Irreducible Infeasible Subsystems of Semidefinite Systems

  • Kai Kellner
  • Marc E. Pfetsch
  • Thorsten TheobaldEmail author


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.


Semidefinite system Irreducible infeasible subsystem Alternative system Spectrahedron 

Mathematics Subject Classification

52A20 90C22 14P05 



We thank the anonymous referees for helpful suggestions.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Frankfurt am MainGermany
  2. 2.Department of MathematicsTU DarmstadtDarmstadtGermany
  3. 3.FB 12 – Institut für MathematikGoethe-UniversitätFrankfurt am MainGermany

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