Fenchel-moreau conjugation inequalities with three couplings and application to the stochastic bellman equation

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 twocoupling functions X ^c↔ X^# and Y ^d↔ Y^#. We define a new coupling c + 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) ≥ inf_y∈Y(K(x, y) + g(y))⇒ f^c(x^#) ≤ inf_y#∈Y#(K^c+d(x^#, y^#) + g^-d(y^#)), 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" inequation for the Fenchel conjugates of the value functions.