Infeasibility certificates for linear matrix inequalities

Abstract

Farkas' lemma is a fundamental result from linear programming providing linear certi cates 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. More precisely, we show that a linear matrix inequality $L(x) \succeq 0$ is infeasible if and only if $-1$ lies in the quadratic module associated to $L$. We prove exponential degree bounds for the corresponding algebraic certificate. In order to get a polynomial size certi cate, we use a more involved algebraic certificate motivated by the real radical and Prestel's theory of semiorderings. Completely different methods, namely complete positivity from operator algebras, are employed to consider linear matrix inequality domination.