An Iterative Solver-Based Infeasible Primal-Dual Path-Following Algorithm for Convex Quadratic Programming

Abstract

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.

Matrix-free interior point method

“…From (29) we have (λI − R d )v = Au. We substitute for v in (28) and multiply the equation with u T to obtain…”

“…The use of inexact Newton method goes back to Dembo et al [13] and has had a number of applications including those in the context of IPMs [4,19,27]. A number of interesting developments have been focused on the analysis of conditions which inexact directions should satisfy to guarantee good convergence properties of the IPM [1,28]. However the focus of this paper lies elsewhere.…”

An iterative solver-based long-step infeasible primal-dual path-following algorithm for convex QP based on a class of preconditioners

“…In [15], the authors also developed an inexact IPDPF algorithm for solving Equation (1) with Q assumed to be given in the form Q = V E 2 V T , or equivalently Q = 0. This inexact IPDPF algorithm is essentially the long-step IPDPF algorithm in [10,28], the only difference being that the search directions are computed by means of an iterative linear solver.…”

“…Since the ANE is solved only approximately, it cannot yield a search direction that satisfies all equations of the primal-dual Newton system. Thus, we developed a recipe in [15] for determining an inexact search direction, based on an approximate solution to the ANE and the MWB preconditioner, which accomplishes the following two goals: (i) problem (1) can be solved within a polynomial number of iterations, and (ii) the required approximate solution to the ANE can be found within a uniformly bounded number of inner iterations. This paper extends the authors'previous work [15] in the following two ways.…”

“…We refer to the iterations of the iterative linear solver as the inner iterations, and the iterations performed by the actual path-following method as the outer iterations. The main step in the inexact IPDPF algorithm in [15] is the computation of a primal-dual search direction ( x, s, y, z), whose subvector ( y, z) can be found by solving the so-called augmented normal equation, or ANE. This ANE is of the formÃD 2ÃT ( y, z) = g, whereD is a positive diagonal matrix, andà is a 2 × 2 block matrix whose blocks consist of A, V T , the zero matrix, and the identity matrix.…”