Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems

Abstract

AbstractThis paper is concerned with the numerical solution of a symmetric indefinite system which is a generalization of the Karush–Kuhn–Tucker system. Following the recent approach of Lukšan and Vlček, we propose to solve this system by a preconditioned conjugate gradient (PCG) algorithm and we devise two indefinite preconditioners with good theoretical properties. In particular, for one of these preconditioners, the finite termination property of the PCG method is stated. The PCG method combined with a parallel version of these preconditioners is used as inner solver within an inexact Interior‐Point (IP) method for the solution of large and sparse quadratic programs. The numerical results obtained by a parallel code implementing the IP method on distributed memory multiprocessor systems enable us to confirm the effectiveness of the proposed approach for problems with special structure in the constraint matrix and in the objective function. Copyright © 2002 John Wiley & Sons, Ltd.