Variational Methods in Partially Ordered Spaces

Abstract

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Generalized performance ratios and risk optimization

“…Namely, we pose A = S, D = K n , n ≥ 2. Also, we take some k 0 ∈ K n+2 , k 0 = 0, and according to the same Lemma C = K n+2 , being a cone, satisfying the condition C + int(D) ⊆ C. Hence, the conclusion of [6,Th.2.3.6], (since the recession conditions 2.22, 2.28 also hold for K n , k 0 ), implies the existence of a continuous, convex functional z Kn,k 0 , such that the scalarization of the constraint of the above problem, as follows:…”

“…namely an unconstrained maximization problem of some continuous, concave functional. By following [6], z Kn,k 0 (x) = inf{t ∈ R|x ∈ tk 0 − K n }. Proof: According to [13,Lem.3.2,Cor.3.3,Th.3.4], z Kn,k 0 (x) = inf{t ∈ R|y ∈ tk 0 − K n } = inf{t ∈ R|x + t(−k 0 ) ∈ −K n } = ρ (−Kn,−k 0 ) (x).…”

Henig proper efficiency in vector optimization with variable ordering structure

“…Introduction. Proper efficiency was very much investigated in the case of Pareto optimality defined by a fixed ordering structure (FOS, for short): see for instance the monograph [11] and the references therein.…”

Optimality conditions for a bilevel optimization problem in terms of KKT multipliers and convexificators

“…Under the following hypotheses (H 1 ) , (H 2 ) , (H 3 ) and (H 4 ) , the optimization problem (P ) has at least one optimal solution[9]. (H 1 ) : F (., .)…”