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.
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Pré-publication, Document de travail
2018
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Contributeur : Michel De Lara <>
Soumis le : vendredi 7 septembre 2018 - 13:39:50
Dernière modification le : mercredi 12 septembre 2018 - 01:11:51

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

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Jean-Philippe Chancelier, Michel De Lara. Fenchel-Moreau Conjugation Inequalities with Three Couplings and Application to Stochastic Bellman Equation. 2018. 〈hal-01760462v2〉

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