Parametric Semidefinite Programming: Geometry of the Trajectory of Solutions

TL;DR

This paper analyzes the geometric behavior of solutions in parametric semidefinite programs, revealing that only six distinct local behaviors can occur along the solution trajectory as the parameters change.

Contribution

It provides an exhaustive characterization of the local geometric behaviors of solution trajectories in parametric semidefinite programming.

Findings

Six distinct local behaviors identified in solution trajectories.

Geometric analysis applicable to on-line convex optimization problems.

Illustrative examples demonstrate each behavior.

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.