Theoretical and numerical comparison of seven single-level reformulations for bilevel programs

TL;DR

This study compares seven single-level reformulations of bilevel programs, demonstrating that duality-based reformulations generally outperform the MPCC approach, with new tighter reformulations showing promising results.

Contribution

The paper introduces three new duality-based reformulations with tighter feasible regions and provides a comprehensive numerical comparison of seven reformulations.

Findings

WDP/MDP/TWDP/TMDP outperform MPCC in numerical tests

eMDP/eTMDP are consistently the worst reformulations

Duality-based reformulations perform 3-5 times better than MPCC

Abstract

This paper considers a bilevel program. To solve this bilevel program, it is generally necessary to transform it into some single-level optimization problem. One approach is to replace the lower-level program by its KKT conditions to transform the bilevel program as a mathematical program with complementarity constraints (MPCC). Another approach is to apply the lower-level Wolfe/Mond-Weir/extended Mond-Weir duality to transform the bilevel program into some duality-based single-level reformulations, called WDP, MDP, and eMDP respectively in the literature. In this paper, inspired by a conjecture from a recent publication that the tighter feasible region of a reformulation, the better its numerical performance, we present three new duality-based single-level reformulations, called TWDP/TMDP/eTMDP, with tighter feasible regions. Our main goal is to compare all above-mentioned reformulations by designing some direct and relaxation algorithms with projection and implementing these algorithms on 450 test examples generated randomly. Our numerical experiments show that, whether overall comparison or pairwise comparison, at least in our tests, the WDP/MDP/TWDP/TMDP reformulations were always better than the MPCC reformulation, while the eMDP/eTMDP reformulations were always the worst ones among six duality-based reformulations, which indicates that the above conjecture is incorrect. In particular, for the relaxation algorithms, the WDP/MDP/TWDP/TMDP reformulations performed 3-5 times better than the MPCC reformulation, while the eMDP/eTMDP reformulations performed 2 times better than the MPCC reformulation.