×

Another version of the proximal point algorithm in a Banach space. (English) Zbl 1167.49031

Summary: We study some non-traditional schemes of proximal point algorithm for nonsmooth convex functionals in a Banach space. The proximal approximations to their minimal points and/or their minimal values are considered separately for unconstrained and constrained minimization problems on convex closed sets. For the latter we use proximal point algorithms with the metric projection operators and first establish the estimates of the convergence rate with respect to functionals. We also investigate the perturbed projection proximal point algorithms and prove their stability. Some results concerning the classical proximal point method for minimization problems in a Banach space are also presented in this paper.

MSC:

49N15 Duality theory (optimization)
49M15 Newton-type methods
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alber, Ya. I., Metric and generalized projection operators in Banach spaces: Properties and applications, (Kartsatos, A., Theory and Applications of Nonlinear Operators of Monotone and Accretive Type (1996), Marcel Dekker: Marcel Dekker New York), 15-50 · Zbl 0883.47083
[2] Alber, Ya., Stability of the iterative approximations to fixed points of nonexpansive mappings, Journal of Mathematical Analysis and Applications, 328, 958-971 (2007) · Zbl 1114.47053
[3] Alber, Ya., The Young-Fenchel transformation and some new characteristics of Banach spaces, (Jarosz, Krzysztof, Functional Spaces. Functional Spaces, Contemporary Mathematics, vol. 435 (2007), AMS), 1-19 · Zbl 1146.46009
[4] Alber, Ya. I.; Burachik, R. S.; Iusem, A. N., A proximal point method for nonsmooth convex optimization problems in Banach spaces, Abstract and Applied Analysis, 2, 1-2, 97-120 (1997) · Zbl 0947.90091
[5] Alber, Ya.; Reich, S.; Shoikhet, D., Iterative approximations of null points of uniformly accretive operators with estimates of convergence rate, Communications in Applied Analysis, 1, 315-335 (2002)
[6] Alber, Ya.; Reich, S.; Yao, Jen-Chih, Fixed point problems with nonself-mappings in Banach spaces, Abstract and Applied Analysis, 2003, 4, 194-216 (2003) · Zbl 1028.47049
[7] Alber, Ya.; Ryazantseva, I., Nonlinear Ill-posed Problems of Monotone Type (2006), Springer · Zbl 1086.47003
[8] Alber, Ya.; Yao, Jen-Chih, On projection dynamical systems in Banach spaces, Taiwanese Journal of Mathematics, 11, 3, 819-847 (2007) · Zbl 1174.49004
[9] Figiel, T., On the moduli of convexity and smoothness, Studia Mathematica, 56, 121-155 (1976) · Zbl 0344.46052
[10] Bonnans, J. F.; Shapiro, A., Perturbation Analysis of Optimization Problems (2000), Springer Verlag · Zbl 0966.49001
[11] Giles, J. R., (Convex Analysis with Applications to Differentiation of Convex Function. Convex Analysis with Applications to Differentiation of Convex Function, Research Notes in Mathematics, vol. 58 (1982), Pitman: Pitman Boston)
[12] Gradshtein, S.; Rizhik, I. M., Table of Integrals, Series and Products (1965), Academic: Academic New York
[13] Kassay, J., The proximal points algorithm for reflexive Banach spaces, Studia Univ. Babes-Bolyai, Mathematica, XXX, 9-17 (1985) · Zbl 0649.47040
[14] Lindenstrauss, J.; Tzafriri, L., Classical Banach Spaces II (1979), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0403.46022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.