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 $\PRIMAL$, $\PRIMALBIS$and two "dual" sets $\DUAL$, $\DUALBIS$, together with two {coupling} functions \(\PRIMAL \overset{\coupling}{\leftrightarrow} \DUAL \) and \(\PRIMALBIS \overset{\couplingbis}{\leftrightarrow} \DUALBIS \). We define a new coupling \(\SumCoupling{\coupling}{\couplingbis} \) between the "primal" product set~$\PRIMAL \times \PRIMALBIS$ and the "dual" product set $\DUAL \times \DUALBIS$. Then, we consider any bivariate function \(\kernel : \PRIMAL \times \PRIMALBIS \to \barRR \) and univariate functions \(\fonctionprimal : \PRIMAL \to \barRR \) and \(\fonctionprimalbis : \PRIMALBIS \to \barRR \), all defined on the "primal" sets. We prove that \(\fonctionprimal\np{\primal} \geq \inf\_{\primalbis \in \PRIMALBIS} \Bp{\kernel\np{\primal, \primalbis} \UppPlus \fonctionprimalbis\np{\primalbis}} \) \( \Rightarrow \SFM{\fonctionprimal}{\coupling}\np{\dual} \leq \inf\_{\dualbis \in \DUALBIS} \Bp{\SFM{\kernel}{\SumCoupling{\coupling}{\couplingbis}}\np{\dual,\dualbis} \UppPlus \SFM{\fonctionprimalbis}{-\couplingbis}\np{\dualbis}} \), where we stress that the Fenchel-Moreau conjugates \(\SFM{\fonctionprimal}{\coupling} \) and \(\SFM{\fonctionprimalbis}{-\couplingbis}\) are not necessarily taken with the same coupling. We study the equality case, after having established the classical Fenchel inequality but with a general coupling. % 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.