Abstract Convexity and the Monge-Kantorovich Duality

Vladimir Levin

A chapter in Generalized Convexity and Related Topics, 2007, pp 33-72 from Springer

Abstract: Summary In the present survey, we reveal links between abstract convex analysis and two variants of the Monge-Kantorovich problem (MKP), with given marginals and with a given marginal difference. It includes: (1) the equivalence of the validity of duality theorems for MKP and appropriate abstract convexity of the corresponding cost functions; (2) a characterization of a (maximal) abstract cyclic monotone map F: X → L ⊂ IRX in terms connected with the constraint set $$ Q_0 (\varphi ): = \{ u \in \mathbb{R}^z :u(z_1 ) - u(z_2 ) \leqslant \varphi (z_1 ,z_2 ){\text{ }}\forall z_1 ,z_1 \in Z = dom{\text{ }}F\} $$ of a particular dual MKP with a given marginal difference and in terms of L-subdifferentials of L-convex functions; (3) optimality criteria for MKP (and Monge problems) in terms of abstract cyclic monotonicity and non-emptiness of the constraint set Q 0(ϕ), where ϕ is a special cost function on X × X determined by the original cost function c on X × Y. The Monge-Kantorovich duality is applied then to several problems of mathematical economics relating to utility theory, demand analysis, generalized dynamics optimization models, and economics of corruption, as well as to a best approximation problem.

Keywords: H-convex function; infinite linear programs; duality relations; Monge-Kantorovich problems (MKP) with given marginals; MKP with a given marginal difference; abstract cyclic monotonicity; Monge problem; utility theory; demand analysis; dynamics models; economics of corruption; approximation theory (search for similar items in EconPapers)
Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:spr:lnechp:978-3-540-37007-9_2

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DOI: 10.1007/978-3-540-37007-9_2

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