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Closing the Gap in Linear Bilevel Optimization: A New Valid Primal-Dual Inequality

Thomas Kleinert 1 Martine Labbé 2 Fränk Plein 2 Martin Schmidt 3 
2 INOCS - Integrated Optimization with Complex Structure
Inria Lille - Nord Europe, ULB - Université libre de Bruxelles, CRIStAL - Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189
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.
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Submitted on : Wednesday, September 16, 2020 - 10:54:42 AM
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Thomas Kleinert, Martine Labbé, Fränk Plein, Martin Schmidt. Closing the Gap in Linear Bilevel Optimization: A New Valid Primal-Dual Inequality. Optimization Letters, 2021, ⟨10.1007/s11590-020-01660-6⟩. ⟨hal-02889276v2⟩



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