Irreducible infeasible subsystems of semidefinite systems

TL;DR

This paper extends Farkas' lemma for semidefinite programming to characterize infeasible subsystems, exploring the structure of extreme points in spectrahedra and providing criteria for solution uniqueness in block systems.

Contribution

It generalizes a known theorem from linear programming to semidefinite systems, clarifies limitations, and offers new criteria for solution uniqueness in block systems.

Findings

Generalization of Gleeson and Ryan's theorem to spectrahedra

Counterexamples showing the reverse implication does not hold

Criterion for uniqueness of solutions in semidefinite block systems

Abstract

Farkas' lemma for semidefinite programming characterizes semidefinite feasibility of linear matrix pencils in terms of an alternative spectrahedron. In the well-studied special case of linear programming, a theorem by Gleeson and Ryan states that the index sets of irreducible infeasible subsystems are exactly the supports of the vertices of the corresponding alternative polyhedron. We show that one direction of this theorem can be generalized to the nonlinear situation of extreme points of general spectrahedra. The reverse direction, however, is not true in general, which we show by means of counterexamples. On the positive side, an irreducible infeasible block subsystem is obtained whenever the extreme point has minimal block support. Motivated by results from sparse recovery, we provide a criterion for the uniqueness of solutions of semidefinite block systems.