An iterative working-set method for large-scale nonconvex quadratic programming

Authors:
Nicholas I. M. Gould;Philippe L. Toint
Affiliations:
Computational Science and Engineering Department, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, UK;Department of Mathematics, Facultés Universitaires ND de la Paix, 61, rue de Bruxelles, B-5000 Namur, Belgium
Venue:
Applied Numerical Mathematics
Year:
2002

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Abstract

We consider a working-set method for solving large-scale quadratic programming problems for which there is no requirement that the objective function be convex. The methods are iterative at two levels, one level relating to the selection of the current working set, and the second due to the method used to solve the equality-constrained problem for this working set. A preconditioned conjugate gradient method is used for this inner iteration, with the preconditioner chosen especially to ensure feasibility of the iterates. The preconditioner is updated at the conclusion of each outer iteration to ensure that this feasibility requirement persists. The well-known equivalence between the conjugate-gradient and Lanczos methods is exploited when finding directions of negative curvature. Details of an implementation--the Fortran 90 package QPA in the forthcoming GALAHAD library--are given.