Pareto Robust optimization on Euclidean vector spaces

TL;DR

This paper extends Pareto efficiency concepts to robust optimization in Euclidean spaces with affine uncertainty, demonstrating tractability and practical benefits in semidefinite programming and max-cut problems.

Contribution

It generalizes Pareto robust optimization to Euclidean spaces, proves tractability for robust SDP, and improves approximation guarantees for robust max-cut.

Findings

Computing Pareto robust solutions for SDP is tractable.

Robust solutions improve eigenvalue problem outcomes.

Modified algorithms enhance approximation in robust max-cut.

Abstract

Pareto efficiency for robust linear programs was introduced by Iancu and Trichakis in [9]. We generalize their approach and theoretical results to robust optimization problems in Euclidean spaces with affine uncertainty. Additionally, we demonstrate the value of this approach in an exemplary manner in the area of robust semidefinite programming (SDP). In particular, we prove that computing a Pareto robustly optimal solution for a robust SDP is tractable and illustrate the benefit of such solutions at the example of the maximal eigenvalue problem. Furthermore, we modify the famous algorithm of Goemans and Williamson [8] in order to compute cuts for the robust max-cut problem that yield an improved approximation guarantee in non-worst-case scenarios.