A New Computational Approach for Solving Linear Bilevel Programs Based on Parameter-Free Disjunctive Decomposition

TL;DR

The paper tackles the computational difficulty of linear bilevel programs caused by the need to select LP-correct big-M parameters. It introduces a disjunctive-based decomposition (DBD) that reformulates the problem into a big-M free framework using a SP and an MP, with a key proposition ensuring SP's equivalence to the DP and thus removing dependence on big-M under certain conditions. Through illustrative and extensive general and large-scale case studies, the authors show that DBD achieves bilevel-optimal solutions with zero duality gap, while dramatically reducing iterations and avoiding the tuning of big-M parameters. The approach demonstrates strong scalability and practical potential for applications in energy networks and bidding strategies, validated against traditional big-M methods across 12 scenarios.

Abstract

Linear bilevel programs (linear BLPs) have been widely used in computational mathematics and optimization in several applications. Single-level reformulation for linear BLPs replaces the lower-level linear program with its Karush-Kuhn-Tucker optimality conditions and linearizes the complementary slackness conditions using the big-M technique. Although the approach is straightforward, it requires finding the big-M whose computation is recently shown to be NP-hard. This paper presents a disjunctive-based decomposition algorithm which does not need finding the big-Ms whereas guaranteeing that obtained solution is optimal. Our experience shows promising performance of our algorithm.

Figures (3)

• Figure 1: The proposed DBD algorithm
• Figure 2: Distribution of solution time and number of iterations for different initial points
• Figure 3: The convergence graph of the proposed DBD algorithm