Robust Bilevel Optimization for Near-Optimal Lower-Level Solutions

TL;DR

This paper introduces a new concept of near-optimality robustness in bilevel optimization, providing theoretical properties, solution methods, and numerical analysis to enhance solution feasibility under deviations.

Contribution

It develops the theoretical foundation and solution techniques for near-optimal robust bilevel problems, a novel approach in the field.

Findings

Duality-based method effective for convex lower levels

Heuristic methods improve solution times

Valid inequalities reduce computational effort

Abstract

Bilevel optimization problems embed the optimality of a subproblem as a constraint of another optimization problem. We introduce the concept of near-optimality robustness for bilevel optimization, protecting the upper-level solution feasibility from limited deviations from the optimal solution at the lower level. General properties and necessary conditions for the existence of solutions are derived for near-optimal robust versions of general bilevel optimization problems. A duality-based solution method is defined when the lower level is convex, leveraging the methodology from the robust and bilevel literature. Numerical results assess the efficiency of exact and heuristic methods and the impact of valid inequalities on the solution time.