A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

Abstract

Bilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, optimal expansion of gas networks, critical infrastructure defense, and machine learning. In this paper, we present a new general purpose branch-and-cut framework for the exact solution of mixed-integer bilevel linear programs (MIBLP), which constitute a very significant subfamily of bilevel optimization problems. Our framework introduces several new classes of valid inequalities to speed-up the solver, along with a very effective bilevel-specific preprocessing procedure. A very extensive computational study is presented, where we evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature-this is by far the most extensive computational analysis ever performed for exact MIBLP solvers. Our new algorithm consistently outperforms (often by a large margin) all alternative state-of-the-art methods from the literature, including methods which exploit problem specific information for special instance classes. In particular, it allows to solve to optimality more than 300 instances previously unsolved instances from literature.

New bounding schemes and algorithmic options for the Branch-and-Sandwich algorithm

“…Proof The set L p,k of sublists to which node V k belongs is a subset of all sublists in inde- From (19) it follows that the set of minimal inner upper bound values for all the sublists containing node V k is also a subset of the set of minimal values for all sublists in L p :…”

“…Here the optimistic (co-operative) formulation is assumed, i.e., the leader can choose among globally optimal lower-level solutions to achieve the best outer objective value. Special cases of bilevel programming have been studied extensively, and many algorithms have been proposed, see e.g., [4,7,8,11,12,[14][15][16]19,28,35,36,43] for reviews. However, the general nonconvex form is very challenging and only recently were the first methods to tackle this class of problems proposed.…”

Closing the gap in linear bilevel optimization: a new valid primal-dual inequality

“…3 via generic CPLEX callbacks. We therefore use the general formulation (9). This allows to add tighter inequalities whenever the required bounds are available.…”

A projection-based reformulation and decomposition algorithm for global optimization of a class of mixed integer bilevel linear programs

“…Most existing MIBLP algorithms are proposed to handle special classes of (P0), such as integer bilevel linear programs [15], MIBLPs with special constraint structures [16], MIBLPs without continuous upper-level variables [7,17,18], and/or MIBLPs without continuous lowerlevel variables [19,20]. Fischetti et al [21,22] introduced a new general-purpose algorithm for MIBLPs based on a branch-and-cut framework, where new classes of valid inequalities and effective preprocessing procedures are introduced. We mention that Zeng and An [23] proposed the reformulation and decomposition method to solve MIBLPs with continuous and integer variables in both upper-and lower-level programs, which provides a rather general strategy and framework to attack those difficult problems.…”