Convex and concave envelopes: revisited and new perspectives

Walid Ben-Ameur (1, 2, 3, 4) , Adam Ouorou (5) , Guanglei Wang (2, 3, 4)

1 METHODES-SAMOVAR - Méthodes et modèles pour les réseaux
2 TSP - RS2M - Département Réseaux et Services Multimédia Mobiles
3 SAMOVAR - Services répartis, Architectures, MOdélisation, Validation, Administration des Réseaux
4 CNRS - Centre National de la Recherche Scientifique
5 Orange Labs [Issy les Moulineaux]

We discuss two approaches to approximate the convex and concave envelopes of bilinear functions over hypercubes. The first approach is based on a semidefinite program. The second approach considers some predefined cover sets of a hypercube and leads to a linear program. Then we establish a connection between the convex envelope of a bilinear function and the concave envelope of a piecewise linear function. Numerical experiments are conducted to compare the two approaches. As an extension, a novel approach is discussed

Mots clés: Convex envelope; Finite cover; Polyhedral approximation

Domaines: Recherche opérationnelle [math.OC]; Optimisation et contrôle [math.OC]

Walid Ben-Ameur, Adam Ouorou, Guanglei Wang. Convex and concave envelopes: revisited and new perspectives. Operations Research Letters, 2017, 45 (5), pp.421 - 426. ⟨10.1016/j.orl.2017.06.008⟩. ⟨hal-01662836⟩