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Parametric Semidefinite Programming: Geometry of the Trajectory of Solutions

Author

Listed:
  • Antonio Bellon

    (Faculty of Electrical Engineering, Czech Technical University, 166 36 Prague 6, Czechia)

  • Didier Henrion

    (LAAS-CNRS, Université de Toulouse, CNRS, 31400 Toulouse, France)

  • Vyacheslav Kungurtsev

    (Faculty of Electrical Engineering, Czech Technical University, 166 36 Prague 6, Czechia)

  • Jakub Mareček

    (Faculty of Electrical Engineering, Czech Technical University, 166 36 Prague 6, Czechia)

Abstract

In many applications, solutions of convex optimization problems are updated on-line, as functions of time. In this paper, we consider parametric semidefinite programs, which are linear optimization problems in the semidefinite cone whose coefficients (input data) depend on a time parameter . We are interested in the geometry of the solution (output data) trajectory, defined as the set of solutions depending on the parameter . We propose an exhaustive description of the geometry of the solution trajectory. As our main result, we show that only six distinct behaviors can be observed at a neighborhood of a given point along the solution trajectory. Each possible behavior is then illustrated by an example.

Suggested Citation

  • Antonio Bellon & Didier Henrion & Vyacheslav Kungurtsev & Jakub Mareček, 2025. "Parametric Semidefinite Programming: Geometry of the Trajectory of Solutions," Mathematics of Operations Research, INFORMS, vol. 50(1), pages 410-430, February.
    • Handle: RePEc:inm:ormoor:v:50:y:2025:i:1:p:410-430
    DOI: 10.1287/moor.2021.0097
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