Chain Rules for a Proper \(\varepsilon \) -Subdifferential of Vector Mappings

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• 作者：César Gutiérrez ; Lidia Huerga ; Vicente Novo…
• 关键词：Vector optimization ; Proper $$\varepsilon $$ ε ; efficiency ; Proper $$\varepsilon $$ ε ; subdifferential ; Strong $$\varepsilon $$ ε ; efficiency ; Strong $$\varepsilon $$ ε ; subdifferential ; Nearly cone ; subconvexlikeness ; Linear scalarization ; 90C48 ; 90C25 ; 90C29 ; 49J52 ; 49K27
• 刊名：Journal of Optimization Theory and Applications
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：167
• 期：2
• 页码：502-526
• 全文大小：575 KB
• 参考文献：1.Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Proper approximate solutions and \(\varepsilon \) -subdifferentials in vector optimization: Basic properties and limit behaviour. Nonlinear Anal. 79, 52-7 (2013)MATH MathSciNet CrossRef
  2.Gutiérrez, C., Huerga, L., Novo, V.: Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems. J. Math. Anal. Appl. 389, 1046-058 (2012)MATH MathSciNet CrossRef
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  4.Gutiérrez, C., Jiménez, B., Novo, V.: On approximate efficiency in multiobjective programming. Math. Methods Oper. Res. 64, 165-85 (2006)MATH MathSciNet CrossRef
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• 作者单位：César Gutiérrez (1)
  Lidia Huerga (2)
  Vicente Novo (2)
  Lionel Thibault (3)
  
  1. Universidad de Valladolid, Valladolid, Spain
  2. Universidad Nacional de Educación a Distancia, Madrid, Spain
  3. Université Montpellier 2, Montpellier, France
  
• 刊物主题：Calculus of Variations and Optimal Control; Optimization; Optimization; Theory of Computation; Applications of Mathematics; Engineering, general; Operations Research/Decision Theory;
• 出版者：Springer US
• ISSN：1573-2878

文摘

In this paper, we derive exact chain rules for a proper epsilon-subdifferential in the sense of Benson of extended vector mappings, recently introduced by ourselves. For this aim, we use a new regularity condition and a new strong epsilon-subdifferential for vector mappings. In particular, we determine chain rules when one of the mappings is linear, obtaining formulations easier to handle in the finite-dimensional case by considering the componentwise order. This Benson proper epsilon-subdifferential generalizes and improves several of the most important proper epsilon-subdifferentials of vector mappings given in the literature and, consequently, the results presented in this work extend known chain rules stated for the last ones. As an application, we derive a characterization of approximate Benson proper solutions of implicitly constrained convex Pareto problems. Moreover, we estimate the distance between the objective values of these approximate proper solutions and the set of nondominated attained values. Keywords Vector optimization Proper \(\varepsilon \)-efficiency Proper \(\varepsilon \)-subdifferential Strong \(\varepsilon \)-efficiency Strong \(\varepsilon \)-subdifferential Nearly cone-subconvexlikeness Linear scalarization