Advertisement

\(\varPhi -\)Weak Slater Constraint Qualification in Nonsmooth Multiobjective Semi-infinite Programming

  • Ali SadeghiehEmail author
  • David Barilla
  • Giuseppe Caristi
  • Nader Kanzi
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

In this paper, we consider a nonsmooth multiobjective semi-infinite programming problem with a feasible set defined by inequality constraints. First we introduce the weak Slater constraint qualification, and derive the Karush-Kuhn-Tucker types necessary conditions for (weakly, properly) efficient solution of the considered problem. Then, we introduce a new gap function for the problem and state sufficient conditions for (weakly, properly) efficient solution of the problem via this gap function.

Keywords

Semi-infinite programming Multiobjective optimization Constraint qualification Optimality conditions Gap function 

References

  1. 1.
    Antczak, T.: Saddle point criteria and Wolfe duality in nonsmooth \((\Phi ,\rho )-\)invex vector optimization problems with inequality and equality constraints. Int. J. Comput. Math. 92(5), 882–907 (2015)Google Scholar
  2. 2.
    Antczak, T., Stasiak, A.: \((\Phi ,\rho )-\)invexity in nonsmooth optimization. Numer. Func. Anal. Optim. 32, 1–25 (2015)Google Scholar
  3. 3.
    Caristi, G., Kanzi, M., Soleimani-damaneh, M.: On gap functions for nonsmooth multiobjective optimization problems. Optim. Lett. (2017).  https://doi.org/10.1007/s11590-017-1110-4MathSciNetCrossRefGoogle Scholar
  4. 4.
    Caristi, G., Ferrara, M., Stefanescu, A.: Semi-infinite multiobjective programming with grneralized invexity. Math. Rep. 62, 217–233 (2010)Google Scholar
  5. 5.
    Caristi, G., Ferrara, M., Stefanescu, A.: Mathematical programming with \((\rho ,\Phi )\)-invexity. In: Konnor, I.V., Luc, D.T., Rubinov, A.M. (eds.) Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, vol. 583, pp. 167–176. Springer, Heidelberg (2006)Google Scholar
  6. 6.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, Interscience (1983)Google Scholar
  7. 7.
    Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)Google Scholar
  8. 8.
    Goberna, M.A., Kanzi, N.: Optimality conditions in convex multiobjective SIP. Math. Program. (2017).  https://doi.org/10.1007/s10107-016-1081-8MathSciNetCrossRefGoogle Scholar
  9. 9.
    Goberna, M., Guerra-Vazquez, F., Todorov, M.I.: Constraint qualifications in linear vector semi-infinite optimization. Eur. J. Oper. Res. 227, 32–40 (2016)Google Scholar
  10. 10.
    Goberna, M.A., Guerra-Vazquez, F., Todorov, M.I.: Constraint qualifications in convex vector semi-infinite optimization. Eur. J. Oper. Res. 249, 12–21 (2013)Google Scholar
  11. 11.
    Gopfert, A., Riahi, H., Tammer, C., Zalinescu, C.: Variational Methods in Partial Ordered Spaces. Springer, New York (2003)Google Scholar
  12. 12.
    Guerraggio, A., Molho, E., Zaffaroni, A.: On the notion of proper efficiency in vector optimization. J. Optim. Theory Appl. 82, 1–21 (1994)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hearn, D.W.: The gap function of a convex program. Oper. Res. Lett. 1, 67–71 (1982)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hettich, R., Kortanek, O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35, 380–429 (1993)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hiriart-Urruty, J.B., Lemarechal, C.: Convex Analysis and Minimization Algorithms. I & II. Springer, Heidelberg (1991)Google Scholar
  16. 16.
    Kanzi, N., Shaker Ardekani, J., Caristi, G.: Optimality scalarization and duality in linear vector semi-infinite programming. Optimization (2018).  https://doi.org/10.1080/02331934.2018.1454921MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kanzi, N.: Necessary and sufficient conditions for (weakly) efficient of nondifferentiable multi-objective semi-infinite programming. Iran. J. Sci. Technol. Trans. A Sci. (2017).  https://doi.org/10.1007/s40995-017-156-6
  18. 18.
    Kanzi, N.: Necessary Optimality conditions for nonsmooth semi-infinite programming problems. J. Global Optim. 49, 713–725 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kanzi, N.: Constraint qualifications in semi-infinite systems and their applications in nonsmooth semi-infinite problems with mixed constraints. SIAM J. Optim. 24, 559–572 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kanzi, N.: On strong KKT optimality conditions for multiobjective semi-infinite programming problems with Lipschitzian data. Optim. Lett. 9, 1121–1129 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kanzi, N., Nobakhtian, S.: Optimality conditions for nonsmooth semi-infinite multiobjective programming. Optim. Lett. (2013).  https://doi.org/10.1007/s11590-013-0683-9MathSciNetCrossRefGoogle Scholar
  22. 22.
    López, M.A., Vercher, E.: Optimality conditions for nondifferentiable convex semi-infinite programming. Math. Program. 27, 307–319 (1983)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsIslamic Azad UniversityYazdIran
  2. 2.Department of EconomicsUniversity of MessinaMessinaItaly
  3. 3.Department of MathematicsPayam Noor UniversityTehranIran

Personalised recommendations