Fenchel-Moreau Conjugation Inequalities with Three Couplings and Application to Stochastic Bellman Equation
Résumé
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.
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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.
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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.
Domaines
Optimisation et contrôle [math.OC]
Origine : Fichiers produits par l'(les) auteur(s)
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