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

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.

Figures (8)

• Figure 1: Schematic diagram of PRS heuristic. (a) Step 1, lower level optimization with a fixed set of upper variables, highlighted in red dashed vertical arrows. (b) Step 2, Critical region generation, highlighting in dashed border lines. (c) Step 3, regional upper level optimization, highlighted by the red arrow. (d) Repeat Step 1, Step 2, and Step 3 in a new iteration. (e) Repeat Step 1 and converge.
• Figure 2: Schematic diagram for the numerical example.
• Figure 3: Algorithms time performance on comparison of different sizes: Tiny (5), Small (10), Mid (25), Large (50). Detailed size information is in Table \ref{['tab:bilevel_sizes']}.
• Figure 4: Algorithms comparison on number of iterations(PRS), and evaluations(COBYLA/ISRES) for different sizes: Tiny (5), Small (10), Mid (25), and Large (50). Detailed size information is in Table \ref{['tab:bilevel_sizes']}.
• Figure 5: Algorithms gaps performance on comparison of different sizes: Tiny (5) and Small (10). Detailed size information is in Table \ref{['tab:bilevel_sizes']}.