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Pré-publication, Document de travail (preprint/prepublication)

Duality Between Lagrangians and Rockafellians

Abstract : In his monograph Conjugate Duality and Optimization, 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 so-called 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 so-called 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. In this paper, 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 Lagrangian-Rockafellian 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.
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Pré-publication, Document de travail (preprint/prepublication)
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https://hal.archives-ouvertes.fr/hal-03867004
Contributeur : Michel De Lara Connectez-vous pour contacter le contributeur
Soumis le : mercredi 23 novembre 2022 - 09:35:08
Dernière modification le : jeudi 24 novembre 2022 - 03:30:33

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  • HAL Id : hal-03867004, version 1
  • ARXIV : 2211.13083

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Michel de Lara. Duality Between Lagrangians and Rockafellians. 2022. ⟨hal-03867004⟩

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