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Optimality Conditions for a Class of Major Constraints Nonsmooth Multi-objective Programming Problems

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Advances in Intelligent Systems and Interactive Applications (IISA 2019)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1084))

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Abstract

The optimality conditions of major constraints nonsmooth programming with multi-objectives are studied. The objective function of this class of multi-objective programming is the sum of a convex vector function and a differentiable vector function. The constraint condition is major cone constraint in Euclid space. By using of the structure representation of major cone constraint of the given problem, the Fritz John condition of Pareto weakly effective solution is obtained by Gordan theorem. Meanwhile, the Kuhn-Tucker condition for Pareto weakly effective solution under Slater constraint qualification is given. The results are very useful to design its numerical methods.

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References

  1. Bao, T.Q., Gupta, P., Mordukhovich, B.S.: Necessary conditions in multiobjective optimization with equilibrium constraints. J. Optim. Theory Appl. 135, 179–203 (2007)

    Article  MathSciNet  Google Scholar 

  2. Bao, T.Q., Mordukhovich, B.S.: Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cybernet. 36, 531–562 (2007)

    MathSciNet  MATH  Google Scholar 

  3. Bao, T.Q., Mordukhovich, B.S.: Necessary conditions for super minimizers in constrained multiobjective optimization. J. Glob. Optim. 43, 533–552 (2009)

    Article  MathSciNet  Google Scholar 

  4. Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122, 301–347 (2010)

    Article  MathSciNet  Google Scholar 

  5. Luc, D.T.: On duality theorems in multiobjective programming. J. Optim. Theory Appl. 48, 557–582 (1984)

    Article  Google Scholar 

  6. Ben-Tal, A., Zowe, J.A.: Unified theory of first and second order conditions for extremum problems in topological vector spaces. Math. Program. Study 19, 39–76 (1982)

    Article  MathSciNet  Google Scholar 

  7. Michel, V.: Theorems of the alternative for multivalued mappings and applications to mixed convex\concave systems of inequalities. Set-Valued Anal. 18, 601–616 (2010)

    Article  MathSciNet  Google Scholar 

  8. Yang, X.M.: Generalized Preinvexity and Second Order Duality in Multiobjective Programming. Springer, Singapore (2018)

    Book  Google Scholar 

  9. Luc, D.T.: Multiobjective Linear Programming: An Introduction. Springer, Cham (2016)

    Book  Google Scholar 

  10. Mond, B.: A class of nondifferentiable mathematical programming problems. J. Math. Anal. Appl. 46, 169–174 (1974)

    Article  MathSciNet  Google Scholar 

  11. Mond, B., Schechter, M.A.: On a constraint qualification in a nonlinear programming problems. New Res. Logic Q. 23, 611–613 (1976)

    Article  Google Scholar 

  12. Xu, Z.K.: Constraint qualifications in a class of nondifferentiable mathematical programming problems. J. Math. Anal. Appl. 302, 282–290 (2005)

    Article  MathSciNet  Google Scholar 

  13. Luo, H.Z., Wu, H.X.: K-T necessary conditions for a class of nonsmooth multiobjective programming. OR Trans. 7, 62–68 (2003)

    Google Scholar 

  14. Wu, H.X., Luo, H.Z.: Necessary optimality conditions for a class of nonsmooth vector optimization problems. Acta Math. Appl. Sinica 25, 87–94 (2009)

    Article  MathSciNet  Google Scholar 

  15. Yang, X.M., Yang, X.Q., Teo, K.L.: Higher-order generalized convexity and duality in nondifferentiable multiobjective mathematical programming. J. Math. Anal. Appl. 297, 48–55 (2004)

    Article  MathSciNet  Google Scholar 

  16. Hu, Y.D.: Classes of major order of vector space. Chin. Ann. Math. 11, 269–280 (1990)

    MATH  Google Scholar 

  17. Hu, Y.D., Zhou, X.W.: Major constraint programming and its optimality conditions. OR Trans. 6, 69–75 (2002)

    Google Scholar 

  18. Rockfellar, R.T.: Convex Analysis. Princeton University Press, New Jersey (1972)

    Google Scholar 

  19. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, Hoboken (2006)

    Book  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Basic Courses, Zhejiang Shuren University, Hangzhou, 310015, China

    Xuanwei Zhou

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  1. Xuanwei Zhou
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Correspondence to Xuanwei Zhou .

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Editors and Affiliations

  1. Dept De Ciències De La Computació, Universitat Politècnica De Catalunya, Barcelona, Spain

    Fatos Xhafa

  2. Department of Computer Science and Engineering, SOA University, Bhubaneswar, Odisha, India

    Srikanta Patnaik

  3. Department of Business Systems and Analytics, La Salle University, Philadelphia, PA, USA

    Madjid Tavana

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Zhou, X. (2020). Optimality Conditions for a Class of Major Constraints Nonsmooth Multi-objective Programming Problems. In: Xhafa, F., Patnaik, S., Tavana, M. (eds) Advances in Intelligent Systems and Interactive Applications. IISA 2019. Advances in Intelligent Systems and Computing, vol 1084. Springer, Cham. https://doi.org/10.1007/978-3-030-34387-3_97

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