Advertisement

Journal of Global Optimization

, Volume 40, Issue 1–3, pp 71–85 | Cite as

Exhausters, optimality conditions and related problems

  • V. F. Demyanov
  • V. A. Roshchina
Article

Abstract

The notions of exhausters were introduced in (Demyanov, Exhauster of a positively homogeneous function, Optimization 45, 13–29 (1999)). These dual tools (upper and lower exhausters) can be employed to describe optimality conditions and to find directions of steepest ascent and descent for a very wide range of nonsmooth functions. What is also important, exhausters enjoy a very good calculus (in the form of equalities). In the present paper we review the constrained and unconstrained optimality conditions in terms of exhausters, introduce necessary and sufficient conditions for the Lipschitzivity and Quasidifferentiability, and also present some new results on relationships between exhausters and other nonsmooth tools (such as the Clarke, Michel-Penot and Fréchet subdifferentials).

Keywords

Positively homogeneous function Optimality conditions Upper and lower exhausters Proper and adjoint exhausters Unconstrained optimization problems Quasidifferentiability The Michel-Penot subdifferential The Clarke subdifferential The Fréchet subdifferential 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Castellani M. (2000). A dual characterization for proper positively homogeneous functions. J. Glob. Optim. 16: 393–400 CrossRefGoogle Scholar
  2. 2.
    Clarke F.H. (1983). Optimization and Nonsmooth Analysis. Wiley Interscience, New York Google Scholar
  3. 3.
    Demyanov V.F. (2000). Exhausters and convexificators—new tools in nonsmooth analysis. In: Demyanov, V., Rubinov, A. (eds) Quasidifferentiability and Related Topics, pp 85–137. Kluwer, Dordrecht Google Scholar
  4. 4.
    Demyanov, V.F., Malozemov, V.N.: Introduction to Minimax, p. 368. Nauka, Moscow 1972 (English translation by J. Wiley, 1974, 2nd edn, 1990)Google Scholar
  5. 5.
    Demyanov, V.F., Roshchina, V.A.: Optimality conditions in terms of upper and lower exhausters. Forthcoming in OptimizationGoogle Scholar
  6. 6.
    Demyanov, V.F., Rubinov, A.M.: Elements of quasidifferential calculus (In Russian). In: Demyanov, V.F. (ed.) Nonsmooth Problems of Optimization Theory and Control, Ch. 1, pp. 5–127. Leningrad University Press (1982)Google Scholar
  7. 7.
    Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis. Verlag Peter Lang, Frankfurt, a/M. (1995)Google Scholar
  8. 8.
    Demyanov V.F. (1999). Exhausters of a positively homogeneous function. Optimization 45: 13–29 CrossRefGoogle Scholar
  9. 9.
    Demyanov V.F. (1999). Conditional derivatives and exhausters in nonsmooth analysis. Dokl. Russ. Acad. Sci. 338(6): 730–733 Google Scholar
  10. 10.
    Demyanov V.F., Roshchina V.A. (2005). Constrained optimality conditions in terms of upper and lower exhausters. Appl. Comput. Math. 4(2): 25–35 Google Scholar
  11. 11.
    Demyanov V.F., Rubinov A.M. (2001). 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, Dordrecht Google Scholar
  12. 12.
    Glover B.M., Ishizuka Y., Jeyakumar V., Tuan H.D. (1996). Complete characterizations of global optimality for problems involving the pointwise minimum of sublinear functions. SIAM J.Optim. 6(2): 362–372 CrossRefGoogle Scholar
  13. 13.
    Ioffe A.D. (1993). A Lagrange multiplier rule with small convex-valued subdifferentials for nonsmooth problems of mathematical programming involving equality and nonfunctional constraints. Math. Program. 58: 137–145 CrossRefGoogle Scholar
  14. 14.
    Kruger A.Y. (2003). On Fréchet subdifferentials Optimization and related topics, 3. J. Math. Sci. (N. Y.) 116(3): 3325–3358 CrossRefGoogle Scholar
  15. 15.
    Michel P., Penot J.-P. (1984). Calcus sous-differential pour les fonctions lipschitzienness et non-lipschitziennes. C.R. Acad. Sci. Paris, Ser. I 298: 269–272 Google Scholar
  16. 16.
    Mordukhovich B.S. (2004). Necessary conditions in nonsmooth minimization via lower and upper subgradients. Set-Valued Anal. 12(1–2): 163–193 CrossRefGoogle Scholar
  17. 17.
    Mordukhovich, B.S.: Variational analysis and generalized differentiation I. Basic theory. Grundlehren der Mathematischen Wissenschaften, vol. 330. Springer, Berlin (2006)Google Scholar
  18. 18.
    Pallaschke D., Scholtes S., Urbanski R. (1991). On minimal pairs of compact convex sets. Bull. Acad. Polon. Sci. Ser. Math. 39: 1–5 Google Scholar
  19. 19.
    Pallaschke D., Urbanski R. (2002). Pairs of Compact Convex Sets. Kluwer, Dordrecht Google Scholar
  20. 20.
    Polyakova, L.N.: Necessary conditions for an extremum of quasidifferentiable functions. (Russian) Vestnik Leningrad. Univ. Mat. Mekh. Astronom. (13), 57–62 (1980)Google Scholar
  21. 21.
    Pschenichnyi, B.N.: Convex Analysis and Extremal Problems (In Russian). Nauka Publishers, Moscow (1980)Google Scholar
  22. 22.
    Rockafellar R.T. (1976). Convex Analysis. Princeton University Press, Princeton, NJ Google Scholar
  23. 23.
    Roschina, V.A.: Bounded exhausters and optimality conditions (in Russian). In: Control Processes and Stability. Proceedings of XXXVI Scientific Conference of students and Ph.D. students of Applied Mathematics Dept. of St.-Petersburg State University, 10–14 April 2005, St.-Petersburg University Press, St.-Petersburg (2005)Google Scholar
  24. 24.
    Roshchina, V.A.: Reducing exhausters. J. Optimiz. Theory. App. 135(3) (2007)Google Scholar
  25. 25.
    Uderzo, A.: Convex approximators, convexificators and exhausters: applications to constrained extremumproblems. In:Demyanov,V.,Rubinov, A. (eds.) Quasidifferentiability andRelated Topics, pp. 279–327. Kluwer, Dordrecht (2000)Google Scholar

Copyright information

© Springer Science+Business Media LLC 2007

Authors and Affiliations

  1. 1.Applied Mathematics DepartmentSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Department of MathematicsCity University of Hong KongKowloon TongHong Kong S.A.R

Personalised recommendations