Abstract
In this paper we continue the study of the subgradient method for nonsmooth convex constrained minimization problems in a uniformly convex and uniformly smooth Banach space. We consider the case when the stepsizes satisfy ∑ k=1 ∞α k =∞, lim k→∞α k =0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alber, Ya. I.: Metric and generalized projection operators in Banach spaces: Properties and applications, In: A. Kartsatos (ed.), Theory and Applications of Nonlinear Operators of Monotone and Accretive Type, Marcel Dekker, New York, 1996, pp. 15-50.
Alber, Ya. I.: The problem of the minimization of smooth functionals by gradient methods, USSR Comput. Math. Math. Phys. 11 (1971), 263-272.
Alber, Ya. I.: Minimization of functionals of class C 1,μ on bounded sets, Econom. Mat. Met. 16 (1980), 185-190.
Alber, Ya. I.: Recurrence relations and variational inequalities, Soviet Mat. Doklady 27 (1983), 511-517.
Alber, Ya. I.: On average convergence of the iterative projection methods (to be published).
Alber, Ya. I., Iusem, A. N. and Solodov, M. V.: On the projection subgradient method for nonsmooth convex optimization in a Hilbert space, Math. Programming 81 (1998), 23-35.
Alber, Ya. I., Iusem, A. N. and Solodov, M. V.: Minimization of nonsmooth convex functionals in Banach spaces, J. Convex Anal. 4 (1997), 235-254.
Alber, Ya. I. and Reich, S.: An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. Math. J. 4 (1994), 39-54.
Clarke, F. H.: Optimization and Nonsmooth Analysis, SIAM Publications, Philadelphia, 1983.
Correa, R. and Lemaréchal, L.: Convergence of some algorithms for convex minimization, Math. Programming 62 (1993), 261-275.
Diestel, J.: The Geometry of Banach Spaces, Lecture Notes in Math. 485, Springer, Berlin, 1975.
Dunn, J. C.: Global and asymptotic convergence rates estimates for a class of projected gradient processes, SIAM J. Control Optim. 19 (1981), 368-400.
Ermoliev, Yu.: Methods for solving nonlinear extremal problems, Cybernetics 2 (1966), 1-17.
Figiel, T.: On the moduli of convexity and smoothness, Studia Math. 56 (1976), 121-155.
Golshtein, E. and Tretyakov, N.: Modified Lagrange Functions, Moscow, 1989.
Knopp, K.: Theory and Applications of Infinite Series, Dover, New York, 1990.
Lindenstrauss, J. and Tzafriry, L.: Classical Banach Spaces II, Springer, Berlin, 1979.
Polyak, B. T.: A general method of solving extremum problems, Soviet Mat. Doklady 8 (1967), 593-597.
Serb, I.: Estimates for the modulus of smoothness and convexity of a Banach space, Mathematica 34 (1992), 61-70.
Vasil'ev, F. P.: Methods of Solving Extremal Problems, Nauka, Moscow, 1981.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Alber, Y.I., Iusem, A.N. Extension of Subgradient Techniques for Nonsmooth Optimization in Banach Spaces. Set-Valued Analysis 9, 315–335 (2001). https://doi.org/10.1023/A:1012665832688
Issue Date:
DOI: https://doi.org/10.1023/A:1012665832688