Primal-Dual Affine Scaling Interior Point Methods for Linear Complementarity Problems

Abstract

Abstract. A first order affine scaling method and two mth order affine scaling methods for solving monotone linear complementarity problems (LCP) are presented. All three methods produce iterates in a wide neighborhood of the central path. The first order method has O(nL 2 (log nL 2 )(log log nL 2 )) iteration complexity. If the LCP admits a strict complementary solution then both the duality gap and the iteration sequence converge superlinearly with Q-order two. If m = Ω(log( √ nL)), then both higher order methods have O( √ n)L iteration complexity. The Q-order of convergence of one of the methods is (m + 1) for problems that admit a strict complementarity solution while the Q-order of convergence of the other method is (m + 1)/2 for general monotone LCPs.

On a Class of Superlinearly Convergent Polynomial Time Interior Point Methods for Sufficient LCP

“…When σ = 0 the algorithms from this class reduce to the corresponding algorithms from [27], while for σ > 0 they reduce to a variant of the algorithm of [34]. The main contribution of the paper is that it generalizes the results of [27] to sufficient LCP with infeasible starting points and it proposes a variant of the algorithm of [34] that has both superlinear convergence and polynomial complexity. Moreover, we hope that our analysis will also contribute to the better understanding of the behavior of interior point methods in the large neighborhood of the central path.…”

“…In a recent paper [27], the first author has analyzed several feasible interior point methods for the monotone LCP acting in the N − ∞ neighborhood of the central path that require one matrix factorization and m backsolves per iteration and have Q-order m + 1 in the nondegenerate case and (m + 1)/2 in the degenerate case, the same as the algorithm proposed by the second author in [34]. The algorithms from [27] and [34] share the property that the N − ∞ neighborhood of the central path is expanded at each iteration.…”

“…In the present paper we propose a class of polynomial time infeasible interior point methods for sufficient LCP that have the same Q-order and computational cost per iteration as the methods from [34] and [27] (i.e., the same asymptotic efficiency index). The class depends on a parameter σ ≥ 0.…”

“…The rate of expansion used in [34] ensures the high Q-order results mentioned above, but does not guarantee polynomial complexity. However by using the rate of expansion from [27] it is possible to modify the method of [34] in such a way that Q-order results are retained and polynomial complexity is guaranteed.…”

“…The algorithms from [27] and [34] share the property that the N − ∞ neighborhood of the central path is expanded at each iteration. The rate of expansion used in [34] ensures the high Q-order results mentioned above, but does not guarantee polynomial complexity.…”