An exact duality theory for semidefinite programming based on sums of squares

TL;DR

This paper develops a new exact duality theory for semidefinite programming using sums of squares and real algebraic geometry, providing certificates for infeasibility in all cases.

Contribution

It introduces nonlinear algebraic certificates for infeasibility and a novel duality framework based on sums of squares, extending beyond strongly infeasible cases.

Findings

Provides algebraic certificates for all infeasible LMIs

Establishes an exact duality theory for SDP

Connects SDP infeasibility with real algebraic geometry concepts

Abstract

Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality is infeasible if and only if -1 lies in the quadratic module associated to it. We also present a new exact duality theory for semidefinite programming, motivated by the real radical and sums of squares certificates from real algebraic geometry.