Time-Varying Semidefinite Programming: Geometry of the Trajectory of Solutions

Antonio Bellon, 26 Sep 2022

In many applications, solutions of convex optimization problems must be updated on-line, as functions of time. In this paper, we consider time-varying semidefinite programs (TV-SDP), which are linear optimization problems in the semidefinite cone whose coefficients (input data) depend on time. We are interested in the geometry of the solution (output data) trajectory, defined as the set of solutions depending on time. We propose an exhaustive description of the geometry of the solution trajectory. As our main result, we show that only 6 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.

This is a joint work with Didier Henrion, Vyacheslav Kungurtsev, Jakub Mareček.