Necessary Conditions for Nondominated Solutions in Vector Optimization
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Necessary Conditions for Nondominated Solutions in Vector Optimization Truong Q. Bao1 · Lidia Huerga2
· Bienvenido Jiménez2
· Vicente Novo2
Received: 28 August 2019 / Accepted: 28 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we study characterizations and necessary conditions for nondominated points of sets and nondominated solutions of vector-valued functions in vector optimization with variable domination structure. We study not only the case, where the intersection of all the involved domination sets has a nonzero element, but also the case, where it might be the singleton. While the first case has been studied earlier, the second case has not, to the best of our knowledge, done yet. Our results extend and improve the existing results in vector optimization with a fixed ordering cone and with a variable ordering structure. Keywords Vector optimization · Domination structures · Nondominated solutions · Generalized differentiation · Nonlinear scalarization functions Mathematics Subject Classification 49J52 · 49J53 · 90C29 · 90C30
Communicated by Alfredo N. Iusem.
B
Vicente Novo [email protected] Truong Q. Bao [email protected] Lidia Huerga [email protected] Bienvenido Jiménez [email protected]
1
Department of Mathematics and Computer Science, Northern Michigan University, Marquette, MI 49855, USA
2
Departamento de Matemática Aplicada I, E.T.S.I. Industriales, UNED, c/ Juan del Rosal 12, Ciudad Universitaria, 28040 Madrid, Spain
123
Journal of Optimization Theory and Applications
1 Introduction This paper addresses problems of set and vector-valued functions in optimization with a variable domination structure. It can be viewed as an extension of vector optimization in which domination sets vary in decision spaces. The nondomination concept in multiobjetive optimization, with respect to domination structures, was introduced by Yu [1] and studied in many publications; see, e.g., [2–4] and recent papers [5–9]. This concept is more general than efficiency, and it is applicable to decision making, games, image registration in medical engineering, etc. There are a few necessary conditions for this kind of optimal solutions in the literature obtained for several special classes of ordering structures: – In [10], Engau formulated necessary conditions for nondominated points to sets with respect to ordering structures, whose domination factor sets are idealsymmetric convex cones. His technique heavily relies on the geometric angles in R2 and R3 . – In [9], Eichfelder and Ha established generalized Fermat and Lagrange multiplier rules for nondominated solutions for a special case, where the cones describing this ordering structure are Bishop-Phelps cones. – In [5], Bao and Mordukhovich obtained necessary conditions for nondominated points of sets and for nondominated solutions of vector optimization problems for the class of convex-cone-valued ordering structures with a nontrivial intersection. In [6], Bao extended them to ordering structures whose
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