Characterizing bad semidefinite programs: normal forms and short proofs

TL;DR

This paper introduces a simple, elementary normal form for semidefinite programs (SDPs) that helps identify pathological cases where primal and dual solutions differ or are unattainable, enhancing understanding and verification of SDPs.

Contribution

It provides a new, easy-to-apply normal form for SDPs based on elementary row operations, simplifying the detection of pathological behaviors and the analysis of linear maps on symmetric matrices.

Findings

Normal form makes pathology verification straightforward

Characterization of pathological SDPs via excluded matrices

Method to check if the image of the semidefinite cone under a linear map is closed

Abstract

Semidefinite programs (SDPs) -- some of the most useful and versatile optimization problems of the last few decades -- are often pathological: the optimal values of the primal and dual problems may differ and may not be attained. Such SDPs are both theoretically interesting and often impossible to solve; yet, the pathological SDPs in the literature look strikingly similar. Based on our recent work \cite{Pataki:17} we characterize pathological semidefinite systems by certain {\em excluded matrices}, which are easy to spot in all published examples. Our main tool is a normal (canonical) form of semidefinite systems, which makes their pathological behavior easy to verify. The normal form is constructed in a surprisingly simple fashion, using mostly elementary row operations inherited from Gaussian elimination. The proofs are elementary and can be followed by a reader at the advanced undergraduate level. As a byproduct, we show how to transform any linear map acting on symmetric matrices into a normal form, which allows us to quickly check whether the image of the semidefinite cone under the map is closed. We can thus introduce readers to a fundamental issue in convex analysis: the linear image of a closed convex set may not be closed, and often simple conditions are available to verify the closedness, or lack of it.