Higher Order Properly Efficient Points in Vector Optimization

Ivan Ginchev (), Angelo Guerraggio () and Matteo Rocca ()
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Ivan Ginchev: Technical University of Varna
Angelo Guerraggio: University of Insubria
Matteo Rocca: University of Insubria

A chapter in Generalized Convexity and Related Topics, 2007, pp 227-245 from Springer

Abstract: Summary We consider the constrained vector optimization problem minC f(x), g(x) ∈ − K, where f: ℝn → ℝm and g : ℝn → ℝp are given functions and C ∈ ℝm and K ∈ ℝp are closed convex cones. Two type of solutions are important for our considerations, namely i-minimizers (isolated minimizers) of order k and pminimizers (properly efficient points) of order k (see e.g. [11]). Every i-minimizer of order k ≥ 1 is a p-minimizer of order k. For k = 1, conditions under which the reversal of this statement holds have been given in [11]. In this paper we investigate the possible reversal of the implication i-minimizer ⇒ p-minimizer in the case k = 2. To carry on this study, we develop second-order optimality conditions for p-minimizers, expressed by means of Dini derivatives. Together with the optimality conditions obtained in [11] and [12] in the case of i-minimizers, they play a crucial role in the investigation. Further, to get a satisfactory answer to the posed reversal problem, we deal with sense I and sense II solution concepts, as defined in [11] and [5].

Keywords: Vector optimization; locally Lipschitz data; properly efficient points; isolated minimizers; optimality conditions (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/978-3-540-37007-9_13

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