Abstract
We consider the dual formulation of the problem of finding the maximum of E ν [ f ( X ) ] , where ν is allowed to vary over all the probability measures on a Polish space X for which d c ( μ , ν ) ≤ r , with d c an optimal transport distance, f a real-valued function on X satisfying some regularity, μ a ‘baseline’ measure and r ≥ 0 . Whereas some derivations of the dual rely on Fenchel duality, applied on a vector space of functions in duality with a vector space of measures, we impose compactness on X to allow the use of the minimax theorem of Ky Fan, which does not require vector space structure.
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