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Continuous Subdifferential Approximations and Their Applications

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REFERENCES

  1. L. Armijo, “Minimization of functions having continuous partial derivatives,” Pac. J. Math., 16, 1–13 (1966).

    Google Scholar 

  2. A. M. Bagirov, “A method of approximating a subdifferential,” Zh. Vychisl. Mat. Mat. Fiz., 32, No. 4, 561–566 (1992).

    Google Scholar 

  3. A. M. Bagirov, “Continuous approximation to a subdifferential of a function of a maximum,” Cyber. Syst. Anal., 4, 626–630 (1994).

    Google Scholar 

  4. A. M. Bagirov and A. A. Gasanov, “A method of approximating a quasidifferential,” Zh. Vychisl. Mat. Mat. Fiz., 35, No. 4, 403–409 (1995).

  5. A. M. Bagirov, “Continuous subdifferential approximation and its construction,” Indian J. Pure Appl. Math., 1, 17–29 (1998).

    Google Scholar 

  6. A. M. Bagirov, “A method for minimizing convex functions based on the continuous approximations to a subdifferential,” Optim. Methods Softw., 9, Nos. 1–3, 1–17 (1998).

    Google Scholar 

  7. A. M. Bagirov and N. K. Gadjiev, “A monotonous method for unconstrained Lipschitz optimization,” In: High Performance Algorithms and Software in Nonlinear Optimization(R. De Leone et al., Eds.), Applied Optimization, 24, Kluwer Academic Publishers (1998), pp.25–42.

  8. A. M. Bagirov, “Minimization methods for one class of nonsmooth functions and calculation of semi-equilibrium prices,” In: Progress in Optimization: Contributions from Australasia(A. Eberhard et al., Eds.), Applied Optimization, 30, Kluwer Academic Publishers (1999), pp.147–175.

  9. A. M. Bagirov, “Derivative-free methods for unconstrained nonsmooth optimization and its numerical analysis,” Investigacao Operacional, 19, 75–93 (1999).

    Google Scholar 

  10. A. M. Bagirov, “Numerical methods for minimizing quasidifferentiable functions: A survey and comparison,” In: Quasidifferentiability and Related Topics(V. F. Demyanov and A. M. Rubinov, Eds.), Nonconvex Optimization and Its Applications, 43, Kluwer Academic Publishers (2000), pp. 33–71.

  11. J. Borwein, “A note on the existence of subgradients,” Math. Program., 24, No. 2, 225–228 (1982).

    Google Scholar 

  12. R. P. Brent, Algorithms for Minimization without Derivatives, Prentice-Hall, Englewood Cliffs, New Jersey (1973).

    Google Scholar 

  13. F. H. Clarke, “Generalized gradients and applications,” Trans. Amer. Math. Soc., 205, 247–262 (1975).

    Google Scholar 

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

    Google Scholar 

  15. A. R. Conn and P. L. Toint, “An algorithm using quadratic interpolation for unconstrained derivative free optimization,” In: Nonlinear Optimization and Applications(G. Di Pillo and F. Gianessi, Eds.), Plenum Publishing, New York (1996), pp.27–47.

    Google Scholar 

  16. A. R. Conn, K. Sheinberg, and P. L. Toint, “Recent progress in unconstrained nonlinear optimization without derivatives,” Math. Program., Ser. B, 79, 397–414 (1997).

    Google Scholar 

  17. V. A. Daugavet, “Modification of the Wolfe method,” Zh. Vychisl. Mat. Mat. Fiz., 21, No. 2, 504–508 (81).

  18. V. F. Demyanov, “On the method of codifferential descent,” Vestn. Leningr. Univ., 7(1989).

  19. V. F. Demyanov, A. M. Bagirov, and A. M. Rubinov, “A method of truncated codifferential with application to some problems of cluster analysis,” J. Glob. Optim.(to appear).

  20. V. F. Demyanov, S. Gamidov, and T. I. Sivelina, “An algorithm for minimizing a certain class of quasidifferentiable functions,” Math. Program.Study, 29, 74–85 (1986).

    Google Scholar 

  21. V. F. Demyanov and V. N. Malozemov, Introduction to Minimax, Wiley, New York (1974).

    Google Scholar 

  22. V. F. Demyanov and A. M. Rubinov, “On quasidifferentiable functions,” Dokl. Akad. Nauk SSSR, 250, 21–25 (1980).

    Google Scholar 

  23. V. F. Demyanov and A. M. Rubinov, Quasidifferential Calculus, New York (1986).

  24. V. F. Demyanov and A. M. Rubinov, Foundations of Nonsmooth Analysis and Quasidifferential Calculus[in Russian ], Nauka, Moscow (1990).

    Google Scholar 

  25. V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Peter Lang, Frankfurt am Main (1995).

  26. V. F. Demyanov, G. E. Stavroulakis, L. N. Polyakova, and P. D. Panagiotopoulos, Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics, KluwerAcademic Publishers, Dordrecht (1996).

    Google Scholar 

  27. J. E. Dennis and R. B. Shnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall, Englewood Cliffs, New Jersey (1983).

    Google Scholar 

  28. M. Fukushima and L. Qi, “A globally and superlinearly convergent algorithm for nonsmooth convex minimization,” SIAM J. Optim., 6, 1106–1120 (1996).

    Google Scholar 

  29. J.-B. Hiriart-Urruty, “From convex optimization to nonconvex optimization. Necessary and sufficient conditions for global optimality,” In: Nonsmooth Optimization and Related Topics(F. H. Clarke, V. F. Demyanov, and F. Gianessi, Eds.), Plenum Publ., New York (1989), pp.219–239.

    Google Scholar 

  30. J.-B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, Vol. 1, Springer-Verlag, Heidelberg (1993).

    Google Scholar 

  31. J.-B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms, Vol. 2, Springer Verlag, New York (1993).

    Google Scholar 

  32. K. C. Kiwiel, Methods of Descent for Nondifferentiable Optimization, Lect.Notes Math., Vol. 1133, Springer-Verlag, Berlin (1985).

    Google Scholar 

  33. K. C. Kiwiel, “A linearization method for minimizing certain quasidifferentiable functions,” Math. Program. Study, 29, 86–94 (1986).

    Google Scholar 

  34. K. C. Kiwiel, “Proximal control in bundle methods for convex nondifferentiable minimization,” Math.Program., 29, 105–122 (1990).

    Google Scholar 

  35. C. Lemarechal, Extensions Diverses des Methodes de Gradient et Application, These d'etat, Paris (1980).

  36. C. Lemarechal, “Numerical experiments in nonsmooth optimization,” In: Progress in Nondifferentiable Optimization(E. A. Nurminski, Ed.), CP-82-S8, IIASA, Laxenburg, Austria (1982), pp.61–84.

    Google Scholar 

  37. C. Lemarechal, “Nondifferentiable optimization,” In: Handbook in Operations Research and Management Science(G. L. Nemhauser et al., Eds.), Vol. 1, North Holland, Amsterdam (1989), pp.529–572.

    Google Scholar 

  38. C. Lemarechal and J. Zowe, “A condensed introduction to bundle methods in nonsmooth optimization,” In: Algorithms for Continuous Optimization(E. Spedicato, Ed.), Kluwer Academic Publishers (1994), pp.357–482.

  39. C. Lemarechal and C. A. Sagastizabal, “Practical aspects of the Moreau–Yosida regularization. I. Theoretical preliminaries,” SIAM J. Optim., 7, 367–385 (1997).

    Google Scholar 

  40. R. Mifflin, “Semismooth and semiconvex functions in constrained optimization,” SIAM J. Control and Optimization, 15, No. 6, 959–972 (1977).

    Google Scholar 

  41. R. Mifflin, “A quasi-second-order proximal bundle algorithm,” Math. Program., 73, 51–72 (1996).

    Google Scholar 

  42. R. Miffin, L. Qi, and D. Sun, “Quasi-Newton bundle-type methods for nondifferentiable convex optimization,” SIAM J. Optim., 8, No. 2, 583–603 (1998).

    Google Scholar 

  43. B. F. Mitchell, V. F. Demyanov, and V. N. Malozemov, “Finding the point of a polyhedron closest to the origin,” SIAM J. Control Optimization, 12, 19–26 (1974).

    Google Scholar 

  44. B. Sh. Mordukhovich, Approximation Methods in Problems of Optimization and Control[in Russian ], Nauka, Moscow (1988).

    Google Scholar 

  45. J. J. More, “Recent developments in algorithms and software for trust region methods,” In: Mathematical Programming:The State of the Art(A. Bachem, M. Grotchel, and B. Korte, Eds.), Springer-Verlag, Berlin (1983), pp.258–287.

    Google Scholar 

  46. Yu. E. Nesterov, “The technique of nonsmooth differentiation,” Soviet J. Comput. Syst. Sci., 25, 113–123 (1987).

    Google Scholar 

  47. E. Polak, Optimization.Algorithms and Consistent Approximations, Springer-Verlag, New York (1997).

    Google Scholar 

  48. E. Polak, D. Q. Mayne, and Y. Wardi, “On the extension of constrained optimization algorithms from differentiable to nondifferentiable problems,” SIAM J. Control Optimization, 21, No. 2, 179–203 (1983).

    Google Scholar 

  49. E. Polak and D. Q. Mayne, “Algorithm models for nondifferentiable optimization,” SIAM J. Control Optimization, 23, No. 3, 477–491 (1985).

    Google Scholar 

  50. M. J. D. Powell, “An efficient method for finding the minimum of a function of several variables without calculating derivatives,” Comput. J., 17, 155–162 (1964).

    Google Scholar 

  51. M. J. D. Powell, “Recent advances in unconstrained optimization,” Math. Program., 1, 26–57 (1972).

    Google Scholar 

  52. M. J. D. Powell, “A view of unconstrained minimization algorithms that do not require derivatives,” ACM Trans. Math. Softw., 1, No. 2, 97–107 (1975).

    Google Scholar 

  53. M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, Cambridge (1981).

    Google Scholar 

  54. R. T. Rockafellar, Convex Analysis, Princeton University Press (1970).

  55. N. Z. Shor, Minimization Methods for Non-Differentiable Functions, Springer-Verlag, Heidelberg (1985).

    Google Scholar 

  56. M. Studniarski, “An algorithm for calculating one subgradient of a convex function of two variables,” Numer. Math., 55, 685–693 (1989).

    Google Scholar 

  57. M. Studniarski, “On the numerical computation of Clarke 's subgradients for strongly subregular functions,” In: Proc. Fifth Int. Congr. Comput. Appl. Math.Leuven, Belgium, July 27–August 1, 1992.

  58. H. Tuy, Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht (1998).

    Google Scholar 

  59. P. H. Wolfe, “A method of conjugate subgradients of minimizing nondifferentiable convex functions,” Math. Program.Study, 3, 145–173 (1975).

    Google Scholar 

  60. P. H. Wolfe, “Finding the nearest point in a polytope,” Math. Program., 11, No. 2, 128–149 (1976).

    Google Scholar 

  61. H. Xu, A. M. Rubinov, and B. M. Glover, “Continuous approximations to generalized Jacobians,” Optimization, 46, 221–246 (1999).

    Google Scholar 

  62. J. Zowe, “Nondifferentiable optimization: A motivation and a short introduction into the subgradient and the bundle concept,” In: NATO SAI Series, 15, Computational Mathematical Programming(K. Schittkowski, Ed.), Springer-Verlag, New York (1985), pp.323–356.

    Google Scholar 

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Bagirov, A.M. Continuous Subdifferential Approximations and Their Applications. Journal of Mathematical Sciences 115, 2567–2609 (2003). https://doi.org/10.1023/A:1023227716953

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