New Branch-and-Bound Rules for Linear Bilevel Programming

Abstract

A new branch-and-bound algorithm for linear bilevel programming is proposed.Necessary optimality conditions expressed in terms of tightness of the follower's constraints are used to fathom or simplify subproblems, branch and obtain penalties similar to those used in mixedinteger programming. Computational results are reported and compare favorably to those of previous methods. Problems with up to 150 constraints, 250 variables controlled by the leader, and 150 variables controlled by the follower have been solved.

Dantzig-Wolfe Reformulation for the Network Pricing Problem with Connected Toll Arcs

“…Even the simple version of a (BP) where the objective functions and the constraints are linear has been shown to be N P-hard by Jeroslow (1985). Furthermore, Hansen et al (1992) prove the strong N P-hardness of these problems. strengthen these results and prove that merely checking strict local optimality and checking local optimality in linear (BP) are N P-hard problems.…”

Mixed network design using hybrid scatter search

“…It has been proved that even bi-level problems with only linear constraints are NP-Hard (Hansen et al, 1992). Here, a two-step recursive procedure is applied to solve this bi-level problem.…”

A Branch-and-cut Algorithm for Integer Bilevel Linear Programs

“…In fact, in contrast to the ILP case, the question of IBLP feasibility is not even in N P, essentially because the question of whether a pair (x, y) ∈ Z n 1 × Z n 2 is feasible for a given IBLP is itself an ILP. Hansen et al [1992] show that even the continuous version of the problem (a BLP) is strongly N P-hard and Vicente et al [1994] adds that checking local optimality for BLPs is an N P-hard problem. All of this indicates that solving IBLPs in practice is likely to be extremely challenging.…”