By Igor V. Konnov, Dinh The Luc, Alexander M. Rubinov

The publication includes invited papers via famous specialists on quite a lot of themes (economics, variational research, likelihood etc.) heavily regarding convexity and generalized convexity, and refereed contributions of experts from the area on present examine on generalized convexity and purposes, specifically, to optimization, economics and operations study.

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If (15) is valid, ′ (c; e′0 ) holds true whenever c is Hthen the duality relation vI′ (c; e′0 ) = vII ′ ′ ′ convex and e0 ∈ E . If, in addition, e0 ∈ dom vI′ (c; ·) then there exists an optimal solution to program II. L. Levin 3 Abstract Convexity and the Monge - Kantorovich Problems (MKP) In this section, we consider two variants of the Monge—Kantorovich problem (MKP), with given marginals and with a given marginal difference. Both the problems are infinite linear programs, and abstract convexity plays important role in their study.

Clearly, cˆ is lsc, vanishes on the diagonal, and satisfies the triangle inequality, c majorizes the restriction of cˆ onto X × Y , and Q(ˆ c) = Q′ (c). Note that if c coincides with the restriction of cˆ onto X × Y then C(c; σ1 , σ2 ) = C(ˆ c; σ ˆ1 , σ ˆ2 ). Proposition 9. (cf. 5] and [26, Lemma 7]). I. Given a cost function c : X × Y → IR ∪ {+∞}, the following statements are equivalent: (a) c is H-convex relative to H from Example 1; (b) c is the restriction to X × Y of a function cˆ on (X ⊕ Y ) × (X ⊕ Y ), which is H-convex relative to H ⊂ C((X ⊕ Y ) × (X ⊕ Y )) from Example 2.

4]). The following statements are equivalent: (a) c is H-convex; (b) c is bounded below and lsc; (c) the duality relation C(c; σ1 , σ2 ) = D(c; σ1 , σ2 ) holds for all σ1 ∈ C(X)∗+ , σ2 ∈ C(Y )∗+ . Moreover, if these equivalent statements hold true then, for any positive measures σ1 , σ2 with σ1 X = σ2 Y , there exists an optimal solution to the MKP with marginals σ1 , σ2 . Proof. (a) ⇔ (b) See Remark 1. (a) ⇒ (c) Taking into account Example 1, this follows from Corollary 4. (c) ⇒ (a) Since µ = δ(x,y) is the sole positive measure with marginals σ1 = δx , σ2 = δy , one gets C(c; δx , δy ) = c(x, y).

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