Perturbation Analysis of Singular Semidefinite Programs and Its Applications to Control Problems

TL;DR

This paper analyzes how the optimal value of semidefinite programs changes under perturbations, especially in control applications, by examining minimal faces and facial reduction techniques.

Contribution

It provides conditions for the continuity of optimal values under perturbations and characterizes face invariance using facial reduction methods.

Findings

Optimal value changes continuously under certain conditions.

Perturbations can cause discontinuous changes in optimal values.

Classification of minimal face behavior in control-related semidefinite programs.

Abstract

We consider sensitivity of a semidefinite program under perturbations in the case that the primal problem is strictly feasible and the dual problem is weakly feasible. When the coefficient matrices are perturbed, the optimal values can change discontinuously as explained in concrete examples. We show that the optimal value of such a semidefinite program changes continuously under conditions involving the behavior of the minimal faces of the perturbed dual problems. In addition, we determine what kinds of perturbations keep the minimal faces invariant, by using the reducing certificates, which are produced in facial reduction. Our results allow us to classify the behavior of the minimal face of a semidefinite program obtained from a control problem.