Duality Between Lagrangians and Rockafellians

TL;DR

This paper explores the duality relationship between Lagrangians and Rockafellians in convex optimization, providing a symmetric framework and new formulas that connect these two fundamental concepts.

Contribution

It introduces a formal definition of Lagrangian-Rockafellian couples and characterizes their duality, extending the understanding of perturbation and dual functions in convex analysis.

Findings

Established a duality between Lagrangians and Rockafellians.

Provided formulas linking Lagrangians and Rockafellians in both directions.

Characterized these couples as dual functions with respect to a coupling.

Abstract

In his monograph \emph{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. 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 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.