A Projection-Based Reformulation and Decomposition Algorithm for Global Optimization of a Class of Mixed Integer Bilevel Linear Programs

TL;DR

This paper introduces a novel projection-based reformulation and decomposition algorithm for solving a specific class of mixed-integer bilevel linear programs, ensuring finite convergence to the global optimum and demonstrating effectiveness through computational experiments.

Contribution

The paper presents an extended reformulation and a projection strategy that improves upon existing methods for solving mixed-integer bilevel linear programs, with proven finite convergence.

Findings

Algorithm converges to global optimal solution in finite steps

Effective on large-scale random and supply chain instances

Projection strategy eliminates need for relatively complete response property

Abstract

We propose an extended variant of the reformulation and decomposition algorithm for solving a special class of mixed-integer bilevel linear programs (MIBLPs) where continuous and integer variables are involved in both upper- and lower-level problems. In particular, we consider MIBLPs with upper-level constraints that involve lower-level variables. We assume that the inducible region is nonempty and all variables are bounded. By using the reformulation and decomposition scheme, an MIBLP is first converted into its equivalent single-level formulation, then computed by a column-and-constraint generation based decomposition algorithm. The solution procedure is enhanced by a projection strategy that does not require the relatively complete response property. To ensure its performance, we prove that our new method converges to the global optimal solution in a finite number of iterations. A large-scale computational study on random instances and instances of hierarchical supply chain planning are presented to demonstrate the effectiveness of the algorithm.