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Journal of Systems Science and Complexity

, Volume 28, Issue 5, pp 1148–1163 | Cite as

Multiobjective program with support functions under (G, C, ρ)-convexity assumptions

  • Dehui YuanEmail author
  • Xiaoling Liu
Article
  • 87 Downloads

Abstract

This paper deals with multiobjective programming problems with support functions under (G, C, ρ)-convexity assumptions. Not only sufficient but also necessary optimality conditions for this kind of multiobjective programming problems are established from a viewpoint of (G, C, ρ)-convexity. When the sufficient conditions are utilized, the corresponding duality theorems are derived for general Mond-Weir type dual program.

Keywords

(G, C, ρ)-convex G-duality theory G-necessary conditions G-sufficient conditions multiobjective program 

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References

  1. [1]
    Soleimani-damaneh M, On some multiobjective optimization problems arising in biology, International Journal of Computer Mathematics, 2011, 88: 1103–1119.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Soleimani-damaneh M, An optimization modelling for string selection in molecular biology using Pareto optimality, Applied Mathematical Modelling, 2011, 35: 3887–3892.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Handl J, Kell D, and Knowles J, Multiobjective optimization in bioinformatics and computational biology, IEEE/ACM Transactions on Computational Biology and Bioinformatics, 2007, 4: 279–292.CrossRefGoogle Scholar
  4. [4]
    Hakanen J, Miettinen K, and Sahlstedt K, Wastewater treatment: New insight provided by interactive multiobjective optimization, Decision Support Systems, 2011, 51: 328–337.CrossRefGoogle Scholar
  5. [5]
    Abakarov A, Sushkov Y, and Sunpson R, Multiobjective optimization approach: Thermal food processing, Journal of Food Science, 2009, 74(9): E471–E487.Google Scholar
  6. [6]
    Ekins S, Honeycutt J, and Metz J, Evolving molecules using multi-objective optimization: Applying to ADME/Tox, Drug Discovery Today, 2010, 15(11–12): 451–460.CrossRefGoogle Scholar
  7. [7]
    Chandra S, Craven B, and Mond B, Generalized concavity and duality with a square root term, Optimization, 1985, 16(5): 653–662.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Chandra S and Husain I, Symmetric dual nondifferentiable programs, Bulletin of the Australian Mathematical Society, 1981, 24(2): 295–307.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Mond B and Schechter M, Nondifferentiable symmetric duality, Bulletin of the Australian Mathematical Society, 1996, 53(2): 177–188.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Kim S, Kim M, and Kim D, Optimality and duality for a class of nondifferentiable multiobjective fractional programming problems, Journal of Optimization Theory and Applications, 2006, 129(1): 131–146.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Kim D and Bae K, Optimality conditions and duality for a class of nondifferentiable multiobjective programming problems, Taiwanese Journal of Mathematics, 2009, 13(2B): 789–804.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Sang K and Jung L, Optimality conditions and duality in nonsmooth multiobjective programs, Journal of Inequalities and Applications, 2010, 2010: Article ID 939537.Google Scholar
  13. [13]
    Bae K, Kang Y, and Kim D, Efficiency and generalized convex duality for nondifferentiable multiobjective programs, Journal of Inequalities and Applications, 2010, 2010: Article ID 930457.Google Scholar
  14. [14]
    Bae K and Kim D, Optimality and duality theorems in nonsmooth multiobjective optimization, Fixed Point Theory and Applications, 2011, 2011(42): 1–11.MathSciNetGoogle Scholar
  15. [15]
    Liang Z, Huang H, and Pardalos P, Optimality conditions and duality for a class of nonlinear fractional programming problems, Journal of Optimization Theory and Applications, 2001, 110(3): 611–619.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Liang Z, Huang H, and Pardalos P, Efficiency conditions and duality for a class of multiobjective fractional programming problems, Journal of Global Optimization, 2003, 27(4): 447–471.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Preda V, On sufficiency and duality for multiobjective programs, Journal of Mathematical Analysis and Applications, 1992, 166(2): 365–377.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Bhatia D and Jain P, Generalized (F, ρ)-convexity and duality for nonsmooth multiobjective programs, Optimization, 1994, 31(2): 153–164.CrossRefzbMATHGoogle Scholar
  19. [19]
    Yuan D, Liu X, Chinchuluun A, and Pardalos P, Nondifferentiable minimax fractional programming problems with (C, a, α, d)-convexity, Journal of Optimization Theory and Applications, 2006, 129(1): 185–199.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Chinchuluun A, Yuan D, and Pardalos P, Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity, Annals of Operations Research, 2007, 154(1): 133–147.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Long X, Optimality conditions and duality for nondifferentiable multiobjective fractional programming problems with (C, α, a, d)-convexity, Journal of Optimization Theory and Applications, 2011, 148(1): 197–208.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Soleimani-Damaneh M, Nonsmooth optimization using Mordukhovich’s subdifferential, SIAM Journal on Control and Optimization, 2010, 48(5): 3403–3436.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Soleimani-Damaneh M, Characterizations and applications of generalized invexity and monotonicity in Asplund spaces, TOP, 2012, 20(3): 592–613.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Soleimani-damaneh M, Optimality and invexity in optimization problems in Banach algebras (spaces), Nonlinear Analysis: Theory, Methods and Applications, 2009, 71(11): 5522–5530.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Soleimani-Damaneh M, Multiple-objective programs in Banach spaces: Sufficiency for (proper) optimality, Nonlinear Analysis: Theory, Methods and Applications, 2007, 67(3): 958–962.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Soleimani-Damaneh M, Duality for optimization problems in Banach algebras, Journal of Global Optimization, 2012, 54(2): 375–388.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Antczak T, New optimality conditions and duality results of G-type in differentiable mathematical programming, Nonlinear Analysis: Theory, Methods and Applications, 2007, 66(7): 1617–1632.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Antczak T, On G-invex multiobjective programming. Part I. Optimality, Journal of Global Optimization, 2009, 43(1): 97–109.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Antczak T, On G-invex multiobjective programming. Part II. Duality, Journal of Global Optimization, 2009, 43(1): 111–140.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Yuan D, Liu X, Yang S, and Lai G, Multiobjective programming problems with (G, C, ρ)- convexity, Journal of Applied Mathematics and Computing, 2012, 40(1–2): 383–397.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Yang X, Teo K, and Yang X, Duality for a class of nondifferentiable multiobjective programming problems, Journal of Mathematical Analysis and Applications, 2000, 252(2): 999–1005.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsHanshan Normal UniversityChaozhouChina

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