Skip to main content
Log in

Proper and adjoint exhausters in nonsmooth analysis: optimality conditions

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

The notions of upper and lower exhausters represent generalizations of the notions of exhaustive families of upper convex and lower concave approximations (u.c.a., l.c.a.). The notions of u.c.a.’s and l.c.a.’s were introduced by Pshenichnyi (Convex Analysis and Extremal Problems, Series in Nonlinear Analysis and its Applications, 1980), while the notions of exhaustive families of u.c.a.’s and l.c.a.’s were described by Demyanov and Rubinov in Nonsmooth Problems of Optimization Theory and Control, Leningrad University Press, Leningrad, 1982. These notions allow one to solve the problem of optimization of an arbitrary function by means of Convex Analysis thus essentially extending the area of application of Convex Analysis. In terms of exhausters it is possible to describe extremality conditions, and it turns out that conditions for a minimum are expressed via an upper exhauster while conditions for a maximum are formulated in terms of a lower exhauster (Abbasov and Demyanov (2010), Demyanov and Roshchina (Appl Comput Math 4(2): 114–124, 2005), Demyanov and Roshchina (2007), Demyanov and Roshchina (Optimization 55(5–6): 525–540, 2006)). This is why an upper exhauster is called a proper exhauster for minimization problems while a lower exhauster is called a proper one for maximization problems. The results obtained provide a simple geometric interpretation and allow one to construct steepest descent and ascent directions. Until recently, the problem of expressing extremality conditions in terms of adjoint exhausters remained open. Demyanov and Roshchina (Appl Comput Math 4(2): 114–124, 2005), Demyanov and Roshchina (Optimization 55(5–6): 525–540, 2006) was the first to derive such conditions. However, using the conditions obtained (unlike the conditions expressed in terms of proper exhausters) it was not possible to find directions of descent and ascent. In Abbasov (2011) new extremality conditions in terms of adjoint exhausters were discovered. In the present paper, a different proof of these conditions is given and it is shown how to find steepest descent and ascent conditions in terms of adjoint exhausters. The results obtained open the way to constructing numerical methods based on the usage of adjoint exhausters thus avoiding the necessity of converting the adjoint exhauster into a proper one.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abbasov, M.E.: Extremality conditions in terms of adjoint exhausters. (In Russian) Vestnik of Saint-Petersburg University; Ser. 10. Applied mathematics, informatics, control processes. N. 2. pp. 3–8. (2011)

  2. Abbasov, M.E., Demyanov, V.F.: Extremum Conditions for a Nonsmooth Function in Terms of Exhausters and Coexhausters, (In Russian) Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2009, vol. 15, No. 4. English translation: Proceedings of the Steklov Institute of Mathematics, 2010, Suppl. 2, pp. S1–S10. Pleiades Publishing, Ltd., (2010)

  3. Castellani M.: A dual representation for proper positively homogeneous functions. J. Glob. Optim. 16(4), 393–400 (2000)

    Article  Google Scholar 

  4. Clarke, F.H.: Optimization and Nonsmooth Analysis. Reprint of the 1983 original. Universite’ de Montre’al, Centre de Recherches Mathe’matiques, Montreal, QC, pp. xiv+312. (1989)

  5. Demyanov, V.F.: Exhausters and Convexificators—New Tools in Nonsmooth Analysis, Quasidifferentiability and Related Topics, Nonconvex Optim. Appl., vol. 43, pp. 85–137. Kluwer, Dordrecht (2000)

  6. Demyanov V.F.: Exhausters of a positively homogeneous function. Optimization 45(1–4), 13–29 (1999)

    Article  Google Scholar 

  7. Demyanov, V.F., Malozemov, V.N.: Introduction to Minimax. Translated from the Russian by D. Louvish. Halsted Press [Wiley], New York-Toronto, Ont.; Israel Program for Scientific Translations, Jerusalem-London, pp. vii+307. (1974)

  8. Demyanov V.F., Roshchina V.A.: Constrained optimality conditions in terms of proper and adjoint exhausters. Appl. Comput. Math. 4(2), 114–124 (2005)

    Google Scholar 

  9. Demyanov V.F., Roshchina V.A.: Exhausters and subdifferentials in non-smooth analysis. Optimization 57(1), 41–56 (2008)

    Article  Google Scholar 

  10. Demyanov, V.F., Roshchina, V.A.: Generalized subdifferentials and exhausters in nonsmooth analysis, (Russian) Dokl. Akad. Nauk 416 (2007), no. 1, 18–21; translation in Dokl. Math. 76, no. 2, pp. 652–655, (2007)

  11. Demyanov V.F., Roshchina V.A.: Optimality conditions in terms of upper and lower exhausters. Optimization 55(5–6), 525–540 (2006)

    Article  Google Scholar 

  12. Demyanov, V.F., Rubinov, A.M.: “Constructive Nonsmooth Analysis”, Approximation & Optimization, 7. Peter Lang, Frankfurt am Main, pp. iv+416. (1995)

  13. Demyanov V.F., Rubinov A.M.: Elements of quasidifferential calculus. In: Demyanov, V.F. (eds) Nonsmooth Problems of Optimization Theory and Control, pp. 5–127. Leningrad University Press, Leningrad (1982)

    Google Scholar 

  14. Demyanov, V.F., Rubinov, A.M.: Exhaustive Families of Approximations Revisited, From Convexity to Nonconvexity, Nonconvex Optim. Appl., vol. 55, pp. 43–50. Kluwer, Dordrecht (2001)

  15. Demyanov, V.F., Rubinov A.M.: Quasidifferential Calculus. Springer, Optimization Software, New York, NY (1986)

  16. Demyanov V.F., Ryabova J.A.: Exhausters, coexhausters and converters in nonsmooth analysis. Discret. Continuous Dyn. Syst. 31(4), 1273–1292 (2011)

    Article  Google Scholar 

  17. Demyanov, V.F., Vasilev, L.V.: “Nondifferentiable Optimization”, Translated from the Russian by Tetsushi Sasagawa. Translation Series in Mathematics and Engineering, pp. xvii+452. Optimization Software, Inc., Publications Division, New York (1985)

  18. Hiriart-Urruty, J.-B., Lemare’chal, C.: “Convex Analysis and Minimization Algorithms. I. Fundamentals”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 305, pp. xviii+417. Springer, Berlin (1993)

  19. Hiriart-Urruty, J.-B., Lemare’chal, C.: “Convex Analysis and Minimization Algorithms. II. Advanced theory and bundle methods”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 306, pp. xviii+346. Springer, Berlin (1993)

  20. Ioffe A.D.: Nonsmooth analysis: differential calculus of nondifferentiable mappings. Trans. Am. Math. Soc. 266(1), 1–56 (1981)

    Article  Google Scholar 

  21. Kusraev, A.G., Kutateladze, S.S.: Subdifferentials: Theory and Applications. Translated from the Russian. Mathematics and its Applications, vol. 323, pp. x+398. Kluwer, Dordrecht, (1995)

  22. Mordukhovich, B.S.: “Variational analysis and generalized differentiation I. Basic theory”, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330, pp. xxii+579. Springer, Berlin (2006)

  23. Penot J.-P.: Sous-diffe’rentiels de fonctions nume’riques non convexes. (French) C. R. Acad. Sci. Paris Se’r. A 278, 1553–1555 (1974)

    Google Scholar 

  24. Pschenichnyi, B.N.: Convex Analysis and Extremal Problems, Series in Nonlinear Analysis and its Applications, “Nauka”, Moscow, p. 320 (1980)

  25. Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton, NJ (1970)

    Google Scholar 

  26. Uderzo, A.: Convex approximators, convexificators and exhausters: applications to constrained extremum problems, Quasidifferentiability and related topics, Nonconvex Optim. Appl., vol. 43, 297–327. Kluwer, Dordrecht (2000)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. E. Abbasov.

Additional information

The work is supported by the Russian Foundation for Basic Research (RFFI) under Grant No 09-01-00360.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abbasov, M.E., Demyanov, V.F. Proper and adjoint exhausters in nonsmooth analysis: optimality conditions. J Glob Optim 56, 569–585 (2013). https://doi.org/10.1007/s10898-012-9873-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-012-9873-8

Keywords

Navigation