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 Ax ≤K b badly behaved if the value of sup { c, x |Ax ≤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 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 N P ∩ co-N P 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.

A Structural Geometrical Analysis of Weakly Infeasible SDPS

“…See the excellent article by Pataki [15] for an elementary exposition of FRA, where he points out the relation between facial reduction and extended duals. Pataki also found that all "badly behaved" semidefinite programs can be reduced to a common 2 × 2 semidefinite program [16]. Finally, we mention that Waki showed that weakly infeasible instances can be obtained from the semidefinite relaxation of polynomial optimization problems [25].…”

Characterizing Bad Semidefinite Programs: Normal Forms and Short Proofs

“…Our recent paper [28] was motivated by the curious similarity of pathological SDPs in the literature. To build intuition, we recall two examples; they or their variants appear in a number of papers and surveys.…”

“…Then V is in the closure of dir(Z, S n + ), but it is not a feasible direction (see [28,Lemma 3]). That is, for small > 0 the matrix Z + V is "almost" psd, but not quite.…”

Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs

“…Basic-Recovery is inspired by the dual solution recovery procedure in [35], which builds on the ideas in [32], and it assumes that the dual problem (D) is reduced 5 .…”