Skip to main content
SpringerLink
Log in

The Rise of Nonsmooth Analysis: Its Main Tools

  • Published:
Cybernetics and Systems Analysis Aims and scope

Abstract

The main notion in the classical (smooth) mathematical analysis is that of gradient. Nonsmooth analysis deals with nondifferentiable functions, therefore, the problem is to find a proper replacement for the concept of gradient. Different scientific schools use different tools and, as a result, they construct different theories that allow treating different nonsmooth function classes and to solve different nonsmooth problems. Several notions and tools used in nonsmooth analysis and nondifferentiable optimization are discussed.

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

Access this article

Log in via an institution

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. F. Clarke, “Generalized gradients and applications,” Trans. Amer. Math. Soc., 205, No. 2, 247–262 (1975).

    Google Scholar 

  2. F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley Interscience, New York (1983).

    Google Scholar 

  3. J. M. Danskin, “The theory of max-min, with applications,” SIAM J. Appl. Math., 14, 641–664 (1966).

    Google Scholar 

  4. J. M. Danskin, The Theory of Max-Min and its Application to Weapons Allocation Problems, Springer-Verlag, New York (1967).

    Google Scholar 

  5. V. F. Dem'yanov, “On the minimization of a maximal deviation,” Vestnik Leningr. Univ., Ser. 1, No. 7, 21–28 (1966).

  6. V. F. Dem'yanov, “Exhausters and convexificators — new tools in nonsmooth analysis,” in: V. Dem'yanov and A. Rubinov (eds.), Nonconvex Optimization and its Applications, Vol. 43, Kluwer Academic Publishers, Dordrecht-Boston-London (2000), pp. 85–137.

    Google Scholar 

  7. V. F. Dem'yanov and V. N. Malozemov, Introduction to Minimax [in Russian], Nauka, Moscow (1972), English translation: John Wiley and Sons (1974).

    Google Scholar 

  8. V. F. Dem'yanov and A. M. Rubinov, “On quasidifferentiable functionals,” Dokl. Akad. Nauk USSR, 250, No.1, 21–25 (1980).

    Google Scholar 

  9. V. F. Dem'yanov and A. M. Rubinov, “Elements of quasidifferential calculus,” in: V. F. Dem'yanov (ed.), Nonsmooth Problems of Optimization Theory and Control [in Russian], Ch.1, Izd. Leningradskogo Univ., Leningrad (1982), pp. 5–127. English revised version see in: V. Dem'yanov and A. Rubinov (eds.), Nonconvex Optimization and its Applications, Vol. 43, Kluwer Academic Publishers, Dordrecht-Boston-London (2000), pp. 1- 31.

    Google Scholar 

  10. V. F. Dem'yanov and L. V. Vasil'ev, Nondifferentiable Optimization, Springer-Optimization Software, New York (1986).

    Google Scholar 

  11. V. F. Dem'yanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Verlag Peter Lang, Frankfurt a/M (1995).

    Google Scholar 

  12. V. F. Dem'yanov and V. Jeyakumar, “Hunting for a smaller convex subdifferential,” J. Global Optimiz., 10, No. 3, 305–326 (1997).

    Google Scholar 

  13. V. F. Dem'yanov, G. Di Pillo, and F. Facchinei, “Exact penalization via Dini and Hadamard conditional derivatives,” Optimization Methods and Software, 9, 19–36 (1998).

    Google Scholar 

  14. V. F. Dem'yanov and A. M. Rubinov, Quasidifferential Calculus, Optimization Software, New York (1986).

    Google Scholar 

  15. V. F. Dem'yanov, “Exhausters of a positively homogeneous function,” Optimization, 45, 13-29 (1999).

    Google Scholar 

  16. V. F. Dem'yanov, “Convexification of a positively homogeneous function,” Dokl. Ros. Akad. Nauk, 366, No. 6, 734–737 (1999).

    Google Scholar 

  17. Yu. M. Ermol'ev, “Methods for solving nonlinear extremum problems,” Kibernetika, No. 4, 1–17 (1966).

  18. I.V. Girsanov, Lectures on the Mathematical Theory of Extremum Problems [in Russian], Izd. Moskovskogo Univ., Moscow (1970).

    Google Scholar 

  19. J.-B. Hiriart-Urruty, “New concepts in nondifferentiable programming,” Bul. Soc. Math. France, Mem. 60, 57-85 (1979)

    Google Scholar 

  20. J.-B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms., Pts I, II, Springer, Berlin (1993).

    Google Scholar 

  21. A. D. Ioffe, “Necessary conditions in nonsmooth optimization,” Mathem. Oper. Res., 9, 159–189 (1984).

    Google Scholar 

  22. A. D. Ioffe, “A Lagrange multiplier rule with small convex-valued subdifferentials for nonsmooth problems of mathematical programming involving equality and non-functional constraints,” Mathematical Programming, 58, 137–145 (1993).

    Google Scholar 

  23. A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems [in Russian], Nauka, Moscow (1974), English transl. by North-Holland (1979).

    Google Scholar 

  24. A. D. Ioffe, “Nonsmooth analysis: differential calculus of nondifferentiable mappings,” Trans. Amer. Math. Soc., 266, No. 1, 1–56 (1981).

    Google Scholar 

  25. A. Ya. Kruger, “On properties of generalized differentials,” Syberian Math. J., 26, 54-66 (1985).

    Google Scholar 

  26. S. S. Kutateladze and A. M. Rubinov, “Minkowski duality and its applications,” Russian Mathem. Surveys, 27, 131–191 (1972).

    Google Scholar 

  27. P. Michel and J.-P. Penot, “Calcus sous-differential pour les fonctions lipschitzienness et non-lipschitziennes,” C. R. Acad. Sci. Paris, Ser. I, 298, 269–272 (1984).

  28. B. S. Mordukhovich,“ Nonsmooth analysis with nonconvex generalised differentials and conjugate mappings,” Dokl. Akad. Nauk BSSR, 28, No. 11, 976–979 (1984).

    Google Scholar 

  29. B. S. Mordukhovich, Approximation Methods in Optimization and Control Problems [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  30. J.-J. Moreau, “Fonctionelles sous-differentiables,” C. R. Acad. Sci. Paris, Ser. A-B, 257, 4117–4119 (1963).

    Google Scholar 

  31. J.-P. Penot, “Sous-differentiels de functions numeriques non convexes,” C. R. Acad. Sci. Paris, Ser. A, 278, 1553-1555 (1974).

    Google Scholar 

  32. B. T. Polyak, “A general method for solving extremum problems,” Dokl. AN SSSR, 174, No. 1, 33–36 (1967).

    Google Scholar 

  33. B. T. Polyak, “Minimization of nonsmooth functionals,” Zh. Vych. Mat. Mat. Fiz., 9, No. 3, 509–521 (1969).

    Google Scholar 

  34. B. N. Pshenichnyi, Convex Analysis and Extremal Problems [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  35. B. N. Pshenichnyi, Necessary Conditions for Extrema [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  36. B. N. Pshenichnyi, A Method of Linearization [in Russian] Nauka, Moscow (1983).

    Google Scholar 

  37. “Quasidifferentiability and related topics,” in: V. Dem'yanov and A. Rubinov (eds.), Nonconvex Optimization and its Applications, Vol. 43, Kluwer Academic Publishers, Dordrecht-Boston-London (2000).

  38. R. T. Rockafellar, “Theorems for convex functions,” Bull. Amer. Math. Soc., 70, 189-192 (1964).

    Google Scholar 

  39. R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, N.J. (1970).

    Google Scholar 

  40. R. T. Rockafellar, The Theory of Subgradients and its Application to Problems of Optimization, Holdermann, Berlin (1981).

    Google Scholar 

  41. N. Z. Shor, “The development of numerical methods for nonsmooth optimization in the USSR,” in: J. K. Lenstra, A. H. G. Rinnoy Kan, and A. Shrijver (eds.), History of Mathematical Programming, CWI, North-Holland, Amsterdam, (1991), pp. 135–139.

    Google Scholar 

  42. N. Z. Shor, “On a method for minimizing almost-differentiable functions,” Kibernetika, No. 4, 65–70 (1972).

  43. N. Z. Shor, Methods for Minimizing Nondifferentiable Functions and their Applications [in Russian], Naukova Dumka, Kiev (1979).

    Google Scholar 

  44. A. Uderzo, “Convex approximators, convexificators and exhausters: applications to constrained extremum problems,” in: V. Dem'yanov and A. Rubinov (eds.), Nonconvex Optimization and its Applications, Vol. 43, Kluwer Academic Publishers, Dordrecht-Boston-London (2000), pp. 297–327.

    Google Scholar 

  45. A. Zaffaroni “Continuous approximations, codifferentiable functions and minimization methods,” in: V. Dem'yanov and A. Rubinov (eds.), Nonconvex Optimization and its Applications, Vol. 43, Kluwer Academic Publishers, Dordrecht-Boston-London (2000), pp. 361–391.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St. Petersburg State University, Russia

    V. F. Demyanov

Authors
  1. V. F. Demyanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Demyanov, V.F. The Rise of Nonsmooth Analysis: Its Main Tools. Cybernetics and Systems Analysis 38, 527–547 (2002). https://doi.org/10.1023/A:1021154101946

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1021154101946

Search

Navigation