A structural geometrical analysis of weakly infeasible SDPs

TL;DR

This paper provides a geometric analysis of weakly infeasible semidefinite programming problems, revealing how such infeasibility arises and how it can be characterized through a decomposition approach.

Contribution

It introduces a decomposition technique for SDFPs that preserves feasibility properties and offers a systematic understanding of weak infeasibility in semidefinite programming.

Findings

At most n-1 directions are needed to approach the positive semidefinite cone in n×n matrices.

Provides a detailed geometric characterization of weak infeasibility in SDFPs.

Discusses feasibility certificates and complexity aspects of SDFPs.

Abstract

In this article, we present a geometric theoretical analysis of semidefinite feasibility problems (SDFPs). This is done by decomposing a SDFP into smaller problems, in a way that preserves most feasibility properties of the original problem. With this technique, we develop a detailed analysis of weakly infeasible SDFPs to understand clearly and systematically how weak infeasibility arises in semidefinite programming. In particular, we show that for a weakly infeasible problem over $n\times n$ matrices, at most $n-1$ directions are required to approach the positive semidefinite cone. We also present a discussion on feasibility certificates for SDFPs and related complexity results.