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Báo cáo hóa học: Research Article Second-Order Contingent Derivative of the Perturbation Map in Multiobjective Optimization

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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Second-Order Contingent Derivative of the Perturbation Map in Multiobjective Optimization
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Báo cáo hóa học: " Research Article Second-Order Contingent Derivative of the Perturbation Map in Multiobjective Optimization"Hindawi Publishing CorporationFixed Point Theory and ApplicationsVolume 2011, Article ID 857520, 13 pagesdoi:10.1155/2011/857520Research ArticleSecond-Order Contingent Derivative of thePerturbation Map in Multiobjective Optimization Q. L. Wang1 and S. J. Li2 1 College of Sciences, Chongqing Jiaotong University, Chongqing 400074, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China Correspondence should be addressed to Q. L. Wang, wangql97@126.com Received 14 October 2010; Accepted 24 January 2011 Academic Editor: Jerzy Jezierski Copyright q 2011 Q. L. Wang and S. J. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Some relationships between the second-order contingent derivative of a set-valued map and its profile map are obtained. By virtue of the second-order contingent derivatives of set-valued maps, some results concerning sensitivity analysis are obtained in multiobjective optimization. Several examples are provided to show the results obtained.1. IntroductionIn this paper, we consider a family of parametrized multiobjective optimization problems ⎧ ⎨min f u, x f1 u, x , f2 u, x , . . . , fm u, x , PVOP 1.1 ⎩s.t. u ∈ X x ⊆ Rp .Here, u is a p-dimensional decision variable, x is an n-dimensional parameter vector, X is anonempty set-valued map from Rn to Rp , which specifies a feasible decision set, and f is anobjective map from Rp × Rn to Rm , where m, n, p are positive integers. The norms of all finitedimensional spaces are denoted by · . C is a closed convex pointed cone with nonemptyinterior in Rm . The cone C induces a partial order ≤C on Rm , that is, the relation ≤C is definedby y ≤C y ←→ y − y ∈ C, ∀y, y ∈ Rm . 1.22 Fixed Point Theory and ApplicationsWe use the following notion. For any y, y ∈ Rm , y Fixed Point Theory and Applications 3order tangent sets. To the best of our knowledge, second-order contingent derivatives ofperturbation map in multiobjective optimization have not been studied until now. Motivatedby the work reported in 5–11, 14 , we discuss some second-order quantitative resultsconcerning the behavior of the perturbation map for PVOP . The rest of the paper is organized as follows. In Section 2, we collect some importantconcepts in this paper. In Section 3, we discuss some relationships between the second-ordercontingent derivative of a set-valued map and its profile map. In Section 4, by the second-order contingent derivative, we discuss the quantitative information on the behavior of theperturbation map for PVOP .2. PreliminariesIn this section, we state several important concepts. m Let F : Rn → 2R be nonempty set-valued maps. The efficient domain and graph of Fare defined by {x ∈ Rn | F x / ∅}, dom F 2.1 x, y ∈ R × R | y ∈ F x , x ∈ R , n m n gph F C, for every x ∈ dom F ,respectively. The profile map F of F is defined by F x Fxwhere C is the order cone of Rm .Definition 2.1 see 18 . A base for C is a nonempty convex subset Q of C with 0Rm ∈ clQ, /such that every c ∈ C, c / 0Rm , has a unique representation of the form αb, where b ∈ Q andα > 0.Definition 2.2 see 19 . F is said to be locally Lipschitz at x0 ∈ Rn if there exist a real numberγ > 0 and a neighborhood U x0 of x0 , such that F x1 ⊆ F x2 γ x1 − x2 BRm , ∀x1 , x2 ∈ U x0 , 2.2where BRm denotes the closed unit ball of the origin in Rm .3. Second-Order Contingent Derivatives for Set-Valued MapsIn this section, let X be a normed space supplied with a distance d, and let A be a subset of infy∈A d x, y the distance from x to A, where we set d x, ∅ ∞.X . We d ...

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