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Generalized differentials of nonsmooth functions, and necessary conditions for an extremum

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Literature Cited

  1. L. W. Neustadt, “A general theory of extremals,” J. Comput. System Sci.,3, No. 1, 57–92 (1969).

    Google Scholar 

  2. L. W. Neustadt, Optimization: A Theory of Necessary Conditions, Princeton Univ. Press (1976).

  3. A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  4. V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control [in Russian], Nauka, Moscow (1979).

    Google Scholar 

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

    Google Scholar 

  6. F. H. Clarke, “A new approach to Lagrange multipliers,” Math. Oper. Res.,1, No. 2, 165–174 (1976).

    Google Scholar 

  7. J. Warga, “Controllability and a multiplier rule for nondifferentiable optimization problems,” SIAM J. Control Optim.,16, No. 5, 803–812 (1978).

    Google Scholar 

  8. J.-B. Hiriart-Urruty, “Refinements of necessary optimality conditions in nondifferentiable programming. I,” Appl. Math. Optim.,5, No. 1, 63–82 (1979).

    Google Scholar 

  9. A. Ya. Dubovitskii and A. A. Milyutin, “Extremum problems in the presence of constraints,” Zh. Vychisl. Mat. Mat. Fiz.,5, No. 3, 395–453 (1965).

    Google Scholar 

  10. H. Halkin, “Necessary conditions in mathematical programming and optimal control theory,” Lecture Notes Econ. Math. Syst.,105, Springer-Verlag, Berlin (1974), pp. 113–165.

    Google Scholar 

  11. V. G. Boltyanskii, “The method of tents in the theory of extremum problems,” Usp. Mat. Nauk,30, No. 3, 3–65 (1975).

    Google Scholar 

  12. A. G. Kusraev, “Convex approximations of nonsmooth operators and necessary conditions for an extremum,” in: Optimization [in Russian], No. 21, Novosibirsk (1978), pp. 41–54.

  13. A. Ya. Kruger and B. Sh. Mordukhovich, “Extremal points and Euler's equation in nonsmooth optimization problems,” Dokl. Akad. Nauk BSSR,24, No. 8, 684–687 (1980).

    Google Scholar 

  14. A. Ya. Kruger, Generalized Differentials of Nonsmooth Functions [in Russian], Minsk (1980), Dep. at VINITI, No. 1332-81.

  15. B. Sh. Mordukhovich, “The maximum principle in an optimal speed of response problem with nonsmooth constraints,” Prikl. Mat. Mekh.,40, No. 6, 1014–1023 (1976).

    Google Scholar 

  16. B. Sh. Mordukhovich and A. Ya. Kruger, “Necessary conditions for optimality in a terminal control problem with nonfunctional constraints,” Dokl. Akad. Nauk BSSR,20, No. 12, 1064–1067 (1976).

    Google Scholar 

  17. A. Ya. Kruger and B. Sh. Mordukhovich, “The minimization of nonsmooth functionals in optimal control problems,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 4, 176–182 (1978).

    Google Scholar 

  18. A. Ya. Kruger and B. Sh. Morkukhovich, Generalized Normals and Derivatives, and Necessary Conditions for an Extremum in Problems of Nondifferentiable Programming. I [in Russian], Minsk (1980), Dep. at VINITI, No. 408-80.

  19. A. Ya. Kruger and B. Sh. Mordukhovich, Generalized Normals and Derivatives, and Necessary Conditions for an Extremum in Problems of Nondifferentiable Programming. II [in Russian], Minsk (1980), Dep. at VINITI, No. 494-80.

  20. B. Sh. Mordukhovich, “Metric approximations and necessary conditions for optimality for general classes of nonsmooth extremal problems,” Dokl. Akad. Nauk SSSR,254, No. 5, 1072–1076 (1980).

    Google Scholar 

  21. A. Ya. Kruger, Necessary Conditions for an Extremum in Problems of Nonsmooth Optimization [in Russian], Minsk (1981), Dep. at VINITI, No. 1333-81.

  22. F. H. Clarke, “Generalized gradients and applications,” Trans. Am. Math. Soc.,204, 247–262 (1975).

    Google Scholar 

  23. F. H. Clarke, “Generalized gradients of Lipschitz functions,” Adv. Math.,40, No. 1, 52–67 (1981).

    Google Scholar 

  24. J.-H. Hiriart-Urruty, “Tangent cones, generalized gradients and mathematical programming in Banach spaces,” Math. Oper. Res.,4, No. 1, 79–97 (1979).

    Google Scholar 

  25. J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York (1972).

    Google Scholar 

  26. A. Ya. Kruger, ε-Semidifferentials and ε-Normal Elements [in Russian], Minsk (1981), Dep. at VINITI, No. 1331-81.

  27. R. T. Rockafellar, “Generalized directional derivatives and subgradients of nonconvex functions,” Can. J. Math.,32, No. 2, 257–280 (1980).

    Google Scholar 

  28. I. Ekeland, “On the variational principle,” J. Math. Anal. Appl.,47, No. 2, 324–353 (1974).

    Google Scholar 

  29. A. Ya. Kruger, “On a characteristic of the covering property for nonsmooth operators,” in: School on the Theory of Operators in Function Spaces (Abstracts of Papers) [in Russian], Minsk (1982), pp. 94–95.

  30. A. D. Ioffe, “Sous-différentielles approchées de fonctions numériques,” C. R. Acad. Sci. Paris, Sér. 1,292, No. 14, 675–678 (1981).

    Google Scholar 

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Minsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 26, No. 3, pp. 78–90, May–June, 1985.

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Kruger, A.Y. Generalized differentials of nonsmooth functions, and necessary conditions for an extremum. Sib Math J 26, 370–379 (1985). https://doi.org/10.1007/BF00968624

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  • DOI: https://doi.org/10.1007/BF00968624

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