Superlinear Convergence of an Interior Point Algorithm on Linear Semi-definite Feasibility Problems

TL;DR

This paper introduces a practical initial point for an infeasible interior point method that guarantees superlinear convergence when solving linear semi-definite feasibility problems, improving implementation practicality.

Contribution

It proposes a new practical initial iterate for an infeasible interior point algorithm that ensures superlinear convergence on LSDFPs, addressing previous limitations.

Findings

Guarantees superlinear convergence with the new initial point.

Applicable to homogeneous feasibility models of LSDFPs.

Enhances practicality of interior point methods for SDPs.

Abstract

In the literature, besides the assumption of strict complementarity, superlinear convergence of implementable polynomial-time interior point algorithms using known search directions, namely, the HKM direction, its dual or the NT direction, to solve semi-definite programs (SDPs) is shown by (i) assuming that the given SDP is nondegenerate and making modifications to these algorithms [10], or (ii) considering special classes of SDPs, such as the class of linear semi-definite feasibility problems (LSDFPs) and requiring the initial iterate to the algorithm to satisfy certain conditions [26, 27]. Otherwise, these algorithms are not easy to implement even though they are shown to have polynomial iteration complexities and superlinear convergence [14]. The conditions in [26, 27] that the initial iterate to the algorithm is required to satisfy to have superlinear convergence when solving LSDFPs however are not practical. In this paper, we propose a practical initial iterate to an implementable infeasible interior point algorithm that guarantees superlinear convergence when the algorithm is used to solve the homogeneous feasibility model of an LSDFP.