Uniform Boundedness of a Preconditioned Normal Matrix Used in Interior-Point Methods

Abstract

Abstract. Solving systems of linear equations with "normal" matrices of the form AD 2 A T is a key ingredient in the computation of search directions for interior-point algorithms. In this article, we establish that a well-known basis preconditioner for such systems of linear equations produces scaled matrices with uniformly bounded condition numbers as D varies over the set of all positive diagonal matrices. In particular, we show that when A is the node-arc incidence matrix of a connected directed graph with one of its rows deleted, then the condition number of the corresponding preconditioned normal matrix is bounded above by m(n − m + 1), where m and n are the number of nodes and arcs of the network.Key words. linear programming, interior-point methods, polynomial bound, network flow problems, condition number, preconditioning, iterative methods for linear equations, normal matrix AMS subject classifications. 65F35, 90C05, 90C35, 90C51 DOI. 10.1137/S1052623403426398 Introduction. Consider the linear programming (LP) problem min{cT x : Ax = b, x ≥ 0}, where A ∈ R m×n has full row rank. Interior-point methods for solving this problem require that systems of linear equations of the form AD 2 A T ∆y = r, where D is a positive diagonal matrix, be solved at every iteration. It often occurs that the "normal" matrix AD 2 A T , while positive definite, becomes increasingly ill-conditioned as one approaches optimality. In fact, it has been proven (e.g., see Kovacevic and Asic [3]) that for degenerate LP problems, the condition number of the normal matrix goes to infinity. Because of the ill-conditioned nature of AD 2 A T , many methods for solving the system AD 2 A T ∆y = r become increasingly unstable. The problem becomes even more serious when conjugate gradient methods are used to solve this linear system. Hence, the development of suitable preconditioners which keep the condition number of the coefficient matrix of the scaled system under control is of paramount importance. We should note, however, that in practice the ill-conditioning of AD 2 A T generally does not cause difficulty when the system AD 2 A T ∆y = r is solved using a backward-stable direct solver (see, e.g., [12,13] and references therein).In this paper, we analyze a preconditioner for the normal matrix AD 2 A T that has been proposed by Resende and Veiga [7] in the context of the minimum cost network flow problem and subsequently by Oliveira and Sorensen [6] for general LP problems. The preconditioning consists of pre-and postmultiplying AD 2 A T by D

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