On the sensitivity of the optimal partition for parametric second-order conic optimization

TL;DR

This paper analyzes the stability of the optimal partition in parametric second-order conic optimization, introducing concepts like invariancy sets and nonlinearity intervals, and proposes methods to compute and identify these regions.

Contribution

It introduces the notions of invariancy sets and nonlinearity intervals for stability analysis and provides algorithms to compute and identify these regions under certain conditions.

Findings

Characterization of invariancy sets and nonlinearity intervals.

An iterative procedure to compute nonlinearity intervals.

Numerical methods to identify boundary points of nonlinearity intervals.

Abstract

In this paper, using an optimal partition approach, we study the parametric analysis of a second-order conic optimization problem, where the objective function is perturbed along a fixed direction. We characterize the notions of so-called invariancy set and nonlinearity interval, which serve as stability regions of the optimal partition. We then propose, under the strict complementarity condition, an iterative procedure to compute a nonlinearity interval of the optimal partition. Furthermore, under primal and dual nondegeneracy conditions, we show that a boundary point of a nonlinearity interval can be numerically identified from a nonlinear reformulation of the parametric second-order conic optimization problem. Our theoretical results are supported by numerical experiments.