An echelon form of weakly infeasible semidefinite programs and bad projections of the psd cone

TL;DR

This paper introduces a simple echelon form for weakly infeasible semidefinite programs, making their infeasibility evident and enabling the construction of challenging instances and analysis of bad projections of the PSD cone.

Contribution

It presents a new echelon form for weakly infeasible SDPs obtained via elementary operations, facilitating their analysis and generation of difficult problem instances.

Findings

Echelon form makes weak infeasibility evident

Allows construction of all weakly infeasible SDPs

Identifies SDPs in literature, e.g., Motzkin polynomial relaxation

Abstract

A weakly infeasible semidefinite program (SDP) has no feasible solution, but it has approximate solutions whose constraint violation is arbitrarily small. These SDPs are ill-posed and numerically often unsolvable. They are also closely related to "bad" linear projections that map the cone of positive semidefinite matrices to a nonclosed set. We describe a simple echelon form of weakly infeasible SDPs with the following properties: (i) it is obtained by elementary row operations and congruence transformations, (ii) it makes weak infeasibility evident, and (iii) it permits us to construct any weakly infeasible SDP or bad linear projection by an elementary combinatorial algorithm. Based on our echelon form we generate a challenging library of weakly infeasible SDPs. Finally, we show that some SDPs in the literature are in our echelon form, for example, the SDP from the sum-of-squares relaxation of minimizing the famous Motzkin polynomial.