Modified Proximal-Point Algorithm for Maximal Monotone Operators in Banach Spaces

We introduce an iterative sequence for finding the solution to 0∈T(v), where T:E⇉E* is a maximal monotone operator in a smooth and uniformly convex Banach space E. This iterative procedure is a combination of iterative algorithms proposed by Kohsaka and Takahashi (Abstr. Appl. Anal. 3:239–249, 2004) and Kamamura, Kohsaka and Takahashi (Set-Valued Anal. 12:417–429, 2004). We prove a strong convergence theorem and a weak convergence theorem under different conditions respectively and give an estimate of the convergence rate of the algorithm. An application to minimization problems is given.

[1]  Wataru Takahashi,et al.  Weak and Strong Convergence Theorems for Maximal Monotone Operators in a Banach Space , 2004 .

[2]  Y. Alber.  Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications , 1993, funct-an/9311001.

[3]  Hong-Kun Xu,et al.  Strong convergence of modified Mann iterations , 2005 .

[4]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[5]  Wataru Takahashi,et al.  Strong convergence of an iterative sequence for maximal monotone operators in a Banach space , 2004 .

[6]  Hong-Kun Xu.  Inequalities in Banach spaces with applications , 1991 .

[7]  A. Iusem,et al.  Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization , 2000 .

[8]  V. Barbu,et al.  Convexity and optimization in banach spaces , 1972 .

[9]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[10]  Regina Sandra Burachik,et al.  A Proximal Point Method for the Variational Inequality Problem in Banach Spaces , 2000, SIAM J. Control. Optim..

[11]  Hong-Kun Xu.  Iterative Algorithms for Nonlinear Operators , 2002 .

[12]  Osman Güer.  On the convergence of the proximal point algorithm for convex minimization , 1991 .

[13]  S. Dietze.  Barbu, V./Precupanu, Th., Convexity and Optimization in Banach Spaces. Bucuresti. Editura Academiei. Alphen an de Rijn. Sijthoff & Noordhoff Intern. Publ. 1978. XI, 316 S., Dfl. 60.00. $ 28.00 , 1979 .

[14]  Wataru Takahashi,et al.  Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces , 2000 .

[15]  Hong-Kun Xu,et al.  Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process , 1993 .

[16]  Wataru Takahashi,et al.  Strong Convergence of a Proximal-Type Algorithm in a Banach Space , 2002, SIAM J. Optim..

[17]  Benar Fux Svaiter,et al.  Forcing strong convergence of proximal point iterations in a Hilbert space , 2000, Math. Program..