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# 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.
Document type :
Journal articles

Cited literature [22 references]

https://hal.archives-ouvertes.fr/hal-01760462
Contributor : Michel de Lara Connect in order to contact the contributor
Submitted on : Friday, September 7, 2018 - 1:39:50 PM
Last modification on : Friday, June 11, 2021 - 2:12:02 PM

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### Identifiers

• HAL Id : hal-01760462, version 2
• ARXIV : 1804.03034

### Citation

Jean-Philippe Chancelier, Michel de Lara. Fenchel-Moreau Conjugation Inequalities with Three Couplings and Application to Stochastic Bellman Equation. Journal of Convex Analysis, Heldermann, 2019, 26 (3), pp.945--966. ⟨hal-01760462v2⟩

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