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Irreducible Infeasible Subsystems of Semidefinite Systems

Author

Listed:
  • Kai Kellner
  • Marc E. Pfetsch

    (TU Darmstadt)

  • Thorsten Theobald

    (Goethe-Universität)

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.

Suggested Citation

  • Kai Kellner & Marc E. Pfetsch & Thorsten Theobald, 2019. "Irreducible Infeasible Subsystems of Semidefinite Systems," Journal of Optimization Theory and Applications, Springer, vol. 181(3), pages 727-742, June.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:3:d:10.1007_s10957-019-01480-4
    DOI: 10.1007/s10957-019-01480-4
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    References listed on IDEAS

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    Cited by:

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