Infeasibility certificates for linear matrix inequalities

Igor Klep (1, 2) , Markus Schweighofer (3)

1 Fakulteta za naravoslovje in matematiko
2 Fakulteta za matematiko in fiziko
3 Fachbereich Mathematik und Statistik

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. More precisely, we show that a linear matrix inequality is infeasible if and only if -1 lies in the quadratic module associated to it. We prove exponential degree bounds for the corresponding algebraic certificate. In order to get a polynomial size certificate, 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.

Mots clés: linear matrix inequality; LMI; spectrahedron; semidefinite program; quadratic module; infeasibility; duality; complete positivity; Farkas' lemma

Domaines: Optimisation et contrôle [math.OC]; Algèbres d'opérateurs [math.OA]; Géométrie algébrique [math.AG]

Igor Klep, Markus Schweighofer. Infeasibility certificates for linear matrix inequalities. 2011. ⟨hal-00617640⟩