A steepest-descent Krasnosel’skii-Mann algorithm for a class of variational inequalities in Banach spaces. (English) Zbl 1457.47012
Summary: In this paper, in order to solve a variational inequality problem over the fixed point set of a nonexpansive mapping on uniformly smooth or reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm, we investigate an explicit iteration method, based on the steepest-descent and Krasnosel’skii-Mann algorithms. We also show that some modifications of the last and Halpern-type algorithms are special cases of our result.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
Keywords:
nonexpansive mapping; fixed point; variational inequality; uniformly smooth Banach space; reflexive and strictly convex Banach space; uniformly Gâteaux differentiable norm; explicit iteration; steepest-descent method; Krasnosel’skii-Mann algorithm; Halpern-type algorithmsReferences:
| [1] | Agarwal R. P., O’Regan D., Sahu D. R.: Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Springer, New York (2009) · Zbl 1176.47037 |
| [2] | Browder E. F., Petryshin W. V.: Constructions of fixed points of nonlinear mappings. J. Math. Anal. Appl. 20, 197-228 (1967) · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6 |
| [3] | Buong Ng., Duong L. Th.: An explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 151, 513-524 (2011) · Zbl 1251.47059 · doi:10.1007/s10957-011-9890-7 |
| [4] | Byrne C.: A unified treatment of some iterative algorithm in signal processing and image reconstruction. Inverse Problems 20, 103-120 (2004) · Zbl 1051.65067 · doi:10.1088/0266-5611/20/1/006 |
| [5] | Ceng L. C., Ansari Q. H., Yao J. Ch.: Mann-type steepest-descent and modified hybrid steepest descent methods for variational inequalities in Banach spaces. Numer. Funct. Anal. Optim. 29, 987-1033 (2008) · Zbl 1163.49002 · doi:10.1080/01630560802418391 |
| [6] | Cioranescu I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer Acad. Publ., Dordrecht (1990) · Zbl 0712.47043 · doi:10.1007/978-94-009-2121-4 |
| [7] | Goebel K., Reich S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984) · Zbl 0537.46001 |
| [8] | Halpern B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957-961 (1967) · Zbl 0177.19101 · doi:10.1090/S0002-9904-1967-11864-0 |
| [9] | Iiduka I.: Fixed point optimization algorithm and its application to power control in CDMA data networks. Math. Program. 133, 227-242 (2012) · Zbl 1274.90428 · doi:10.1007/s10107-010-0427-x |
| [10] | Iiduka I.: An ergodic algorithm for the power-control games for CDMA data networks. J. Math. Model. Algorithms 8, 1-18 (2009) · Zbl 1170.93020 · doi:10.1007/s10852-008-9099-4 |
| [11] | Iiduka I.: Fixed point optimization algorithm and its application to network bandwidth allocation. J. Comput. Appl. Math. 236, 1733-1742 (2012) · Zbl 1242.65119 · doi:10.1016/j.cam.2011.10.004 |
| [12] | Iiduka I.: Iterative algorithm for triple-hierarchical constrained nonconvex optimization problem and its application to network bandwidth allocation. SIAM J. Optim. 22, 862-878 (2012) · Zbl 1267.90139 · doi:10.1137/110849456 |
| [13] | Iiduka I.: Fixed point optimization algorithms for distributed optimization in network systems. SIAM J. Optim. 23, 1-26 (2013) · Zbl 1266.49067 · doi:10.1137/120866877 |
| [14] | Ishikawa S.: Fixed points by a new iteration method. Proc. Amer. Math. Soc. 44, 147-150 (1974) · Zbl 0286.47036 · doi:10.1090/S0002-9939-1974-0336469-5 |
| [15] | L. V. Kantorovich and G. P. Akilov, Functional Analysis. 2nd ed., Nauka, Moscow, 1977 (in Russian). · Zbl 0555.46001 |
| [16] | Kim T. H., Xu H. K.: Strong convergence of modified Mann iterations. Nonlinear Anal. 61, 51-60 (2005) · Zbl 1091.47055 · doi:10.1016/j.na.2004.11.011 |
| [17] | Krasnosel’skiĭ M. A.: Two remarks on the method of successive approximations. Uspekhi Mat. Nauk 10, 123-127 (1955) · Zbl 0064.12002 |
| [18] | Mann W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506-510 (1953) · Zbl 0050.11603 · doi:10.1090/S0002-9939-1953-0054846-3 |
| [19] | Moudafi A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46-55 (2000) · Zbl 0957.47039 · doi:10.1006/jmaa.1999.6615 |
| [20] | Reich S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274-276 (1979) · Zbl 0423.47026 · doi:10.1016/0022-247X(79)90024-6 |
| [21] | Reich S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287-292 (1980) · Zbl 0437.47047 · doi:10.1016/0022-247X(80)90323-6 |
| [22] | Reich S.: Review of: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, by Ioana Cioranescu [6]. Bull. Amer. Math. Soc. 26, 367-370 (1992) · doi:10.1090/S0273-0979-1992-00287-2 |
| [23] | Shehu Y.: Modified Krasnoselskii-Mann iterative algorithm for nonexpansive mappings in Banach spaces. Arab J. Math. (Springer) 2, 209-219 (2013) · Zbl 1515.47104 · doi:10.1007/s40065-013-0066-1 |
| [24] | Suzuki T.: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 2005, 103-123 (2005) · Zbl 1123.47308 · doi:10.1155/FPTA.2005.103 |
| [25] | Suzuki T.: A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings. Proc. Amer. Math. Soc. 135, 99-106 (2007) · Zbl 1117.47041 · doi:10.1090/S0002-9939-06-08435-8 |
| [26] | Takahashi W., Ueda Y.: On Reich’s strong convergence theorems for resolvents of accretive operators. J. Math. Anal. Appl. 104, 546-553 (1984) · Zbl 0599.47084 · doi:10.1016/0022-247X(84)90019-2 |
| [27] | Vainberg M.M.: Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations. Wiley, New York (1973) · Zbl 0279.47022 |
| [28] | Xu H.-K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. (2) 66, 240-256 (2002) · Zbl 1013.47032 · doi:10.1112/S0024610702003332 |
| [29] | Xu Z., Roach G.F.: A necessary and sufficient condition for convergence of steepest descent approximation to accretive operator equations. J. Math. Anal. Appl. 167, 340-354 (1992) · Zbl 0818.47061 · doi:10.1016/0022-247X(92)90211-U |
| [30] | I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (D. Butnariu, Y. Censor and S. Reich, eds.), North-Holland, Amsterdam, 2001, 473-504. · Zbl 1013.49005 |
| [31] | Yao Y., Zhou H., Liou Y.-Ch.: Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings. J. Appl. Math. Comput. 29, 383-389 (2009) · Zbl 1222.47129 · doi:10.1007/s12190-008-0139-z |
| [32] | Zhou H.: A characteristic condition for convergence of steepest descent approximation to accretive operator equations. J. Math. Anal. Appl. 271, 1-6 (2002) · Zbl 1036.47046 · doi:10.1016/S0022-247X(02)00122-1 |
| [33] | Zhou H., Wang P.: A simpler explicit iterative algorithm for a class of variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 161, 716-727 (2014) · Zbl 1293.49026 · doi:10.1007/s10957-013-0470-x |
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.
