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

ISBN-10: 3540370064

ISBN-13: 9783540370062

ISBN-10: 3540370072

ISBN-13: 9783540370079

The publication comprises invited papers by means of recognized specialists on quite a lot of themes (economics, variational research, chance etc.) heavily on the topic of convexity and generalized convexity, and refereed contributions of experts from the realm on present learn on generalized convexity and purposes, specifically, to optimization, economics and operations examine.

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**Extra info for Generalized convexity and related topics**

**Example text**

Clearly, both the functionals, (13) and (14), are sublinear, and vI′ (c; e′0 ) ≤ ′ vII (c; e′0 ). 1 (see also Remark 1 after it) in [47]. Proposition 5. Given e′0 ∈ dom vI′ (c; ·) := {e′ ∈ E ′ : vI′ (c; e′ ) < +∞}, the following assertions are equivalent: ′ (c; e′0 ); (a) vI′ (c; e′0 ) = vII ′ (b) the functional vII (c; ·) is weakly lower semi-continuous at e′0 . Let us deﬁne H := AK; then H 0 = (A∗ )−1 (K 0 ). Remark 3. Note that if dom vI′ (c; ·) ⊆ A∗ C(Ω)∗+ , (15) then, for every e′0 ∈ dom vI′ (c; ·), vI′ (c; e′0 ) = vI (c; µ0 ) and ′ vII (c; e′0 ) = vII (c; µ0 ) / dom vI′ (c; ·), one has whenever µ0 ∈ (A∗ )−1 (e′0 ).

Theorem 4. 1]). A multifunction F : X → L is L-cyclic monotone if and only if Q0 (ϕF ) is nonempty. Theorem 5. 2]). Suppose F : X → L is L-cyclic monotone. Given a function u : Z = dom F → IR ∪ {+∞}, the following statements are equivalent: (a) u ∈ Q0 (ϕF ); (b) u is a restriction to Z of some L-convex function U : X → IR ∪ {+∞}, and F (z) ⊆ ∂L U (z) for all z ∈ Z. Abstract Convexity and the Monge–Kantorovich Duality 51 The next result extending a classical convex analysis theorem due to Rockafellar is an immediate consequence of Theorems 4 and 5.

Abstract Convexity and the Monge–Kantorovich Duality 37 The original program is to maximize the linear functional h, µ0 := h(ω) µ0 (dω) subject to constraints: h ∈ H, h(ω) ≤ c(ω) for all ω ∈ Ω. Ω The optimal value of this program will be denoted as vI (c; µ0 ). , µ ∈ C(Ω)∗+ ) and µ ∈ µ0 − H 0 , where H 0 stands for the conjugate (polar) cone in C(Ω)∗+ , H 0 := {µ ∈ C(Ω)∗ : h, µ ≤ 0 for all h ∈ H}. The optimal value of this program will be denoted as vII (c; µ0 ). Thus, for any µ0 ∈ C(Ω)∗+ , one has vI (c; µ0 ) = sup{ h, µ0 : h ∈ H(c)}, (1) vII (c; µ0 ) = inf{c(µ) : µ ≥ 0, µ ∈ µ0 − H 0 }.

### Generalized convexity and related topics by Igor V. Konnov, Dinh The Luc, Alexander M. Rubinov

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