Fenchel-Moreau Conjugation Inequalities with Three Couplings and Application to Stochastic Bellman Equation

Abstract

Given two couplings between "primal" and "dual" sets, we prove a general implication that relates an inequality involving "primal" sets to a reverse inequality involving the "dual" sets. More precisely, let be given two "primal" sets X, Y and two "dual" sets X ♯ , Y ♯ , together with two coupling functions+ d between the "primal" product set X × Y and the "dual" product set X ♯ × Y ♯ . Then, we consider any bivariate function K : X × Y → [−∞, +∞] and univariate functions f : X → [−∞, +∞] and g : Y → [−∞, +∞], all defined on the "primal" sets. We prove that f (x), where we stress that the Fenchel-Moreau conjugates f c and g −d are not necessarily taken with the same coupling. We study the equality case. We display several applications. We provide a new formula for the Fenchel-Moreau conjugate of a generalized inf-convolution. We obtain formulas with partial Fenchel-Moreau conjugates. Finally, we consider the Bellman equation in stochastic dynamic programming and we provide a "Bellman-like" equation for the Fenchel conjugates of the value functions.

Learning Energy Networks with Generalized Fenchel-Young Losses

“…The Φ-convex conjugate is an important tool in abstract convex analysis [63,60]. Recently, it has been used to provide "Bellman-like" equations in stochastic dynamic programming [24] and to provide tropical analogues of reproducing kernels [5]. We assume that the maximum is feasible for all v ∈ V = R d , meaning that dom(Ω Φ ) = V. We emphasize again that, unlike with (1), v and p do not need to have compatible dimensions.…”