Fenchel-Moreau Conjugation Inequalities with Three Couplings and Application to 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 $\text{PRIMAL}$, $\text{PRIMALBIS}$and two “dual” sets $\text{DUAL}$, $\text{DUALBIS}$, together with two {coupling} functions (\PRIMAL \overset{\coupling}{\leftrightarrow} \DUAL ) and (\PRIMALBIS \overset{\couplingbis}{\leftrightarrow} \DUALBIS ). We define a new coupling $\text{SumCoupling}\text{coupling}\text{couplingbis}$ between the “primal” product set~$\text{PRIMAL}\times \text{PRIMALBIS}$ 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 $\text{fonctionprimalbis}:\text{PRIMALBIS}\to \text{barRR}$, 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 $\text{SFM}\text{fonctionprimal}\text{coupling}$ and $\text{SFM}\text{fonctionprimalbis}-\text{couplingbis}$ 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.