On Lagrangian Duality in Infinite Dimension and Its Applications

Antonio Causa (), Giandomenico Mastroeni () and Fabio Raciti ()
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Antonio Causa: Dipartimento di Matematica e Informatica dell’Università di Catania
Giandomenico Mastroeni: Dipartimento di Informatica dell’Università di Pisa
Fabio Raciti: Dipartimento di Matematica e Informatica dell’Università di Catania

A chapter in Applications of Nonlinear Analysis, 2018, pp 37-60 from Springer

Abstract: Abstract The aim of this contribution is to review some recent results on Lagrangian duality in infinite dimensional spaces which permit to deal with problems where the ordering cone describing the inequality constraints has empty topological interior. For instance, the topological interior of the cone of the nonnegative L p functions (p > 1) is empty, as it is the cone of nonnegative functions in many Sobolev spaces. To point out where the difficulty comes from, we first review the classical theory which requires the nonemptiness of the ordering cone and then describe the main results obtained by some authors in the last decade, based on what they called “Assumption S”. At last, we show how the new theory can be applied to extend a classical result by Rosen on Nash equilibria, from ℝ n $$\mathbb {R}^n$$ to infinite dimensional spaces.

Date: 2018
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DOI: 10.1007/978-3-319-89815-5_3

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