Closing the Gap in Linear Bilevel Optimization: A New Valid Primal-Dual Inequality
Abstract: Linear bilevel optimization problems are often tackled by replacing the linear lower-level problem with its Karush–Kuhn–Tucker (KKT) conditions. The resulting single-level problem can be solved in a branch-and-bound fashion by branching on the complementarity constraints of the lower-level problem’s optimality conditions. While in mixed-integer single-level optimization branch- and-cut has proven to be a powerful extension of branch-and-bound, in linear bilevel optimization not too many bilevel-tailored valid inequalities exist. In this paper, we briefly review existing cuts for linear bilevel problems and introduce a new valid inequality that exploits the strong duality condition of the lower level. We further discuss strengthened variants of the inequality that can be derived from McCormick envelopes. In a computational study, we show that the new valid inequalities can help to close the optimality gap very effectively on a large test set of linear bilevel instances.
Keywords: Bilevel optimization, Valid inequalities, Branch-and-cut, Computational analysis.
Category 1: Integer Programming ((Mixed) Integer Linear Programming )
Category 2: Complementarity and Variational Inequalities
Category 3: Integer Programming (Cutting Plane Approaches )
Entry Submitted: 06/05/2020
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