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Hybrid methods for a class of monotone variational inequalities. (English) Zbl 1176.90462
The paper deals with the study of certain hybrid methods for a special class of ill-posed monotone variational inequality problems in a Hilbert space setting, where the underlying operator is the complement of a nonexpansive mapping and the constraint set equals the set of fixed points of another nonexpansive mapping. Problems of this type include in particular monotone inclusions and convex optimization problems with a constraint set of the latter type. It is shown that both implicit and explicit iterative schemes are strongly convergent, where the employed regularization technique uses contractions of the nonexpansive operator in the variational inequality and allows a proof under considerably less restrictive conditions than they were needed for a related earlier approach by other authors. The paper is completed with an application to the class of hierarchical optimization problems where a proper lower semi-continuous convex function on a Hilbert space is minimized over the set of minimizers of another function of this type.

MSC:
90C25Convex programming
47H05Monotone operators (with respect to duality) and generalizations
47H09Mappings defined by “shrinking” properties
65J15Equations with nonlinear operators (numerical methods)
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References:
[1] Yamada, I.: The hybrid steepest descent for the variational inequality problems over the intersection of fixed point sets of nonexpansive mappings. Inherently parallel algorithms in feasibility and optimization and their applications, 473-504 (2001) · Zbl 1013.49005
[2] Mainge, P. E.; Moudafi, A.: Strong convergence of an iterative method for hierarchical fixed-points problems. Pacific J. Optim. 3, 529-538 (2007) · Zbl 1158.47057
[3] A. Moudafi, P-E. Mainge, Towards viscosity approximations of hierarchical fixed-points problems, Fixed Point Theory Appl., vol. 2006, Article ID 95453, 1--10
[4] Xu, H. K.: Viscosity approximation methods for nonexpansive mappings. J. math. Anal. appl. 298, 279-291 (2004) · Zbl 1061.47060
[5] Geobel, K.; Kirk, W. A.: Topics in metric fixed point theory. Cambridge studies in advanced mathematics 28 (1990)
[6] Xu, H. K.: Iterative algorithms for nonlinear operators. J. London math. Soc. 66, 240-256 (2002) · Zbl 1013.47032
[7] Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. math. Anal. appl. 241, 46-55 (2000) · Zbl 0957.47039
[8] H.K. Xu, Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math. 13 (December 2009) (in press). http://www.tjm.nsysu.edu.tw/myweb/FrameToAppear.htm
[9] Browder, F. E.: Convergence of approximation to fixed points of nonexpansive nonlinear mappings in Hilbert spaces. Arch. ration. Mech. anal. 24, 82-90 (1967) · Zbl 0148.13601
[10] Halpern, B.: Fixed points of nonexpanding maps. Bull. amer. Math. soc. 73, 957-961 (1967) · Zbl 0177.19101
[11] Lions, P. L.: Approximation de points fixes de contractions. CR acad. Sci. sèr. A-B Paris 284, 1357-1359 (1977) · Zbl 0349.47046
[12] Marino, G.; Xu, H. K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. math. Anal. appl. 318, 43-52 (2006) · Zbl 1095.47038
[13] Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. math. Anal. appl. 75, 287-292 (1980) · Zbl 0437.47047
[14] Reich, S.: Approximating fixed points of nonexpansive mappings. Panamer. math. J. 4, No. 2, 23-28 (1994) · Zbl 0856.47032
[15] Shioji, N.; Takahashi, W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. amer. Math. soc. 125, 3641-3645 (1997) · Zbl 0888.47034
[16] Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. math. 58, 486-491 (1992) · Zbl 0797.47036
[17] Xu, H. K.: Another control condition in an iterative method for nonexpansive mappings. Bull. austral. Math. soc. 65, 109-113 (2002) · Zbl 1030.47036
[18] Xu, H. K.: Remarks on an iterative method for nonexpansive mappings. Comm. appl. Nonlinear anal. 10, No. 1, 67-75 (2003) · Zbl 1035.47035
[19] Xu, H. K.: An iterative approach to quadratic optimization. J. optim. Theory appl. 116, 659-678 (2003) · Zbl 1043.90063
[20] G. Marino, H.K. Xu, Explicit hierarchical fixed point approach to variational inequalities. Preprint · Zbl 1221.49012
[21] Baillon, J. B.; Haddad, G.: Quelques proprietes des operateurs angle-bornes et n-cycliquement monotones. Isr. J. Math. 26, 137-150 (1977) · Zbl 0352.47023
[22] Cabot, A.: Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization. SIAM J. Optim. 15, 555-572 (2005) · Zbl 1079.90098