intnza (Input) Number of nonzero entries in constraint matrix A.
int nzq (Input) Number of nonzero entries in the lower triangular part of the matrix Q of the objective function.
Imsl_d_sparse_elema[] (Input) An array of length nza containing the location and value of each nonzero coefficient in the constraint matrix A.
doubleb[] (Input) An array of length m containing the right-hand side of the constraints; if there are limits on both sides of the constraints, then b contains the lower limit of the constraints.
doublec[] (Input) An array of length n containing the coefficients of the linear term of the objective function.
Imsl_d_sparse_elem q[] (Input) Array of length nzq containing the location and value of each nonzero coefficient in the lower triangular part of the matrix Q of the objective function. The matrix must be symmetric positive semidefinite.
Return Value
A pointer to an array of length n containing the solution x of the convex QP problem. To release this space, use imsl_free. If no solution can be computed, then NULL is returned.
IMSL_CONSTR_TYPE, intirtype[] (Input) An array of length m indicating the types of general constraints in the matrix A. Let ri = ai1x1 + … + ainxn. Then, the value of irtype[i] signifies the following:
irtype[i]
Constraint
0
ri =bi
1
ri≤bi
2
ri≥bi
3
bi≤ri≤bui
4
Ignore this constraint
Note that irtype[i] = 3 should only be used for constraints i with both finite lower and finite upper bound. For one-sided constraints, use irtype[i] = 1 or irtype[i] = 2. For free constraints, use irtype[i] = 4. Default: irtype = 0
IMSL_UPPER_LIMIT, doublebu[] (Input) An array of length m containing the upper limit of the constraints that have both a lower and an upper bound. If such a constraint exists, then optional argument IMSL_CONSTR_TYPE must be used to define the type of constraints. If irtype[i]≠ 3, i.e. if constraint i is not two-sided, then the corresponding entry in bu, bu[i], is ignored. Default: None of the constraints has an upper bound.
IMSL_LOWER_BOUND, doublexlb[] (Input) An array of length n containing the lower bound on the variables. If there is no lower bound on a variable, then −1030 should be set as the lower bound. Default: xlb = 0.
IMSL_UPPER_BOUND, doublexub[] (Input) An array of length n containing the upper bound on the variables. If there is no upper bound on a variable, then 1030 should be set as the upper bound. Default: None of the variables has an upper bound.
IMSL_OBJ_CONSTANT, doublec0 (Input) Value of the constant term in the objective function. Default: c0 = 0.
IMSL_PREORDERING, intpreorder (Input) The variant of the Minimum Degree Ordering (MDO) algorithm used in the preordering of the normal equations or augmented system matrix:
preorder
Method
0
A variant of the MDO algorithm using pivotal cliques.
1
A variant of George and Liu’s Quotient Minimum Degree algorithm.
Default: preorder = 0.
IMSL_MAX_ITERATIONS, intmax_iterations (Input) The maximum number of iterations allowed for the primal-dual solver. Default: max_iterations = 200.
Prints statistics on the QP problem and the solution process.
Default: iprint = 0.
IMSL_PRESOLVE, intpresolve (Input) Presolve the QP problem in order to reduce the problem size or to detect infeasibility or unboundedness of the problem. Depending on the number of presolve techniques used different presolve levels can be chosen:
presolve
Description
0
No presolving.
1
Eliminate singleton rows
2
Additionally to 1, eliminate redundant (and forcing) rows.
3
Additionally to 2, eliminate dominated variables.
4
Additionally to 3, eliminate singleton columns.
5
Additionally to 4, eliminate doubleton rows.
6
Additionally to 5, eliminate aggregate columns.
Default: presolve = 0.
IMSL_CSC_FORMAT, inta_colptr[], inta_rowind[], doublea_values[], intq_colptr[], intq_rowind[], doubleq_values[] (Input) Accept the constraint matrix A (via vectors a_colptr, a_rowind and a_values) and the matrix Q of the objective function (via vectors q_colptr, q_rowind and q_values) in Harwell-Boeing format. See (Compressed Sparse Column (CSC) Format) in the Introduction to this manual for a discussion of this storage scheme.
If this optional argument is used, then required arguments a and q are ignored.
IMSL_TERMINATION_STATUS, int*status (Output) The termination status for the problem.
status
Description
0
Optimal solution found.
1
The problem is primal infeasible (or dual unbounded).
2
The problem is primal unbounded (or dual infeasible).
3
Suboptimal solution found (accuracy problems).
4
Iterations limit max_iterations exceeded.
5
An error outside of the solution phase of the algorithm, e.g. a user input or a memory allocation error.
IMSL_OBJ, double*obj (Output) Optimal value of the objective function.
IMSL_ITERATION_COUNT, int*iterations (Output) The number of iterations required by the primal-dual solver.
IMSL_DUAL, double**y (Output) The address of a pointer y to an internally allocated array of length m containing the dual solution.
IMSL_DUAL_USER, doubley[] (Output) A user-allocated array of size m containing the dual solution.
IMSL_PRIMAL_INFEAS, double*err_b, double*err_u (Output) The violation of the primal constraints, described by err_b, the primal infeasibility of the solution, and by err_u, the violation of the variable bounds.
IMSL_DUAL_INFEAS, double*err_c (Output) The violation of the dual constraints, described by err_c, the dual infeasibility of the solution.
IMSL_CP_RATIO_SMALLEST, double*cp_smallest (Output) The ratio of the smallest complementarity product to the average.
IMSL_CP_RATIO_LARGEST, double*cp_largest (Output) The ratio of the largest complementarity product to the average.
IMSL_RETURN_USER, doublex[] (Output) A user-allocated array of length n containing the primal solution.
Description
The function imsl_d_sparse_quadratic_prog uses an infeasible primal-dual interior-point method to solve convex quadratic programming problems, i.e., problems of the form
where c is the objective coefficient vector, Q is the symmetric positive semidefinite coefficient matrix, A is the constraint matrix and the vectors bl, bu, xl, and xu are the lower and upper bounds on the constraints and the variables, respectively.
Internally, imsl_d_sparse_quadratic_prog transforms the problem given by the user into a simpler form that is computationally more tractable. After redefining the notation, the new form reads
Here, is a partition of the index set into upper bounded and standard variables.
In order to simplify the description it is assumed in the following that the problem above contains only variables with upper bounds, i.e. is of the form
The corresponding dual problem is then
The Karush-Kuhn-Tucker (KKT) optimality conditions for (P) and (D) are
where are diagonal matrices and is a vector of ones.
Function imsl_d_sparse_quadratic_prog, like all infeasible interior point methods, generates a sequence
of iterates, that satisfy for all , but are in general not feasible, i.e. the linear constraints (1.1)-(1.3) are only satisfied in the limiting case .
The barrier parameter , defined by
measures how good the complementarity conditions (1.4), (1.5) are satisfied.
Mehrotra’s predictor-corrector algorithm is a variant of Newton’s method applied to the KKT conditions (1.1)-(1.5). Function imsl_d_sparse_quadratic_prog uses a modified version of this algorithm to compute the iterates . In every step of the algorithm, the search direction vector
is decomposed into two parts, where and denote the affine-scaling and a weighted centering component, respectively. Here,
where the scalar denotes the corrector weight.
The vectors and are determined by solving the linear system
for two different right-hand sides.
For , the right-hand side is defined as
Here, and are the violations of the primal constraints and defines the violations of the dual constraints.
The resulting direction is the pure Newton step applied to the system (1.1)-(1.5).
In order to obtain the corrector direction , the maximum stepsize in the primal and dual space preserving nonnegativity of , is determined and the predicted complementarity gap
is computed. It is then used to determine the barrier parameter
where denotes the current complementarity gap.
The direction is then computed by choosing
as the right-hand side in the linear system (2).
Function imsl_d_sparse_quadratic_prog now uses a linesearch to find the optimal weight that maximizes the stepsize in the primal and dual direction of .
A new iterate is then computed using a step reduction factor :
In addition to the weighted Mehrotra predictor-corrector, imsl_d_sparse_quadratic_prog also uses multiple centrality correctors to enlarge the primal-dual stepsize per iteration step and to reduce the overall number of iterations required to solve a QP problem. The maximum number of centrality corrections depends on the ratio of the factorization and solve efforts for system (2) and is therefore problem dependent. For a detailed description of multiple centrality correctors, refer to Gondzio(1994).
The linear system (2) can be reduced to more compact forms, the augmented system (AS)
or further by elimination of to the normal equations (NE) system
where
The matrix on the left-hand side of (3), which is symmetric indefinite, can be transformed into a symmetric quasidefinite matrix by regularization. Since these types of matrices allow for a Cholesky-like factorization, the resulting linear system can be solved easily for by triangular substitutions. For more information on the regularization technique, see Altman and Gondzio (1998). For the NE system, matrix is positive definite, and therefore a sparse Cholesky algorithm can be used to factor and solve the system for by triangular substitutions with the Cholesky factor .
In function imsl_d_sparse_quadratic_prog, both approaches are implemented. The AS approach is chosen if contains dense columns, if there is a considerable number of columns in that are much denser than the remaining ones or if there are many more rows than columns in the structural part of . Otherwise, the NE approach is selected.
Function imsl_d_sparse_quadratic_prog stops with optimal termination status if the current iterate satisfies the following three conditions:
where prinf_tol, dlinf_tol and opt_tol are primal infeasibility, dual infeasibility and optimality tolerances, respectively. The default value is 1.0e-10 for opt_tol and 1.0e-8 for the two other tolerances.
Function imsl_d_sparse_quadratic_prog is based on the code HOPDM developed by Jacek Gondzio et al., see the HOPDM User’s Guide (1995).
Examples
Example 1
The convex quadratic programming problem
is solved.
#include <imsl.h>
#include <stdio.h>
int main()
{
int m = 2, n = 3, nza = 5, nzq = 4;
Imsl_d_sparse_elem a[] = { 0, 0, 2.0,
0, 1, 1.0,
0, 2, -8.0,
1, 0, 2.0,
1, 1, 3.0 };
Imsl_d_sparse_elem q[] = { 0, 0, 2.0,
1, 1, 32.0,
2, 2, 4.0,
1, 0, -4.0 };
double b[] = { 0.0, 6.0 };
double c[] = { 10.0, 0.0, 3.0 };
double xlb[] = { 0.0, -3.0, -5.0 };
double xub[] = { 7.0, 2.0, 20.0 };
int irtype[] = { 2, 1 };
double *x = NULL;
double obj;
x = imsl_d_sparse_quadratic_prog(m, n, nza, nzq, a, b, c, q,
IMSL_CONSTR_TYPE, irtype,
IMSL_LOWER_BOUND, xlb,
IMSL_UPPER_BOUND, xub,
IMSL_OBJ, &obj,
0);
imsl_d_write_matrix("x", 1, n, x, 0);
printf("\nObjective: %lf\n", obj);
}
Output
x
1 2 3
0.00 0.00 -0.75
Objective: -1.125000
Example 2
This example demonstrates how the function imsl_d_read_mps can be used with imsl_d_sparse_quadratic_prog to solve a convex quadratic programming problem defined in an MPS file. The MPS file used in this example is the file ‘qafiro’, available from the QP problems collection QPDATA2 on István Maros’ home page under http://www.doc.ic.ac.uk/~im/#DATA/.
#include <imsl.h>
#include <stdio.h>
#include <stdlib.h>
int main()
{
Imsl_d_mps *problem;
int i, m, n, *irtype, nza, nzq;
double *x, objective, *bl, *bu, *xlb, *xub;
Imsl_d_sparse_elem *a = NULL, *q = NULL;
/* Read the QPS file. */
problem = imsl_d_read_mps("QAFIRO.QPS", 0);
m = problem->nrows;
n = problem->ncolumns;
/*
* Setup the constraint matrix.
*/
nza = problem->nonzeros;
a = problem->constraint;
/*
* Setup the Hessian.
*/
nzq = problem->nhessian;
q = problem->hessian;
/*
* Setup constraint bounds and constraint type array.
*/
irtype = (int*) malloc(m*sizeof(int));
bl = (double*) malloc(m*sizeof(double));
bu = (double*) malloc(m*sizeof(double));
for (i = 0; i < m; i++) {
if (problem->lower_range[i] == problem->negative_infinity &&
