Parametric Region Search: A Mixed-Integer Bilevel Optimization Problem Primal Heuristic

TL;DR

This paper introduces the Parametric Region Search heuristic for mixed-integer bilevel optimization, leveraging parametric insights to efficiently find high-quality solutions in complex hierarchical problems.

Contribution

It develops a novel primal heuristic based on parametric region exploration, filling a gap in specialized methods for solving MIBO problems.

Findings

PRS consistently finds high-quality solutions compared to existing metaheuristics.

The heuristic effectively integrates with other methods like DOMINO-COBYLA.

Computational results validate the efficiency and solution quality of PRS.

Abstract

Bilevel optimization is a mathematical modeling formulation for hierarchical systems and two-player interactions, with wide-ranging applications in environmental, energy, and control engineering. Despite its utility, the mixed-integer bilevel optimization (MIBO) problem is exceptionally challenging to solve. While numerous exact and metaheuristic methods exist, the development of specialized primal heuristics for MIBO, aimed at quickly identifying high-quality feasible solutions, remains an underexplored area. This paper introduces the Parametric Region Search (PRS), a new primal heuristic for MIBO. The PRS method leverages insights from multi-parametric optimization by iteratively exploring regions defined by the lower-level problem's critical regions. We formally define the MIBO structure and the necessary parametric region formulations, and then detail the proposed heuristic's initialization and iterative search mechanism. Computational results demonstrate that the PRS heuristic consistently locates high-quality primal solutions compared to established derivative-free metaheuristics, including DOMINO-COBYLA and DOMINO-ISRES. Furthermore, we illustrate how the PRS can be effectively integrated with other heuristics like DOMINO-COBYLA to enhance the overall solution discovery process for MIBO.