Solving bilevel programs based on lower-level Mond-Weir duality

TL;DR

This paper introduces a new reformulation for bilevel programs using lower-level Mond-Weir duality, which can satisfy constraint qualifications and improve solution feasibility and efficiency over existing methods.

Contribution

The paper presents the MDP reformulation based on Mond-Weir duality, demonstrating its advantages over MPCC and WDP in solving bilevel programs.

Findings

MDP may satisfy MFCQ at feasible points unlike MPCC

Numerical experiments show MDP outperforms MPCC and WDP in feasibility and efficiency

MDP achieves better average CPU time in solving test problems

Abstract

This paper focuses on developing effective algorithms for solving bilevel program. The most popular approach is to replace the lower-level problem by its Karush-Kuhn-Tucker conditions to generate a mathematical program with complementarity constraints (MPCC). However, MPCC does not satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. In this paper, inspired by a recent work using the lower-level Wolfe duality (WDP), we apply the lower-level Mond-Weir duality to present a new reformulation, called MDP, for bilevel program. It is shown that, under mild assumptions, they are equivalent in globally or locally optimal sense. An example is given to show that, different from MPCC, MDP may satisfy the MFCQ at its feasible points. Relations among MDP, WDP, and MPCC are investigated. Furthermore, in order to compare the new MDP approach with the MPCC and WDP approaches, we design a procedure to generate 150 tested problems randomly and comprehensive numerical experiments showed that MDP has evident advantages over MPCC and WDP in terms of feasibility to the original bilevel programs, success efficiency, and average CPU time.