Exact duality in semidefinite programming based on elementary reformulations

TL;DR

This paper introduces an elementary reformulation approach to establish exact duality and infeasibility certificates in semidefinite programming, overcoming limitations of Farkas' lemma in SDP.

Contribution

It provides a simple, elementary method to reformulate SDP systems, ensuring strong duality or infeasibility certificates without complex procedures.

Findings

Elementary reformulations yield exact infeasibility certificates.

Strong duality holds for reformulated systems with feasible SDPs.

Algorithms are developed for identifying all infeasible and certain feasible SDPs.

Abstract

In semidefinite programming (SDP), unlike in linear programming, Farkas' lemma may fail to prove infeasibility. Here we obtain an exact, short certificate of infeasibility in SDP by an elementary approach: we reformulate any semidefinite system of the form Ai*X = bi (i=1,...,m) (P) X >= 0 using only elementary row operations, and rotations. When (P) is infeasible, the reformulated system is trivially infeasible. When (P) is feasible, the reformulated system has strong duality with its Lagrange dual for all objective functions. As a corollary, we obtain algorithms to generate the constraints of {\em all} infeasible SDPs and the constraints of {\em all} feasible SDPs with a fixed rank maximal solution.