A novel approach for bilevel programs based on Wolfe duality

TL;DR

This paper introduces a new reformulation of bilevel programs using Wolfe duality, which can satisfy constraint qualifications that traditional MPEC reformulations cannot, leading to more effective numerical solution methods.

Contribution

The paper proposes a Wolfe duality-based reformulation of bilevel programs that improves constraint qualification satisfaction and offers a new relaxation method for solving these problems.

Findings

WDP reformulation may satisfy Mangasarian-Fromovitz constraint qualification.

The relaxation method based on WDP is efficient in numerical experiments.

WDP provides a theoretically equivalent single-level reformulation under mild conditions.

Abstract

This paper considers a bilevel program, which has many applications in practice. To develop effective numerical algorithms, it is generally necessary to transform the bilevel program into a single-level optimization problem. The most popular approach is to replace the lower-level program by its KKT conditions and then the bilevel program can be reformulated as a mathematical program with equilibrium constraints (MPEC for short). However, since the MPEC does not satisfy the Mangasarian-Fromovitz constraint qualification at any feasible point, the well-developed nonlinear programming theory cannot be applied to MPECs directly. In this paper, we apply the Wolfe duality to show that, under very mild conditions, the bilevel program is equivalent to a new single-level reformulation (WDP for short) in the globally and locally optimal sense. We give an example to show that, unlike the MPEC reformulation, WDP may satisfy the Mangasarian-Fromovitz constraint qualification at its feasible points. We give some properties of the WDP reformulation and the relations between the WDP and MPEC reformulations. We further propose a relaxation method for solving WDP and investigate its limiting behavior. Comprehensive numerical experiments indicate that, although solving WDP directly does not perform very well in our tests, the relaxation method based on the WDP reformulation is quite efficient.