Parametric analysis of semidefinite optimization

TL;DR

This paper analyzes how semidefinite optimization solutions change with objective function perturbations, focusing on the optimal partition's behavior and sensitivity within nonlinearity intervals, supported by theoretical bounds and experiments.

Contribution

It provides new bounds on the approximation of optimal partitions under perturbations and explores their behavior within nonlinearity intervals in semidefinite optimization.

Findings

Derived an upper bound on the distance between original and perturbed optimal partitions.

Analyzed the behavior of the optimal partition on nonlinearity intervals.

Validated theoretical bounds through experimentation.

Abstract

In this paper, we study parametric analysis of semidefinite optimization problems w.r.t. the perturbation of the objective function. We study the behavior of the optimal partition and optimal set mapping on a so-called nonlinearity interval. Furthermore, we investigate the sensitivity of the approximation of the optimal partition in a nonlinearity interval, which has been recently studied by Mohammad-Nezhad and Terlaky. The approximation of the optimal partition was obtained from a bounded sequence of interior solutions on, or in a neighborhood of the central path. We derive an upper bound on the distance between the approximations of the optimal partitions of the original and perturbed problems. Finally, we examine the theoretical bounds by way of experimentation.