Advertisement

Characterization of generalized FJ and KKT conditions in nonsmooth nonconvex optimization

  • Javad Koushki
  • Majid Soleimani-damanehEmail author
Article
  • 42 Downloads

Abstract

In this paper, we investigate new generalizations of Fritz John (FJ) and Karush–Kuhn–Tucker (KKT) optimality conditions for nonconvex nonsmooth mathematical programming problems with inequality constraints and a geometric constraint set. After defining generalized FJ and KKT conditions, we provide some alternative-type characterizations for them. We present characterizations of KKT optimality conditions without assuming traditional Constraint Qualification (CQ), invoking strong duality for a sublinear approximation of the problem in question. Such characterizations will be helpful when traditional CQs fail. We present the results with more details for a problem with a single-inequality constraint, and address an application of the derived results in mathematical programming problems with equilibrium constraints. The objective function and constraint functions of the dealt with problem are nonsmooth and we establish our results in terms of the Clarke generalized directional derivatives and generalized gradient. The results of the current paper cover classic optimality conditions existing in the literature and extend the outcomes of Flores-Bazan and Mastroeni (SIAM J Optim 25:647–676, 2015).

Keywords

FJ conditions KKT conditions Strong duality Nonconvex optimization Nonsmooth optimization 

Notes

References

  1. 1.
    Asadi, M.B., Soleimani-damaneh, M.: Infinite alternative theorems and nonsmooth constraint qualification conditions. Set-Valued Var. Anal. 20, 551–566 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bagirov, A., Karmitsa, N., Makela, M.: Introduction to Nonsmooth Optimization: Theory, Practice and Software. Springer, Basel (2014)CrossRefGoogle Scholar
  3. 3.
    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (2006)CrossRefGoogle Scholar
  4. 4.
    Bazaraa, M.S., Shetty, C.M.: Foundations of Optimization. Springer, Berlin (1976)CrossRefGoogle Scholar
  5. 5.
    Bertsekas, D. P.: Control of Uncertain Systems with a Set-Membership Description of the Uncertainty, Ph.D. Dissertation, Massachusetts Institute of Technology, Cambridge (1971)Google Scholar
  6. 6.
    Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control. Springer, London (2013)CrossRefGoogle Scholar
  7. 7.
    Flores-Bazan, F., Hadjisavvas, N., Vera, C.: An optimal alternative theorem and applications to mathematical programming. J. Glob. Optim. 37, 229–243 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Flores-Bazan, F., Mastroeni, G.: Characterizing FJ and KKT conditions in nonconvex mathematical programming with applications. SIAM J. Optim. 25, 647–676 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Flores-Bazan, F., Mastroeni, G.: Strong duality in cone constrained nonconvex optimization. SIAM J. Optim. 23, 153–169 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53, 99–110 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Giorgi, G., Guerraggio, A., Thierfelder, J.: Mathematics of Optimization: Smooth and Nonsmooth Case. Elsevier, Amsterdam (2004)zbMATHGoogle Scholar
  12. 12.
    Gould, F.J., Tolle, J.W.: A necessary and sufficient qualification for constrained optimization. SIAM J. Appl. Math. 20, 164–172 (1971)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gould, F.J., Tolle, J.W.: Geometry of optimality conditions and constraint qualifications. Math. Program. 2, 1–18 (1972)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Guignard, M.: Generalized Kuhn–Tucker conditions for mathematical programming problems in a Banach space. SIAM J. Control 7, 232–241 (1969)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hoheisel, T., Kanzowon, C.: On the Abadie and Guignard constraint qualification for mathematical programmes with vanishing constraints optimization. Optimization 58, 431–448 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jahn, J.: Introduction to the Theory of Nonlinear Optimization. Springer, Berlin (1996)CrossRefGoogle Scholar
  17. 17.
    Kabgani, A., Soleimani-damaneh, M.: Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators. Optimization 67, 217–235 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Karush, W.: Minima of Functions of Several Variables with Inequalities as Side Conditions, M.Sc. thesis, Department of Mathematics. University of Chicago, Chicago (1939)Google Scholar
  19. 19.
    Kuhn, H.W., Tucker, A.W.: Nonlinear Programming. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492. University of California Press, Berkeley (1951)Google Scholar
  20. 20.
    Makela, M.M., Neittaanmaki, P.: Nonsmooth Optimization: Analysis and Algorithms with Applications to Optimal Control. World Scientific, Singapore (1992)CrossRefGoogle Scholar
  21. 21.
    Mangasarian, O.L.: Nonlinear Programming. McGraw Hill, New York (1969)zbMATHGoogle Scholar
  22. 22.
    Mordukhovich, B.S., Nghia, T.T.A.: Constraint qualification and optimality conditions in semi-infinite and infinite programming. Math. Program. 139, 271–300 (2012)CrossRefGoogle Scholar
  23. 23.
    Penot, J.P.: Optimality conditions in mathematical programming and composite optimization. Math. Program. 67, 225–245 (1994)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefGoogle Scholar
  25. 25.
    Sekiguchi, Y., Takahashi, W.: Tangent and normal vectors to feasible regions with geometrically derivable sets. Sci. Math. Jpn. 64, 61–71 (2006) MathSciNetzbMATHGoogle Scholar
  26. 26.
    Soleimani-damaneh, M.: Nonsmooth optimization using Mordukhovich’s subdifferential. SIAM J. Control Optim. 48, 3403–3432 (2010)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ye, J.J., Zhang, J.: Enhanced Karush–Kuhn–Tucker condition and weaker constraint qualifications. Math. Program. B 139, 353–381 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics, Statistics and Computer Science, College of ScienceUniversity of TehranTehranIran

Personalised recommendations