Convergence of Interior Point Algorithms for the Monotone Linear Complementarity Problem

Abstract

The literature on interior point algorithms shows impressive results related to the speed of convergence of the objective values, but very little is known about the convergence of the iterate sequences. This paper studies the horizontal linear complementarity problem, and derives general convergence properties for algorithms based on Newton iterations. This problem provides a simple and general framework for most existing primal-dual interior point methods. The conclusion is that most of the published algorithms of this kind generate convergent sequences. In many cases (whenever the convergence is not too fast in a certain sense), the sequences converge to the analytic center of the optimal face.

Q-superlinear convergence of the iterates in primal-dual interior-point methods

“…The superlinear convergence of the duality gap for the feasible largest step algorithm was first proved by McShane [12] for nondegenerate monotone SLCPs under the assumption that the iteration sequence converges. The results of Bonnans and Gonzaga [2] show that this assumption holds. The largest step method was extended for the infeasible case by Bonnans and Potra [4].…”

“…Gonzaga and Tapia [10] also proposed a modification of the MTY predictor-corrector method (called the simplified MTY method) that asymptotically requires only one matrix factorization per iteration while retaining the O( √ n L) iteration complexity and the Q-quadratic convergence of the duality gap, and showed that the iteration sequence produced by this method converges, but not necessarily to the analytic center. The above results about the convergence of the iteration sequence were extended for a very general class of feasible interior-point methods by Bonnans and Gonzaga [2] and generalized for infeasible interior-point methods by Bonnans and Potra [4]. From the last two papers it follows that virtually all interior-point methods using an l 2 neighborhood of the central path produce convergent iteration sequences.…”

“…It is known (see, e.g. [2]) that this problem trivially includes the linear programming problem (LP) and the convex quadratic programming problem (QP) in their usual formulations, and thus provides a quite general framework for the study of algorithms. Of course any LP and QP can also be written as a standard linear complementarity problem (SLCP) which is an HLCP where R is the identity matrix and −Q is positive semidefinite.…”

On the Local Convergence of a Predictor-Corrector Method for Semidefinite Programming

“…MTY was generalized to (monotone) LCP in [8], and the resulting algorithm was proved to have O( √ nL) iteration complexity under general conditions and superlinear convergence, under the assumption that the LCP has a strictly complementary condition (i.e., when the LCP is nondegenerate) and the iteration sequence converges. From [3] it follows that the latter assumption always holds. Subsequently Ye and Anstreicher [19] proved that MTY converges quadratically assuming only that the LCP is nondegenerate.…”

“…Its extension for the HLCP is immediate, using for example the equivalences proved in [1] (see also [3]). …”