
Journal of Convex Analysis 30 (2023), No. 3, 887896 Copyright Heldermann Verlag 2023 Duality Between Lagrangians and Rockafellians Michel De Lara CERMICS, École des Ponts, MarnelaVallée, France, MarnelaVallée, France michel.delara@enpc.fr In his monograph Conjugate Duality and Optimization [CBMSNSF Regional Conference Series in Applied Mathematics Vol. 16, Society for Industrial and Applied Mathematics, Philadelphia (1974)], R. T. Rockafellar puts forward a "perturbation + duality" method to obtain a dual problem for an original minimization problem. First, one embeds the minimization problem into a family of perturbed problems (thus giving a socalled perturbation function); the perturbation of the original function to be minimized has recently been called a Rockafellian. Second, when the perturbation variable belongs to a primal vector space paired, by a bilinear form, with a dual vector space, one builds a Lagrangian from a Rockafellian; one also obtains a socalled dual function (and a dual problem). The method has been extended from Fenchel duality to generalized convexity: when the perturbation belongs to a primal set paired, by a coupling function, with a dual set, one also builds a Rockafellian from a Lagrangian. Following these paths, we highlight a duality between Lagrangians and Rockafellians. Where the material mentioned above mostly focuses on moving from Rockafellian to Lagrangian, we treat them equally and display formulas that go both ways. We propose a definition of LagrangianRockafellian couples. We characterize these latter as dual functions, with respect to a coupling, and also in terms of generalized convex functions. The duality between perturbation and dual functions is not as clear cut. Keywords: Lagrangian, Rockafellian, duality, generalized convexity. MSC: 49N15, 52A01. [ Fulltextpdf (106 KB)] for subscribers only. 