There's No Free Lunch: On the Hardness of ...
Type de document :
Pré-publication ou Document de travail
Titre :
There's No Free Lunch: On the Hardness of Choosing a Correct Big-M in Bilevel Optimization
Auteur(s) :
Kleinert, Thomas [Auteur]
Friedrich-Alexander Universität Erlangen-Nürnberg = University of Erlangen-Nuremberg [FAU]
Labbé, Martine [Auteur]
Integrated Optimization with Complex Structure [INOCS]
Plein, Fränk [Auteur]
Integrated Optimization with Complex Structure [INOCS]
Schmidt, Martin [Auteur]
Trier University
Friedrich-Alexander Universität Erlangen-Nürnberg = University of Erlangen-Nuremberg [FAU]
Labbé, Martine [Auteur]
Integrated Optimization with Complex Structure [INOCS]
Plein, Fränk [Auteur]
Integrated Optimization with Complex Structure [INOCS]
Schmidt, Martin [Auteur]
Trier University
Discipline(s) HAL :
Mathématiques [math]/Optimisation et contrôle [math.OC]
Computer Science [cs]/Operations Research [math.OC]
Informatique [cs]/Complexité [cs.CC]
Computer Science [cs]/Operations Research [math.OC]
Informatique [cs]/Complexité [cs.CC]
Résumé en anglais : [en]
One of the most frequently used approaches to solve linear bilevel optimization problems consists in replacing the lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions and by reformulating the KKT complementarity ...
Lire la suite >One of the most frequently used approaches to solve linear bilevel optimization problems consists in replacing the lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions and by reformulating the KKT complementarity conditions using techniques from mixed-integer linear optimization. The latter step requires to determine some big-M constant in order to bound the lower level's dual feasible set such that no bilevel-optimal solution is cut off. In practice, heuristics are often used to find a big-M although it is known that these approaches may fail. In this paper, we consider the hardness of two proxies for the above mentioned concept of a bilevel-correct big-M. First, we prove that verifying that a given big-M does not cut off any feasible vertex of the lower level's dual polyhedron cannot be done in polynomial time unless P=NP. Second, we show that verifying that a given big-M does not cut off any optimal point of the lower level's dual problem (for any point in the projection of the high-point relaxation onto the leader's decision space) is as hard as solving the original bilevel problem.Lire moins >
Lire la suite >One of the most frequently used approaches to solve linear bilevel optimization problems consists in replacing the lower-level problem with its Karush-Kuhn-Tucker (KKT) conditions and by reformulating the KKT complementarity conditions using techniques from mixed-integer linear optimization. The latter step requires to determine some big-M constant in order to bound the lower level's dual feasible set such that no bilevel-optimal solution is cut off. In practice, heuristics are often used to find a big-M although it is known that these approaches may fail. In this paper, we consider the hardness of two proxies for the above mentioned concept of a bilevel-correct big-M. First, we prove that verifying that a given big-M does not cut off any feasible vertex of the lower level's dual polyhedron cannot be done in polynomial time unless P=NP. Second, we show that verifying that a given big-M does not cut off any optimal point of the lower level's dual problem (for any point in the projection of the high-point relaxation onto the leader's decision space) is as hard as solving the original bilevel problem.Lire moins >
Langue :
Anglais
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