Globally Solving a Class of Bilevel Programs with Spatial Price Equilibrium Constraints

TL;DR

This paper introduces a new, stronger single-level formulation for bilevel programs with spatial price equilibrium constraints, enabling more efficient global solutions and handling larger instances, with applications in facility location, power networks, and renewable energy integration.

Contribution

The paper presents an enhanced single-level formulation based on duality that improves computational efficiency and scalability for solving bilevel programs with equilibrium constraints.

Findings

The new formulation outperforms existing models in computational experiments.

The heuristic effectively finds high-quality feasible solutions for large instances.

Numerical studies demonstrate the approach's applicability to real-world energy and facility location problems.

Abstract

Bilevel programs with spatial price equilibrium constraints are strategic models that consider a price competition at the lower level. These models find application in facility location-price models, optimal bidding in power networks, and integration of renewable energy sources in distribution networks. In this paper, for the case where the equilibrium at the lower level can be formulated as an optimization problem, we introduce an enhanced single-level formulation based on duality and show that its relaxation is stronger than the single-level formulation obtained using KKT conditions. Compared to the literature [1, 2], this new formulation (i) is computationally friendly to global solution strategies using branch-and-bound, and (ii) can tackle instances of larger size. Further, we develop a heuristic procedure to find feasible solutions inside of the branch-and-bound tree that is effective on instances of large size and produces solutions whose objective values are close to the relaxation bound. We demonstrate the benefits of this formulation and heuristic through an extensive numerical study on synthetic instances of Equilibrium Facility Location [3] and on standard IEEE bus networks for planning renewable generation capacity under uncertainty.