On computing the nonlinearity interval in parametric semidefinite optimization

TL;DR

This paper analyzes how the optimal solutions of semidefinite optimization problems change with objective perturbations, characterizes transition points, and introduces a numerical method to compute nonlinearity intervals and transition points.

Contribution

It provides a theoretical characterization of transition points and nonlinearity intervals, and develops a numerical algebraic geometry-based method to compute them efficiently.

Findings

The set of transition points is finite.

Continuity of the optimal set mapping can fail on a nonlinearity interval.

The proposed method successfully computes nonlinearity intervals and transition points in numerical examples.

Abstract

This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction. We review the notions of invariancy set, nonlinearity interval, and transition point of the optimal partition, and we investigate their characterizations. We show that the set of transition points is finite and the continuity of the optimal set mapping, on the basis of Painlev\'e-Kuratowski set convergence, might fail on a nonlinearity interval. Under a local nonsingularity condition, we then develop a methodology, stemming from numerical algebraic geometry, to efficiently compute nonlinearity intervals and transition points of the optimal partition. Finally, we support the theoretical results by applying our procedure to some numerical examples.