On Lagrangian Duality in Infinite Dimension and Its Applications

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.