Duality related to approximate proper solutions of vector optimization problems

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• 作者：C. Gutiérrez ; L. Huerga ; V. Novo ; C. Tammer
• 关键词：Vector optimization ; Approximate duality ; Proper \(\varepsilon \) ; efficiency ; Nearly cone ; subconvexlikeness ; Linear scalarization
• 刊名：Journal of Global Optimization
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：64
• 期：1
• 页码：117-139
• 全文大小：571 KB
• 参考文献：1.Boţ, R.I., Grad, S.-M.: Duality for vector optimization problems via a general scalarization. Optimization 60, 1269–1290 (2011)MATH MathSciNet CrossRef
  2.Boţ, R.I., Grad, S.-M., Wanka, G.: Duality in Vector Optimization. Springer, Berlin (2009)MATH
  3.Boţ, R.I., Wanka, G.: An analysis of some dual problems in multiobjective optimization (I). Optimization 53, 281–300 (2004)MATH MathSciNet CrossRef
  4.Boţ, R.I., Wanka, G.: An analysis of some dual problems in multiobjective optimization (II). Optimization 53, 301–324 (2004)CrossRef
  5.Dutta, J., Vetrivel, V.: On approximate minima in vector optimization. Numer. Funct. Anal. Optim. 22, 845–859 (2001)MATH MathSciNet CrossRef
  6.Ekeland, I., Témam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)MATH
  7.Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)MATH
  8.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, 52–67 (2013)MATH MathSciNet CrossRef
  9.Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Proper approximate solutions and \(\varepsilon \) -subdifferentials in vector optimization: Moreau–Rockafellar type theorems. J. Convex Anal. 21, 857–886 (2014)MATH MathSciNet
  10.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, 1046–1058 (2012)MATH MathSciNet CrossRef
  11.Gutiérrez, C., Jiménez, B., Novo, V.: Multiplier rules and saddle-point theorems for Helbig’s approximate solutions in convex Pareto problems. J. Glob. Optim. 32, 367–383 (2005)
  12.Gutiérrez, C., Jiménez, B., Novo, V.: On approximate efficiency in multiobjective programming. Math. Methods Oper. Res. 64, 165–185 (2006)MATH MathSciNet CrossRef
  13.Gutiérrez, C., Jiménez, B., Novo, V.: A unified approach and optimality conditions for approximate solutions of vector optimization problems. SIAM J. Optim. 17, 688–710 (2006)MATH MathSciNet CrossRef
  14.Gutiérrez, C., Jiménez, B., Novo, V.: A generic approach to approximate efficiency and applications to vector optimization with set-valued maps. J. Glob. Optim. 49, 313–342 (2011)MATH CrossRef
  15.Jahn, J.: Duality in vector optimization. Math. Program. 25, 343–353 (1983)MATH MathSciNet CrossRef
  16.Jahn, J.: Vector Optimization. Theory, Applications and Extensions. Springer, Berlin (2011)MATH
  17.Jia, J.-H., Li, Z.-F.: \(\varepsilon \) -Conjugate maps and \(\varepsilon \) -conjugate duality in vector optimization with set-valued maps. Optimization 57, 621–633 (2008)MATH MathSciNet CrossRef
  18.Kutateladze, S.S.: Convex \(\varepsilon \) -programming. Soviet Math. Dokl. 20, 391–393 (1979)MATH
  19.Lemaire, B.: Approximation in multiobjective optimization. J. Glob. Optim. 2, 117–132 (1992)MATH MathSciNet CrossRef
  20.Li, Z.F.: Benson proper efficiency in the vector optimization of set-valued maps. J. Optim. Theory Appl. 98, 623–649 (1998)MATH MathSciNet CrossRef
  21.Luc, D.T.: On duality theory in multiobjective programming. J. Optim. Theory Appl. 43, 557–582 (1984)MATH MathSciNet CrossRef
  22.El Maghri, M.: Pareto-Fenchel \(\varepsilon \) -subdifferential sum rule and \(\varepsilon \) -efficiency. Optim. Lett. 6, 763–781 (2012)MATH MathSciNet CrossRef
  23.El Maghri, M.: (\(\varepsilon \) -)Efficiency in difference vector optimization. J. Glob. Optim. 61, 803–812 (2015)CrossRef
  24.Qiu, J.H.: Dual characterization and scalarization for Benson proper efficiency. SIAM J. Optim. 19, 144–162 (2008)MATH MathSciNet CrossRef
  25.Rong, W.D., Wu, Y.N.: \(\varepsilon \) -Weak minimal solutions of vector optimization problems with set-valued maps. J. Optim. Theory Appl. 106, 569–579 (2000)MATH MathSciNet CrossRef
  26.Sach, P.H., Tuan, L.A., Minh, N.B.: Approximate duality for vector quasi-equilibrium problems and applications. Nonlinear Anal. 72, 3994–4004 (2010)MATH MathSciNet CrossRef
  27.Son, T.Q., Kim, D.S.: \(\varepsilon \) -Mixed type duality for nonconvex multiobjective programs with an infinite number of constraints. J. Glob. Optim. 57, 447–465 (2013)MATH MathSciNet CrossRef
  28.Tanino, T., Sawaragi, Y.: Duality theory in multiobjective programming. J. Optim. Theory Appl. 27, 509–529 (1979)MATH MathSciNet CrossRef
  29.Vályi, I.: Approximate saddle-point theorems in vector optimization. J. Optim. Theory Appl. 55, 435–448 (1987)MATH MathSciNet CrossRef
  30.Yang, X.M., Li, D., Wang, S.Y.: Near-subconvexlikeness in vector optimization with set-valued functions. J. Optim. Theory Appl. 110, 413–427 (2001)MATH MathSciNet CrossRef
  
• 作者单位：C. Gutiérrez (1)
  L. Huerga (2)
  V. Novo (2)
  C. Tammer (3)
  
  1. Departamento de Matemática Aplicada, E.T.S. de Ingenieros de Telecomunicación, Universidad de Valladolid, Paseo de Belén 15, Campus Miguel Delibes, 47011, Valladolid, Spain
  2. Departamento de Matemática Aplicada, E.T.S.I. Industriales, UNED, c/ Juan del Rosal 12, Ciudad Universitaria, 28040, Madrid, Spain
  3. Institute of Mathematics, Faculty of Natural Sciences II, Martin-Luther University Halle-Wittenberg, 06099, Halle (Saale), Germany
  
• 刊物类别：Business and Economics
• 刊物主题：Economics
  Operation Research and Decision Theory
  Computer Science, general
  Real Functions
  Optimization
  
• 出版者：Springer Netherlands
• ISSN：1573-2916

文摘

In this work we introduce two approximate duality approaches for vector optimization problems. The first one by means of approximate solutions of a scalar Lagrangian, and the second one by considering \((C,\varepsilon )\)-proper efficient solutions of a recently introduced set-valued vector Lagrangian. In both approaches we obtain weak and strong duality results for \((C,\varepsilon )\)-proper efficient solutions of the primal problem, under generalized convexity assumptions. Due to the suitable limit behaviour of the \((C,\varepsilon )\)-proper efficient solutions when the error \(\varepsilon \) tends to zero, the obtained duality results extend and improve several others in the literature.