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Optimality conditions for an exhausterable function on an exhausterable set

  • Majid E. AbbasovEmail author
Article
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Abstract

Exhausters are families of convex compact sets that allow one to represent directional derivative of the studied function at a point in the form of InfMax or SupMin of linear functions. Functions for which such a representation is valid we call exhausterable. This class of functions is quite wide and contains many nonsmooth ones. The set of exhausterable function is also called exhausterable. In the present paper we describe optimality conditions for an exhausterable function on an exhausterable set. These conditions can be used for solving many nondifferentiable optimization problems. An example that illustrate obtained results is provided.

Keywords

Exhausters Nonsmooth analysis Nondifferentiable optimization Constrained optimization Optimality conditions 

Mathematics Subject Classification

49J52 90C46 90C47 

Notes

Acknowledgements

The reported study was supported by Russian Science Foundation, Research Project No. 18-71-00006.

References

  1. 1.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefGoogle Scholar
  2. 2.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  3. 3.
    Mordukhovich, B.S.: Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40, 960–969 (1976)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Michel, P., Penot, J.-P.: Calcul sous-differentiel pour les fonctions lipschitziennes et non lipschitziennes. C. R. Acad. Sci. Paris 298, 269–272 (1984)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Hiriart-Urruty, J.-B., Lewis, A.S.: The Clarke and Michel–Penot subdifferentials of the eigenvalues of a symmetric matrix. Comput. Optim. Appl. 13, 13–23 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Mordukhovich, B.S.: Variational Analysis and Applications. Springer, New York (2018)CrossRefGoogle Scholar
  7. 7.
    Pshenichny, B.N.: Convex Analysis and Extremal Problems. Nauka, Moscow (1980). (in Russian)Google Scholar
  8. 8.
    Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis. Approximation and Optimization, vol. 7, p. iv+416. Peter Lang, Frankfurt (1995)zbMATHGoogle Scholar
  9. 9.
    Demyanov, V.F., Rubinov, A.M.: Exhaustive families of approximations revisited. In: Gilbert, R.P., Panagiotopoulos, P.D., Pardalos, P.M. (eds.) From Convexity to Nonconvexity. Nonconvex Optimization and Its Applications, vol. 55, pp. 43–50. Kluwer Academic, Dordrecht (2001)CrossRefGoogle Scholar
  10. 10.
    Demyanov, V.F.: Exhausters af a positively homogeneous function. Optimization 45, 13–29 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Demyanov, V.F.: Exhausters and convexificators—new tools in nonsmooth analysis. In: Demyanov, V., Rubinov, A. (eds.) Quasidifferentiability and Related Topics, pp. 85–137. Kluwer Academic, Dordrecht (2000)CrossRefGoogle Scholar
  12. 12.
    Demyanov, V.F., Roshchina, V.A.: Optimality conditions in terms of upper and lower exhausters. Optimization 55, 525–540 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Abbasov, M.E., Demyanov, V.F.: Proper and adjoint exhausters in nonsmooth analysis: optimality conditions. J. Glob. Optim. 56, 569–585 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Abbasov, M.E., Demyanov, V.F.: Extremum conditions for a nonsmooth function in terms of exhausters and coexhausters. Proc. Steklov Inst. Math. 269, 6–15 (2010)CrossRefGoogle Scholar
  15. 15.
    Demyanov, V.F., Roshchina, V.A.: Constrained optimality conditions in terms of proper and adjoint exhausters. Appl. Comput. Math. 4, 114–124 (2005)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Castellani, M.: A dual representation for proper positively homogeneous functions. J. Glob. Optim. 16, 393–400 (2000)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Roshchina, V.A.: Reducing exhausters. J. Optim. Theory Appl. 136, 261–273 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Küçük, M., Urbanski, R., Grzybowski, J., et al.: Reduction of weak exhausters and optimality conditions via reduced weak exhausters. J. Optim. Theory Appl. 165, 693–707 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Abbasov, M.E.: Geometric conditions of reduction of exhausters. J. Glob. Optim. (2018).  https://doi.org/10.1007/s10898-018-0683-5 CrossRefzbMATHGoogle Scholar
  20. 20.
    Abbasov, M.E.: Generalized exhausters: existence, construction, optimality conditions. J. Glob. Optim. 11, 217–230 (2015)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Murzabekova, G.E.: Implicit function theorem for nonsmooth systems by means of exhausters. J. Comput. Syst. Sci. Int. 48, 574–580 (2009)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Luderer, B., Weigelt, J.: A solution method for a special class of nondifferentiable unconstrained optimization problems. Comput. Optim. Appl. 24, 83–93 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Gorokhovik, V.V., Trafimovich, M.A.: On methods for converting exhausters of positively homogeneous functions. Optimization 65(3), 589–608 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Abbasov, M.E.: Comparison between quasidifferentials and exhausters. J. Optim. Theory Appl. 175(1), 59–75 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Demyanov, V.F.: Proper exhausters and coexhausters in nonsmooth analysis. Optimization 61, 1347–1368 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ugray, Z., Lasdon, L., et al.: Scatter search and local NLP solvers: a multistart framework for global optimization. Inf. J. Comput. 19(3), 328–340 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.St. Petersburg State University, SPbSUSt. PetersburgRussia

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