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Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems. (English) Zbl 1222.47091

The authors consider an iterative process for finding a common element of the set of common fixed points of N nonexpansive mappings and the set of solutions of the variational inequality for a pseudomonotone, Lipschitz-continuous mapping T in a real Hilbert space. To overcome the difficulty of dealing with an operator T which is not maximal monotone, the idea proposed in this paper is to investigate an iterative scheme based on the combination of the extragradient method and the approximate proximal method introduced in [R. T. Rockafellar, SIAM J. Control Optim. 14, 877–898 (1976; Zbl 0358.90053)]. A necessary and sufficient condition for weak convergence of the sequences produced by this scheme is established. The scheme proposed in this paper is related to the scheme in [N. Nadezhkina and W. Takahashi, SIAM J. Optim. 16, 1230–1241 (2006; Zbl 1143.47047)] utilized for the approximation of a common element of the solution set of a monotone variational inequality problem and the fixed point set of a single nonexpansive mapping.

The present paper considers the more general case of several nonexpansive mappings. More importantly, it drops the maximal monotonicity, using instead the weaker assumption of (algebraic) pseudomonotonicity. The price for this is the assumption that the nonlinear pseudomonotone mapping T, in addition to being k-Lipschitz, must be weak-strong sequentially continuous or completely continuous. In the finite-dimensional case, the assumption of Lipschitz continuity is enough, and the result presented in this paper extends the result of Nadezhkina and Takahashi.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47J20Inequalities involving nonlinear operators
47H05Monotone operators (with respect to duality) and generalizations
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
49J40Variational methods including variational inequalities
49J45Optimal control problems involving semicontinuity and convergence; relaxation
49J53Set-valued and variational analysis
References:
[1]Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vols. I and II. Springer, New York (2003)
[2]Antipin, A.S.: Methods for solving variational inequalities with related constraints. Comput. Math. Math. Phys. 40, 1239–1254 (2000)
[3]Antipin, A.S., Vasiliev, F.P.: Regularized prediction method for solving variational inequalities with an inexactly given set. Comput. Math. Math. Phys. 44, 750–758 (2004)
[4]Popov, L.D.: On a one-stage method for solving lexicographic variational inequalities. Izv. Vyssh. Uchebn. Zaved. Mat. 12, 71–81 (1998)
[5]Takahashi, W.: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama (2000)
[6]Yamada, I.: The hybrid steepest-descent method for the variational inequality problem over the intersection of fixed-point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473–504. Kluwer Academic, Dordrecht (2001)
[7]Berinde, V.: Iterative Approximation of Fixed Points. Lecture Notes in Mathematics, vol. 1912. Springer, New York (2007)
[8]Ceng, L.C., Yao, J.C.: An extragradient-like approximation method for variational inequality problems and fixed point problems. Appl. Math. Comput. 190, 206–215 (2007) · Zbl 1124.65056 · doi:10.1016/j.amc.2007.01.021
[9]Ceng, L.C., Yao, J.C.: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan. J. Math. 10(5), 1293–1303 (2006)
[10]Iiduka, H., Takahashi, W.: Strong convergence theorem by a hybrid method for nonlinear mappings of nonexpansive and monotone type and applications. Adv. Nonlinear Var. Inequal. 9, 1–10 (2006)
[11]Iiduka, H., Takahashi, W., Toyoda, M.: Approximation of solutions of variational inequalities for monotone mappings. Panam. Math. J. 14, 49–61 (2004)
[12]Nadezhkina, N., Takahashi, W.: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 128, 191–201 (2006) · Zbl 1130.90055 · doi:10.1007/s10957-005-7564-z
[13]Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006) · Zbl 1143.47047 · doi:10.1137/050624315
[14]Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Math. Metody 12, 746–756 (1976); [English translation: Matecon 13, 35–49 (1977)]
[15]Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970) · doi:10.1090/S0002-9947-1970-0282272-5
[16]Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976) · Zbl 0358.90053 · doi:10.1137/0314056
[17]Goebel, K., Kirk, W.A.: Topics on Metric Fixed-Point Theory. Cambridge University Press, Cambridge (1990)
[18]Polyak, B.T.: Introduction to Optimization. Optimization Software Inc., New York (1987)
[19]Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967) · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0