Visco-resolvent algorithms for monotone operators and nonexpansive mappings
- Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China.
Shin Min Kang
- Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Korea.
- School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, China.
Two new type of visco-resolvent algorithms for finding a zero of the sum of two monotone operators and a
fixed point of a nonexpansive mapping in a Hilbert space are investigated. The algorithms consist of the
zeros and the fixed points of the considered problems in which one operator is replaced with its resolvent
and a viscosity term is added. Strong convergence of the algorithms are shown. As special cases, we can
approach to the minimum norm common element of the zero of the sum of two monotone operators and the
fixed point of a nonexpansive mapping without using the metric projection. Some applications are included.
Share and Cite
Peize Li, Shin Min Kang, Li-Jun Zhu, Visco-resolvent algorithms for monotone operators and nonexpansive mappings, Journal of Nonlinear Sciences and Applications, 7 (2014), no. 5, 325--344
Li Peize, Kang Shin Min, Zhu Li-Jun, Visco-resolvent algorithms for monotone operators and nonexpansive mappings. J. Nonlinear Sci. Appl. (2014); 7(5):325--344
Li, Peize, Kang, Shin Min, Zhu, Li-Jun. "Visco-resolvent algorithms for monotone operators and nonexpansive mappings." Journal of Nonlinear Sciences and Applications, 7, no. 5 (2014): 325--344
- Monotone operator
- nonexpansive mapping
- zero point
- fixed point
K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, On a strongly nonexpansive sequence in Hilbert spaces, J. Nonlinear Convex Anal., 8 (2007), 471-489.
H. H. Bauschke, J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Review, 38 (1996), 367-426.
H. H. Bauschke, P. L. Combettes, A Dykstra-like algorithm for two monotone operators, Pacific J. Optim., 4 (2008), 383-391.
H. H. Bauschke, P. L. Combettes, S. Reich , The asymptotic behavior of the composition of two resolvents , Nonlinear Anal., 60 (2005), 283-301.
E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-145.
Y. Censor, S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications, Oxford University Press, New York, USA (1997)
P. L. Combettes, Solving monotone inclusions via compositions of nonexpansive averaged operators, Optimization, 53 (2004), 475-504.
P. L. Combettes, S. A. Hirstoaga , Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136.
P. L. Combettes, S. A. Hirstoaga , Approximating curves for nonexpansive and monotone operators, J. Convex Anal., 13 (2006), 633-646.
P. L. Combettes, S. A. Hirstoaga, Visco-penalization of the sum of two monotone operators, Nonlinear Anal., 69 (2008), 579-591.
F. Ding, T. Chen , Iterative least squares solutions of coupled Sylvester matrix equations, Systems Control Lett., 54 (2005), 95-107.
J. Eckstein, D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), 293-318.
Y. P. Fang, N. J. Huang, H-Monotone operator resolvent operator technique for quasi-variational inclusions, Appl. Math. Comput., 145 (2003), 795-803.
K. Geobel, W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press (1990)
L. J. Lin, Variational inclusions problems with applications to Ekeland's variational principle, fixed point and optimization problems , J. Global Optim., 39 (2007), 509-527.
P.-L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal., 16 (1979), 964-979.
X. Liu, Y. Cui, Common minimal-norm fixed point of a finite family of nonexpansive mappings, Nonlinear Anal., 73 (2010), 76-83.
X. Lu, H. K. Xu, X. Yin , Hybrid methods for a class of monotone variational inequalities, Nonlinear Anal., 71 (2009), 1032-1041.
A. Moudafi, On the regularization of the sum of two maximal monotone operators, Nonlinear Anal., 42 (2009), 1203-1208.
G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl., 72 (1979), 383-390.
J. W. Peng, Y. Wang, D. S. Shyu, J. C. Yao, Common solutions of an iterative scheme for variational inclusions, equilibrium problems and fixed point problems, J. Inequal. Appl., Article ID 720371, 2008 (2008), 15 pages.
S. M. Robinson, Generalized equation and their solutions, part I, basic theory, Math Program. Study, 10 (1979), 128-141.
R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math., 33 (1970), 209-216.
R. T. Rockafellar , Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898
A. Sabharwal, L. C. Potter, Convexly constrained linear inverse problems: iterative least-squares and regularization, IEEE Trans. Signal Process, 46 (1998), 2345-2352.
T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 2005 (2005), 103-123.
S. Takahashi, W. Takahashi, M. Toyoda, Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces, J. Optim. Theory Appl., 147 (2010), 27-41.
W. Takahashi, M. Toyoda , Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.
H. K. Xu, Iterative algorithms for nonlinear operators , J. London Math. Soc., 2 (2002), 1-17.
Y. Yao, R. Chen, H. K. Xu , Schemes for finding minimum-norm solutions of variational inequalities, Nonlinear Anal., 72 (2010), 3447-3456.
Y. Yao, Y. C. Liou , Composite algorithms for minimization over the solutions of equilibrium problems and fixed point problems , Abstr. Appl. Anal., Article ID 763506, 2010 (2010), 19 pages.
S. S. Zhang, H. W. Lee Joseph, C. K. Chan, Algorithms of common solutions for quasi variational inclusion and fixed point problems , Appl. Math. Mech. (English Ed.), 29 (2008), 571-581.