Bad semidefinite programs: they all look the same

TL;DR

This paper analyzes pathological behaviors in semidefinite programs, providing geometric characterizations, canonical forms, and criteria to identify badly behaved systems, which are common in literature and have implications for the structure of linear maps on symmetric matrices.

Contribution

It introduces a geometric characterization of badly behaved conic linear systems, especially semidefinite programs, and provides methods to transform and verify their behavior.

Findings

Characterization of badly behaved systems via excluded matrices

Canonical form transformation for semidefinite systems

Proof that such systems are in NP ∩ co-NP

Abstract

Conic linear programs, among them semidefinite programs, often behave pathologically: the optimal values of the primal and dual programs may differ, and may not be attained. We present a novel analysis of these pathological behaviors. We call a conic linear system $Ax <= b$ {\em badly behaved} if the value of $\sup { < c, x > | A x <= b }$ is finite but the dual program has no solution with the same value for {\em some} $c.$ We describe simple and intuitive geometric characterizations of badly behaved conic linear systems. Our main motivation is the striking similarity of badly behaved semidefinite systems in the literature; we characterize such systems by certain {\em excluded matrices}, which are easy to spot in all published examples. We show how to transform semidefinite systems into a canonical form, which allows us to easily verify whether they are badly behaved. We prove several other structural results about badly behaved semidefinite systems; for example, we show that they are in $NP \cap co-NP$ in the real number model of computing. As a byproduct, we prove that all linear maps that act on symmetric matrices can be brought into a canonical form; this canonical form allows us to easily check whether the image of the semidefinite cone under the given linear map is closed.