Parametric Semidefinite Programming: Geometry of the Trajectory of Solutions

Antonio Bellon (), Didier Henrion (), Vyacheslav Kungurtsev () and Jakub Mareček ()
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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

Mathematics of Operations Research, 2025, vol. 50, issue 1, 410-430

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.

Keywords: Primary: 90C22; 90C25; secondary: 90C46; semidefinite programming; parametric optimization; convex optimization; set-valued analysis (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:50:y:2025:i:1:p:410-430

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