given to options.max_df not valid. Correct range is n options.max_df 1 .
Value value given to options.max_iter not valid. Correct range is options.max_iter0 .
NE_INVALID_INT_RANGE_2
Value value given to options.reset_ftol not valid. Correct range is 0 < options.reset_ftol < 10000000 .
NE_INVALID_REAL_RANGE_F
Value value given to options.ftol not valid. Correct range is options.ftol>0.0 .
Value value given to options.inf_bound not valid. Correct range is options.inf_bound>0.0 .
Value value given to options.inf_step not valid. Correct range is options.inf_step>0.0 .
NE_INVALID_REAL_RANGE_FF
Value value given to options.crash_tol not valid. Correct range is 0.0 options.crash_tol 1.0 .
Value value given to options.rank_tol not valid. Correct range is 0.0 options.rank_tol < 1.0 .
NE_NOT_APPEND_FILE
Cannot open file string  for appending.
NE_NOT_CLOSE_FILE
Cannot close file string .
NE_OPT_NOT_INIT
Options structure not initialized.
NE_STATE_VAL
options.state[value]  is out of range. options.state[value] = value.
NE_UNBOUNDED
Solution appears to be unbounded.
This value of fail implies that a step as large as options.inf_step would have to be taken in order to continue the algorithm. This situation can occur only when H  is not positive definite and at least one variable has no upper or lower bound.
NE_USER_STOP
User requested termination, user flag value =value .
This exit occurs if you set commflag  to a negative value in qphess. If fail is supplied the value of fail.errnum will be the same as your setting of commflag .
NE_WARM_START
options.start=Nag_Warm but pointer options.state=  NULL.
NE_WRITE_ERROR
Error occurred when writing to file string .
NW_DEAD_POINT
Iterations terminated at a dead point (check the optimality conditions).
The necessary conditions for optimality have been satisfied but the sufficient conditions are not. (The reduced gradient is negligible, the Lagrange multipliers are optimal, but H r  is singular or there are some very small multipliers.) If H  is not positive definite, x  is not necessarily a local solution of the problem and verification of optimality requires further information.
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 at the final point. You should check that there are no constraint redundancies. If the data for the constraints are accurate only to the absolute precision σ , you should ensure that the value of the optional argument 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, the optional argument 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 necessary conditions for optimality have been satisfied but the sufficient conditions are not. (The reduced gradient is negligible, the Lagrange multipliers are optimal, but H r  is singular or there are some very small multipliers.) If H  is positive semidefinite, x  gives the global minimum value of the objective function, but the final x  is not unique.
NW_TOO_MANY_ITER
The maximum number of iterations, value, have been performed.
The value of the optional argument options.max_iter may be too small. If the method appears to be making progress (e.g., the objective function is being satisfactorily reduced), increase the value of options.max_iter and rerun nag_opt_qp (e04nfc) (possibly using the options.start=Nag_Warm facility to specify the initial working set).

7  Accuracy

nag_opt_qp (e04nfc) 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

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. (1986) for further information and advice.

9  Example

To minimize the quadratic function f x = cT x + 1 2 xT Hx , where
c = -0.02,-0.2,-0.2,-0.2,-0.2,0.04,0.04T
H = 2 0 0 0 0 - 0 - 0 0 2 0 0 0 - 0 - 0 0 0 2 2 0 - 0 - 0 0 0 2 2 0 - 0 - 0 0 0 0 0 2 - 0 - 0 0 0 0 0 0 -2 -2 0 0 0 0 0 -2 -2
subject to the bounds
-0.01 x 1 0.01 -0.10 x 2 0.15 -0.01 x 3 0.03 -0.04 x 4 0.02 -0.10 x 5 0.05 -0.01 x 6 0.05 -0.01 x 7 0.05
and the general constraints
x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 = -0.13 0.15 x 1 + 0.04 x 2 + 0.02 x 3 + 0.04 x 4 + 0.02 x 5 + 0.01 x 6 + 0.03 x 7 -0.0049 0.03 x 1 + 0.05 x 2 + 0.08 x 3 + 0.02 x 4 + 0.06 x 5 + 0.01 x 6 + -0.0064 0.02 x 1 + 0.04 x 2 + 0.01 x 3 + 0.02 x 4 + 0.02 x 5 + -0.0037 0.02 x 1 + 0.03 x 2 + + 0.01 x 5 + -0.0012 -0.0992 0.70 x 1 + 0.75 x 2 + 0.80 x 3 + 0.75 x 4 + 0.80 x 5 + 0.97 x 6 + -0.0020 - 0.003 0 0.02 x 1 + 0.06 x 2 + 0.08 x 3 + 0.12 x 4 + 0.02 x 5 + 0.01 x 6 + 0.97 x 7 -0.002
The initial point, which is infeasible, is
x 0 = -0.01,-0.03,0.0,-0.01,-0.1,0.02,0.01T .
The computed solution (to five figures) is
x * = -0.01,-0.069865,0.018259,-0.024261,-0.062006,0.013054,0.0040665T .
One bound constraint and four general constraints are active at the solution.
This example shows the use of certain optional arguments. Option values are assigned directly within the program text and by reading values from a data file. The options structure is declared and initialized by nag_opt_init (e04xxc). Values are then assigned directly to options.outfile and options.inf_bound and two further options are read from the data file by use of nag_opt_read (e04xyc). nag_opt_qp (e04nfc) is then called to solve the problem using the function qphess1, with the Hessian implicit, for argument qphess. On successful return two further options are set, selecting a warm start and a reduced level of printout, and the problem is solved again using the function qphess2. In this case the Hessian is defined explicitly. Finally the memory freeing function 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 (e04nfce.c)

9.2  Program Data

Program Data (e04nfce.d)

Program Options (e04nfce.opt)

9.3  Program Results

Program Results (e04nfce.r)

10  Further Description

This section gives a detailed description of the algorithm used in nag_opt_qp (e04nfc). 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_qp (e04nfc) 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. (1991). Here we briefly summarize the main features of the method. Where possible, explicit reference is made to the names of variables that are arguments of nag_opt_qp (e04nfc) or appear in the printed output. nag_opt_qp (e04nfc) 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 computations in both phases are performed by the same functions. 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 m lin n . Once any iterate is feasible, all subsequent iterates remain feasible.
nag_opt_qp (e04nfc) 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. 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 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 , (1)
where the steplength 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). The working set is the current prediction of the constraints that hold with equality at a solution of a linearly constrained QP problem. 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 component 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 components of the search direction are zero, the columns of A  corresponding to fixed variables may be ignored.
Let m 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 ( m w  and n fx  are the quantities Lin and Bnd in the printed output from nag_opt_qp (e04nfc)). 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.

10.2  Definition of the Search Direction

Let A fr  denote the m 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 . (2)
In order to compute p fr , the TQ  factorization of A fr  is used:
A fr Q fr = 0 T , (3)
where T  is a non-singular m w  by m w  upper triangular matrix (i.e., t ij = 0  if i>j ), 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 ,
where Y  is n fr × m w , then the n z  ( n z = n fr - m 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 r  denote a matrix whose n r  columns are a subset of the columns of Z . (The integer n r  is the quantity Nrz in the printed output from nag_opt_qp (e04nfc). In many cases, Z r  will include all the columns of Z .) The direction p fr  will satisfy (2) if
p fr = Z r p r , (4)
where p r  is any n r -vector.
Let Q  denote the n  by n  matrix
Q = Q fr I fx ,
where I fx  is the identity matrix of order n fx . Let H q  and g q  denote the n  by n  transformed Hessian and the transformed gradient
H q = QT HQ  and  g q = QT c + Hx
and let the matrix of first n r  rows and columns of H q  be denoted by H r  and the vector of the first n r  elements of g q  be denoted by g r . The quantities H r  and g r  are known as the reduced Hessian and reduced gradient of f x , respectively. Roughly speaking, g r  and H r  describe the first and second derivatives of an unconstrained problem for the calculation of p r .
At each iteration, a triangular factorization of H r  is available. If H r  is positive definite, H r = RT R , where R  is the upper triangular Cholesky factor of H r . If H r  is not positive definite, H r = RT DR , where D = diag1,1,,1,μ , with μ0 .
The computation is arranged so that the reduced gradient vector is a multiple of e r , a vector of all zeros except in the last (i.e., n r th) position. This allows the vector p r  in (4) to be computed from a single back-substitution
R p r = γ e r , (5)
where γ  is a scalar that depends on whether or not the reduced Hessian is positive definite at x . In the positive definite case, x+p  is the minimizer of the objective function subject to the constraints (bounds and general) in the working set treated as equalities. If H r  is not positive definite, p r  satisfies the conditions
prT H r p r < 0  and  grT p r 0 ,
which allow the objective function to be reduced by any positive step of the form x + α p .

10.3  The Main Iteration

If the reduced gradient is zero, x  is a constrained stationary point in the subspace defined by Z . During the feasibility phase, the reduced 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 reduced 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 c  and λ b  for the general and bound constraints are defined from the equations
AfrT λ c = g fr  and  λ b = g fx - AfxT λ c . (6)
Given a positive constant δ  of the order of the machine precision, a 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 you can force nag_opt_qp (e04nfc) to continue until the minimum value of the sum of infeasibilities has been found (see the discussion of the optional argument options.min_infeas in Section 11.2). 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 .
If the reduced gradient is not zero, Lagrange multipliers need not be computed and the nonzero elements of the search direction p  are given by Z r p r  (see (5)). The choice of step length is influenced by the need to maintain feasibility with respect to the satisfied constraints. If H r  is positive definite and x+p  is feasible, α  will be taken as unity. In this case, the reduced 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.
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 , and QT c . The triangular factor R  associated with the reduced Hessian is only updated during the optimality phase.
One of the most important features of nag_opt_qp (e04nfc) 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 diagonal elements 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.
nag_opt_qp (e04nfc) includes a rigorous procedure that prevents the possibility of cycling at a point where the active constraints are nearly linearly dependent (see Gill et al. (1989)). The main feature of the anti-cycling procedure is that the feasibility tolerance is increased slightly at the start of every iteration. This not only allows a positive step to be taken at every iteration, but also provides, whenever possible, a choice of constraints to be added to the working set. 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 a 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.

10.4  Choosing the Initial Working Set

At the start of the optimality phase, a positive definite H r  can be defined 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 that the reduced Hessian is positive definite.
Let H z  denote the matrix of the first n z  rows and columns of the matrix H q = QT HQ  at the beginning of the optimality phase. A partial Cholesky factorization is used to find an upper triangular matrix R  that is the factor of the largest positive definite leading sub-matrix of H z . The use of interchanges during the factorization of H z  tends to maximize the dimension of R . (The condition of R  may be controlled using the optional argument options.rank_tol.) Let Z r  denote the columns of Z  corresponding to R , and let Z  be partitioned as Z = Z r Z a . A working set, for which Z r  defines the null space, can be obtained by including the rows of ZaT  as ‘artificial constraints’. Minimization of the objective function then proceeds within the subspace defined by Z r , as described in Section 10.2.
The artificially augmented working set is given by
A - fr = ZaT A fr , (7)
so that p fr  will satisfy A fr p fr = 0  and ZaT p fr = 0 . By definition of the TQ  factorization, A - fr  automatically satisfies the following:
A - fr Q fr = ZaT A fr Q fr = ZaT A fr Z r Z a Y = 0 T- ,
where
T - = I 0 0 T ,
and hence the TQ  factorization of (7) is available trivially from T  and Q fr  without additional expense.
The matrix Z a  is not 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 a  when ZrT g fr = 0 , since this simply involves repartitioning Q fr . The ‘artificial’ multiplier vector associated with the rows of ZaT  is equal to ZaT g fr , and the multipliers corresponding to the rows of the ‘true’ working set are the multipliers that would be obtained if the artificial constraints were not present. If an artificial constraint is ‘deleted’ from the working set, an A appears alongside the entry in the Jdel column of the printed output (see Section 11.3).
The number of columns in Z a  and Z r , the Euclidean norm of ZrT g fr , and the condition estimator of R  appear in the printed output as Nart, Nrz, Norm Gz and Cond Rz (see Section 11.3).
Under some circumstances, a different type of artificial constraint is used when solving a linear program. Although the algorithm of nag_opt_qp (e04nfc) does not usually perform simplex steps (in the traditional sense), there is one exception: a linear program with fewer general constraints than variables (i.e., m lin 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. If a temporary bound is ‘deleted’ from the working set, an F (for ‘Fixed’) appears alongside the entry in the Jdel column of the printed output (see Section 11.3).

11  Optional Arguments

A number of optional input and output arguments to nag_opt_qp (e04nfc) 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_qp (e04nfc); 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, 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_qp (e04nfc) 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_QP2
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
Boolean min_infeas Nag_FALSE
double crash_tol 0.01
double ftol ε  
double optim_tol ε0.8 
Integer reset_ftol 10000
Integer fcheck 50
double inf_bound 10 20  
double inf_step maxoptions.inf_bound, 10 20
Integer hrows n
Integer max_df n
double rank_tol 100 ε  
Integer *state size n+nclin  
double *ax size nclin
double *lambda size n+nclin  
Integer iter
Integer nf

11.2  Description of the Optional Arguments

prob – Nag_ProblemTypeDefault =Nag_QP2  
On entry: specifies the type of objective function to be minimized during the optimality phase. The following are the six possible values of options.prob and the size of the arrays h and cvec that are required to define the objective function:
Nag_FP h and cvec not accessed;
Nag_LP h not accessed, cvec[n]  required;
Nag_QP1 h[n×tdh]  symmetric, cvec not referenced;
Nag_QP2 h[n×tdh]  symmetric, cvec[n]  required;
Nag_QP3 h[n×tdh]  upper trapezoidal, cvec not referenced;
Nag_QP4 h[n×tdh]  upper trapezoidal, cvec[n]  required.
If H=0 , i.e., the objective function is purely linear, the efficiency of nag_opt_qp (e04nfc) may be increased by specifying options.prob as Nag_LP.
Constraint: options.prob=Nag_FP, Nag_LP, Nag_QP1, Nag_QP2, Nag_QP3 or Nag_QP4.
start – Nag_StartDefault =Nag_Cold  
On entry: specifies how the initial working set is chosen. With options.start=Nag_Cold, nag_opt_qp (e04nfc) 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 options.crash_tol).
With options.start=Nag_Warm, you must provide a valid definition of every element of the array pointer options.state (see below for the definition of this member of options). nag_opt_qp (e04nfc) will override your specification of options.state if necessary, so that a poor choice of the working set will not cause a fatal error. Nag_Warm will be advantageous if a good estimate of the initial working set is available – for example, when nag_opt_qp (e04nfc) 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_qp (e04nfc) will be printed.
print_level – Nag_PrintTypeDefault =Nag_Soln_Iter  
On entry: the level of results printout produced by nag_opt_qp (e04nfc). 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 upper triangular matrix T  associated with the TQ  factorization (3) 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 specifies the maximum number of iterations allowed in the feasibility phase. options.max_iter specifies the maximum number of iterations permitted in the optimality phase.
If you wish to check that a call to nag_opt_qp (e04nfc) 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 the initialization stages prior to the first iteration will be processed and a listing of argument settings output, if options.list=Nag_TRUE  (the default setting).
Constraint: options.fmax_iter0  and options.max_iter0 .
min_infeas – Nag_BooleanDefault =Nag_FALSE  
On entry: options.min_infeas specifies whether nag_opt_qp (e04nfc) should minimize the sum of infeasibilities if no feasible point exists for the constraints.
options.min_infeas=Nag_FALSE
nag_opt_qp (e04nfc) will terminate as soon as it is evident that the problem is infeasible, in which case the final point will generally not be the point at which the sum of infeasibilities is minimized.
options.min_infeas=Nag_TRUE
nag_opt_qp (e04nfc) will continue until the sum of infeasibilities is minimized.
crash_tol – doubleDefault =0.01  
On entry: options.crash_tol is used in conjunction with the optional argument options.start when options.start has the default setting, i.e., options.start=Nag_Cold, nag_opt_qp (e04nfc) selects an initial working set. The initial working set will include 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: options.ftol 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_qp (e04nfc) attempts to find a feasible solution before optimizing the objective function. If the sum of infeasibilities cannot be reduced to zero, options.min_infeas can be used to find 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 tolerance options.ftol.
Constraint: options.ftol>0.0 .
optim_tol – doubleDefault =ε0.8  
On entry: options.optim_tol defines the tolerance used to determine whether the bounds and generated constraints have the correct sign for the solution to be judged optimal.
reset_ftol – IntegerDefault =5  
On entry: this option is part of an anti-cycling procedure designed to guarantee progress even on highly degenerate problems.
The strategy is to force a positive step at every iteration, at the expense of violating the constraints by a small amount. Suppose that the value of the optional argument options.ftol is δ . Over a period of options.reset_ftol iterations, the feasibility tolerance actually used by nag_opt_qp (e04nfc) increases from 0.5 δ  to δ  (in steps of 0.5 δ / options.reset_ftol).
At certain stages the following ‘resetting procedure’ is used to remove constraint infeasibilities. First, all variables whose upper or lower bounds are in the working set are moved exactly onto their bounds. A count is kept of the number of nontrivial adjustments made. If the count is positive, iterative refinement is used to give variables that satisfy the working set to (essentially) machine precision. Finally, the current feasibility tolerance is reinitialized to 0.5 δ .
If a problem requires more than options.reset_ftol iterations, the resetting procedure is invoked and a new cycle of options.reset_ftol iterations is started with options.reset_ftol incremented by 10. (The decision to resume the feasibility phase or optimality phase is based on comparing any constraint infeasibilities with δ .)
The resetting procedure is also invoked when nag_opt_qp (e04nfc) reaches an apparently optimal, infeasible or unbounded solution, unless this situation has already occurred twice. If any nontrivial adjustments are made, iterations are continued.
Constraint: 0 < options.reset_ftol < 10000000 .
fcheck – IntegerDefault =50  
On entry: every options.fcheck iterations, a numerical test is made to see if the current solution x  satisfies the constraints in the working set. If the largest residual of the constraints in the working set is judged to be too large, the current working set is re-factorized and the variables are recomputed to satisfy the constraints more accurately.
Constraint: options.fcheck1 .
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 for a lower bound less than or equal to -options.inf_bound ).
Constraint: options.inf_bound>0.0 .
inf_step – doubleDefault = maxoptions.inf_bound, 10 20
On entry: options.inf_step 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 not positive definite.) 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 .
hrows – IntegerDefault =n  
On entry: specifies m , the number of rows of the quadratic term H  of the QP objective function. The default value of options.hrows is n , the number of variables of the problem, except that if the problem is specified as type options.prob=Nag_FP or Nag_LP, the default value of options.hrows is zero.
If the problem is of type QP, options.hrows will usually be n , the number of variables. However, a value of options.hrows less than n  is appropriate for options.prob=Nag_QP3 or Nag_QP4 if H  is an upper trapezoidal matrix with m  rows. Similarly, options.hrows may be used to define the dimension of a leading block of nonzeros in the Hessian matrices of options.prob=Nag_QP1 or Nag_QP2, in which case the last n-m  rows and columns of H  are assumed to be zero.
Constraint: 0 options.hrows n .
max_df – IntegerDefault =n  
On entry: places a limit on the storage allocated for the triangular factor R  of the reduced Hessian H r . Ideally, options.max_df should be set slightly larger than the value of n r  expected at the solution. It need not be larger than m n + 1 , where m n  is the number of variables that appear nonlinearly in the quadratic objective function. For many problems it can be much smaller than m n .
For quadratic problems, a minimizer may lie on any number of constraints, so that n r  may vary between 1  and n . The default value is therefore normally n but if the optional argument options.hrows is specified then the default value of options.max_df is set to the value in options.hrows.
Constraint: 1 options.max_df n .
rank_tol – doubleDefault = 100 ε  
On entry: options.rank_tol enables you to control the condition number of the triangular factor R  (see Section 10). If ρ i  denotes the function ρ i = max R 11 , R 22 , , R ii , the dimension 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_qp (e04nfc).
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_qp (e04nfc) from the calling program, using the same values of n and nclin and options.start=Nag_Cold. If a previous call has not been made you must be allocate sufficient memory to options.state.
When a warm start is chosen options.state should specify the desired 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 m lin  elements refer to the general linear constraints (if any). Possible values for options.state[j]  are as follows:
options.state[j] Meaning
0 The corresponding 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 1, 2 or 3 all have the same effect when bl[j] = bu[j] .
The values -2 , -1  and 4 are also acceptable but will be reset to zero by the function. In particular, if nag_opt_qp (e04nfc) 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.
On exit: if nag_opt_qp (e04nfc) exits with a value of fail.code=NE_NOERROR , NW_DEAD_POINT, NW_SOLN_NOT_UNIQUE or NW_NOT_FEASIBLE, the values in options.state indicate the status of the constraints in the working set at the solution. Otherwise, options.state indicates the composition of the working set at the final iterate. 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_DEAD_POINT  or NW_SOLN_NOT_UNIQUE.
ax – double *Default memory =nclin
On entry: nclin values of memory will be automatically allocated by nag_opt_qp (e04nfc) 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_qp (e04nfc) 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 m lin  elements contain the multipliers for the general linear constraints (if any). If options.state[j] = 0  (i.e., constraint j  is not in the working set), options.lambda[j]  is zero. If x  is optimal, options.lambda[j]  should be non-negative if options.state[j] = 1 , non-positive if options.state[j] = 2  and zero if options.state[j] = 4 .
iter – Integer 
On exit: the total number of iterations performed in the feasibility phase and (if appropriate) the optimality phase.
nf – Integer 
On exit: the number of times the product Hx  has been calculated (i.e., number of calls of qphess).

11.3  Description of Printed Output

You can control the level of printed output 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_qp (e04nfc) 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_qp (e04nfc).
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 + m lin  refer to the general constraints. When the status of a constraint changes, the index of the constraint is printed, along with the designation L (lower bound), U (upper bound), E (equality), F (temporarily fixed variable) or A (artificial constraint).
When 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 the iteration count.
Jdel the index of the constraint deleted from the working set. If Jdel is zero, no constraint was deleted.
Jadd the index of the constraint added to the working set. If Jadd is zero, no constraint was added.
Step the step taken along the computed search direction. If a constraint is added during the current iteration (i.e., Jadd is positive), Step will be the step to the nearest constraint. During the optimality phase, the step can be greater than 1.0  only if the reduced Hessian is not positive definite.
Ninf the number of violated constraints (infeasibilities). This will be zero during the optimality phase.
Sinf/Obj 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, Obj 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.
Bnd the number of simple bound constraints in the current working set.
Lin the number of general linear constraints in the current working set.
Nart the number of artificial constraints in the working set, i.e., the number of columns of Z a  (see Section 10). At the start of the optimality phase, Nart provides an estimate of the number of non-positive eigenvalues in the reduced Hessian.
Nrz the number of columns of Z r  (see Section 10). Nrz is the dimension of the subspace in which the objective function is currently being minimized. The value of Nrz is the number of variables minus the number of constraints in the working set; i.e., Nrz = n - Bnd + Lin + Nart .
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 Gz ZrT g fr , the Euclidean norm of the reduced gradient with respect to Z r . 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 information. (Note this longer line extends over more than 80 characters.)
NOpt the number of non-optimal Lagrange multipliers at the current point. NOpt is not printed if the current x  is infeasible or no multipliers have been calculated. At a minimizer, NOpt will be zero.
Min LM the value of the Lagrange multiplier associated with the deleted constraint. If Min LM is negative, a lower bound constraint has been deleted; if Min LM is positive, an upper bound constraint has been deleted. If no multipliers are calculated during a given iteration, Min LM will be zero.
Cond T a lower bound on the condition number of the working set.
Cond Rz a lower bound on the condition number of the triangular factor R  (the Cholesky factor of the current reduced Hessian). If the problem is specified to be of type options.prob=Nag_LP, Cond Rz is not printed.
Rzz the last diagonal element μ  of the matrix D  associated with the RT DR  factorization of the reduced Hessian H r  (see Section 10.2). Rzz is only printed if H r  is not positive definite (in which case μ1 ). If the printed value of Rzz is small in absolute value, then H r  is approximately singular. A negative value of Rzz implies that the objective function has negative curvature on the current working set.
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 the value of x  currently held in x.
State the current value of options.state associated with x .
Value of Ax the value of Ax  currently held in options.ax.
State 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 diagonal of T  and Z r  are also output at each iteration.
When options.print_level=Nag_Soln, Nag_Soln_Iter, Nag_Soln_Iter_Const or Nag_Soln_Iter_Full the final printout from nag_opt_qp (e04nfc) 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 feasibility tolerance, State will be ++ or -- respectively.
Value is the value of the variable at the final iteration.
Lower bound is the lower bound specified for the variable. (None indicates that bl[j-1] -options.inf_bound .)
Upper bound is the upper bound specified for the variable. (None indicates that bu[j-1] options.inf_bound.)
Lagr mult is the value of the Lagrange multiplier for the associated bound constraint. This will be zero if State is FR. 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 bounds bl[j-1]  and bu[j-1] .
The meaning of the printout for general constraints is the same as that given above for variables, with ‘variable’ replaced by ‘constraint’, and with the following change in the heading:
LCon is the name (L) and index j , for j=1,2,, m lin , of the constraint.

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 only 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_qp (e04nfc). 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_qp (e04nfc) are set in the following members of st:
firstNag_Boolean
Nag_TRUE on the first call to options.print_fun.
iterInteger
The number of iterations performed.
nInteger
The number of variables.
nclinInteger
The number of linear constraints.
jdelInteger
Index of constraint deleted.
jaddInteger
Index of constraint added.
stepdouble
The step taken along the current search direction.
ninfInteger
The number of infeasibilities.
fdouble
The value of the current objective function.
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.
nrzInteger
Number of columns of Z r .
norm_gzdouble
Euclidean norm of the reduced gradient, ZrT g fr .
noptInteger
Number of non-optimal Lagrange multipliers.
min_lmdouble
Value of the Lagrange multiplier associated with the deleted constraint.
condtdouble
A lower bound on the condition number of the working set.
xdouble
x points to the n memory locations holding the current point x .
axdouble
options.ax points to the nclin memory locations holding the current values Ax .
stateInteger
options.state points to the n+nclin  memory locations holding the status of the variables and general linear constraints. See Section 11.2 for a description of the possible status values.
tdouble
The upper triangular matrix T  with stlin  columns. Matrix element i,j  is held in stt[ i-1 × sttdt + j - 1 ] .
tdtInteger
The trailing dimension for stt .
If strset = Nag_TRUE then the problem is QP, nag_opt_qp (e04nfc) is executing the optimality phase and the following members of st are also set:
rdouble
The upper triangular matrix R  with stnrz  columns. Matrix element i,j  is held in str[ i-1 × sttdr + j - 1 ] .
tdrInteger
The trailing dimension for str .
condrdouble
A lower bound on the condition number of the triangular factor R .
rzzdouble
Last diagonal element μ  of the matrix D .
If commnew_lm = Nag_TRUE then the Lagrange multipliers have been updated and the following members of st are set:
kxInteger
Indices of the bound constraints with associated multipliers. Value of stkx[i]  is the index of the constraint with multiplier stlambda[i] , for i=0,1,,stbnd - 1.
kactiveInteger
Indices of the linear constraints with associated multipliers. Value of stkactive[i]  is the index of the constraint with multiplier stlambda[ stbnd + i ] , for i=0,1,,stlin - 1.
lambdadouble
The multipliers for the constraints in the working set. options.lambda[i] , for i=0,1,,stbnd - 1, hold the multipliers for the bound constraints while the multipliers for the linear constraints are held at indices i = stbnd , , stbnd + stlin - 1 .
gqdouble
stgq[i] , for i=0,1,,stnart - 1, hold the multipliers for the artificial constraints.
The following members of st are also relevant and apply when commit_prt  or commnew_lm  is Nag_TRUE.
refactorNag_Boolean
Nag_TRUE if iterative refinement performed. See Section 11.2 and optional argument options.reset_ftol.
jmaxInteger
If strefactor = Nag_TRUE then stjmax  holds the index of the constraint with the maximum violation.
errmaxdouble
If strefactor = Nag_TRUE then sterrmax  holds the value of the maximum violation.
movedNag_Boolean
Nag_TRUE if some variables have been moved to their bounds. See the optional argument options.reset_ftol.
nmovedInteger
If stmoved = Nag_TRUE then stnmoved  holds the number of variables which were moved to their bounds.
rowerrNag_Boolean
Nag_TRUE if some constraints are not satisfied to within options.ftol.
feasibleNag_Boolean
Nag_TRUE when a feasible point has been found.
If commsol_prt = Nag_TRUE then the final result from nag_opt_qp (e04nfc) is available and the following members of st are set:
iterInteger
The number of iterations performed.
nInteger
The number of variables.
nclinInteger
The number of linear constraints.
xdouble
x points to the n memory locations holding the final point x .
fdouble
The final objective function value or, if x  is not feasible, the sum of infeasibilities. If the problem is of type options.prob=Nag_FP and x  is feasible then stf is set to zero.
axdouble
stax points to the nclin memory locations holding the final values Ax .
stateInteger
ststate points to the n+nclin  memory locations holding the final status of the variables and general linear constraints. See Section 11.2 for a description of the possible status values.
lambdadouble
stlambda points to the n+nclin  final values of the Lagrange multipliers.
bldouble
stbl points to the n+nclin  lower bound values.
budouble
stbu points to the n+nclin  upper bound values.
endstateNag_EndState
The state of termination of nag_opt_qp (e04nfc). Possible values of stendstate and their correspondence to the exit value of fail.code are:
Value of stendstate  Value of fail.code
Nag_Feasible and Nag_Optimal NE_NOERROR
Nag_Deadpoint and Nag_Weakmin If the problem is QP NW_DEAD_POINT otherwise NW_SOLN_NOT_UNIQUE
Nag_Unbounded NE_UNBOUNDED
Nag_Infeasible NW_NOT_FEASIBLE
Nag_Too_Many_Iter NW_TOO_MANY_ITER
Nag_Hess_Too_Big NE_HESS_TOO_BIG
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. You must allocate memory either before entry to nag_opt_qp (e04nfc) or during a call to qphess or options.print_fun. The type Pointer will be void * with a C compiler that defines void * and char * otherwise.

nag_opt_qp (e04nfc) (PDF version)
e04 Chapter Contents
e04 Chapter Introduction
NAG C Library Manual

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