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Two-step iterative algorithms for hierarchical fixed point problems and variational inequality problems. (English) Zbl 1223.47106

Summary: Let \(C\) be a nonempty closed convex subset of a real Hilbert space \(H\). Let \(Q:C\rightarrow C\) be a fixed contraction and \(S,T:C\rightarrow C\) be two nonexpansive mappings such that \(\text{Fix}(T)\neq \emptyset \). Consider the following two-step iterative algorithm:
\[ \begin{gathered} x_{n+1}=\alpha_{n}Qx_{n}+(1-\alpha_{n})Ty_{n},\\y_{n}=\beta_{n}Sx_{n}+(1-\beta_{n})x_{n},\quad n\geq0.\end{gathered} \]
It is proven that under appropriate conditions, the above iterative sequence \(\{x_n\}\) converges strongly to \(\widetilde x\in \text{Fix}(T)\), which solves some variational inequality depending on a given criterion \(S\), namely: find \(\widetilde x\in H\); \(0\in (I-S)\widetilde x+N_{\text{Fix}(T)}\widetilde x\), where \(N_{\text{Fix}(T)}\) denotes the normal cone to the set of fixed points of \(T\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
65J15 Numerical solutions to equations with nonlinear operators
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References:

[1] Moudafi, A.: Viscosity approximations methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000) · Zbl 0957.47039
[2] Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004) · Zbl 1061.47060
[3] Rockafellar, R.T., Wets, R.: Variational Analysis. Springer, Berlin (1988) · Zbl 0888.49001
[4] Yamada, I., Ogura, N.: Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619–655 (2004) · Zbl 1095.47049
[5] Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996) · Zbl 0870.90092
[6] Marino, G., Xu, H.K.: A general iterative method for nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 318, 43–52 (2006) · Zbl 1095.47038
[7] 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
[8] Solodov, M.: An explicit descent method for bilevel convex optimization. J. Convex Anal. 14, 227–237 (2007) · Zbl 1145.90081
[9] Moudafi, A., Mainge, P.E.: Towards viscosity approximations of hierarchical fixed point problems. Fixed Point Theory Appl. 2006, 1–10 (2006) · Zbl 1143.47305
[10] Moudafi, A.: Krasnoselski-Mann iteration for hierarchical fixed-point problems. Inverse Probl. 23, 1635–1640 (2007) · Zbl 1128.47060
[11] Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004) · Zbl 1051.65067
[12] Censor, Y., Motova, A., Segal, A.: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 327, 1244–1256 (2007) · Zbl 1253.90211
[13] Yang, Q., Zhao, J.: Generalized KM theorems and their applications. Inverse Probl. 22, 833–844 (2006) · Zbl 1117.65081
[14] Xu, H.K.: A variable Krasnosel’ski-Mann algorithm and the multiple-set split feasibility problem. Inverse Probl. 22, 2021–2034 (2006) · Zbl 1126.47057
[15] Mainge, P.E., Moudafi, A.: Strong convergence of an iterative method for hierarchical fixed point problems. Pac. J. Optim. 3, 529–538 (2007) · Zbl 1158.47057
[16] Brezis, H.: Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. Math. Studies, vol. 5. Am. Elsevier, New York (1973)
[17] Lions, P.L.: Two remarks on the convergence of convex functions and monotone operators. Nonlinear Anal. 2, 553–562 (1978) · Zbl 0383.47033
[18] Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 2, 240–256 (2002) · Zbl 1013.47032
[19] Attouch, H., Riahi, H., Thera, M.: Somme ponctuelle d’operateurs maximaux monotones. Serdica Math. J. 22, 267–292 (1996) · Zbl 0869.47028
[20] Yao, J.C.: Applications of variational inequalities to nonlinear analysis. Appl. Math. Lett. 4, 89–92 (1991) · Zbl 0734.49003
[21] Yao, J.C.: The unification of the calculus of variations and the theory of nonlinear operators in Banach spaces. Appl. Math. Lett. 5, 81–84 (1992) · Zbl 0777.49012
[22] Zeng, L.C., Schaible, S., Yao, J.C.: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. J. Optim. Theory Appl. 124, 725–738 (2005) · Zbl 1067.49007
[23] Schaible, S., Yao, J.C., Zeng, L.C.: A proximal method for pseudomonotone type variational-like inequalities. Taiwan. J. Math. 10, 497–513 (2006) · Zbl 1116.49004
[24] Zeng, L.C., Lin, L.J., Yao, J.C.: Auxiliary problem method for mixed variational-like inequalities. Taiwan. J. Math. 10, 515–529 (2006) · Zbl 1259.49010
[25] Zeng, L.C., Guu, S.M., Yao, J.C.: Characterization of H-monotone operators with applications to variational inclusions. Comput. Math. Appl. 50, 329–337 (2005) · Zbl 1080.49012
[26] 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 Algorithm for Feasibility and Optimization. Studies in Computational Mathematics, vol. 8, pp. 473–504. Elsevier, Amsterdam (2001) · Zbl 1013.49005
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