Bad Semidefinite Programs: They All Look the Same

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 ${\cal A} x \leq_K b$ badly behaved if the value of $\sup \, \{ \langle c, x \rangle | {\cal A} x \leq_K b \}$ is finite but the dual program has no solution with the same value for 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 excluded matrices, which are easy to spot in all published examples. We prove several other structural results about badly behaved semidefinite systems; for example, we show that they are in ${\cal NP} \cap \mbox{{\rm co-}}{\cal NP}$ in the real number model of computing.