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

Abstract. In this paper we develop a long-step primal-dual infeasible path-following algorithm for convex quadratic programming (CQP) whose search directions are computed by means of a preconditioned iterative linear solver. We propose a new linear system, which we refer to as the augmented normal equation (ANE), to determine the primal-dual search directions. Since the condition number of the ANE coefficient matrix may become large for degenerate CQP problems, we use a maximum weight basis preconditioner introduced in [A. R. L. Oliveira and D. C. Sorensen, Linear Algebra Appl., 394 (2005) (2004), pp. 96-100], we establish a uniform bound, depending only on the CQP data, for the number of iterations needed by the iterative linear solver to obtain a sufficiently accurate solution to the ANE. Since the iterative linear solver can generate only an approximate solution to the ANE, this solution does not yield a primal-dual search direction satisfying all equations of the primal-dual Newton system. We propose a way to compute an inexact primal-dual search direction so that the equation corresponding to the primal residual is satisfied exactly, while the one corresponding to the dual residual contains a manageable error which allows us to establish a polynomial bound on the number of iterations of our method.

In this paper, we present a long-step infeasible primal-dual path-following algorithm for convex quadratic programming (CQP) whose search directions are computed by means of a preconditioned iterative linear solver. In contrast to the authors' previous paper (2006), pp. 287-310] is that, instead of using the maximum weight basis (MWB) preconditioner in the aforesaid recipe for constructing the inexact search direction, this paper proposes the use of any member of a whole class of preconditioners, of which the MWB preconditioner is just a special case. The proposed recipe allows us to: (i) establish a polynomial bound on the number of iterations performed by our path-following algorithm and (ii) establish a uniform bound, depending on the quality of the preconditioner, on the number of iterations performed by the iterative solver.

Social network analysis (SNA) is a rapidly growing field with numerous applications in industry and government. However, the field still lacks means to generate random social networks with certain desired properties, thus inhibiting their ability to test new SNA algorithms and metrics. Available random graph generation algorithms suffer from tendencies to generate disconnected graphs and sometimes induce undesirable network properties. In this paper, we present an algorithm, the prescribed node degree, connected graph (PNDCG) algorithm, designed to generate weakly connected social networks. Extensions to the PNDCG algorithm allow one to create random graphs that control the clustering coefficient and degree correlation within the generated networks. Empirical test results demonstrate the capability of the PNDCG algorithm to produce networks with the desired properties.