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Explicit hierarchical fixed point approach to variational inequalities. (English) Zbl 1221.49012
Summary: An explicit hierarchical fixed point algorithm is introduced to solve monotone variational inequalities, which are governed by a pair of nonexpansive mappings, one of which is used to define the governing operator and the other to define the feasible set. These kinds of variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Strong convergence of the algorithm is proved under different circumstances of parameter selections. Applications in hierarchical minimization problems are also included.

MSC:
49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
90C33Complementarity and equilibrium problems; variational inequalities (finite dimensions)
References:
[1]Baiocchi, C., Capelo, A.: Variational and Quasi-Variational Inequalities. Wiley, New York (1984)
[2]Cottle, R.W., Giannessi, F., Lions, J.L.: Variational Inequalities and Complementarity Problems: Theory and Applications. Wiley, New York (1980)
[3]Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vols. I & II. Springer, Berlin (2003)
[4]Giannessi, F., Maugeri, A.: Variational Inequalities and Network Equilibrium Problems. Plenum, New York (1995)
[5]Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)
[6]Auslender, A., Teboulle, M.: Interior projection-like methods for monotone variational inequalities. Math. Program. 104, 39–68 (2005) · Zbl 1159.90517 · doi:10.1007/s10107-004-0568-x
[7]He, B.S.: A class of projection and contraction methods for monotone variational inequalities. Appl. Math. Optim. 35, 69–76 (1997)
[8]He, B.S.: Inexact implicit methods for monotone general variational inequalities. Math. Program. 86, 113–123 (1999) · Zbl 0979.49006 · doi:10.1007/s101070050086
[9]He, B.S., Li, M., Liao, L.-Z.: An improved contraction method for structured monotone variational inequalities. Optimization 57, 643–653 (2008) · Zbl 1157.65403 · doi:10.1080/02331930802386288
[10]He, B.S., Xu, M.-H.: A general framework of contraction methods for monotone variational inequalities. Pac. J. Optim. 4, 195–212 (2008)
[11]He, B., He, X.Z., Liu, H.X., Wu, T.: Self-adaptive projection method for co-coercive variational inequalities. Eur. J. Oper. Res. 196, 43–48 (2009) · Zbl 1163.58305 · doi:10.1016/j.ejor.2008.03.004
[12]He, B., Wang, X., Yang, J.: A comparison of different contraction methods for monotone variational inequalities. J. Comput. Math. 27, 459–473 (2009) · Zbl 1212.65259 · doi:10.4208/jcm.2009.27.4.013
[13]He, B.S., Yang, Z.H., Yuan, X.M.: An approximate proximal-extragradient type method for monotone variational inequalities. J. Math. Anal. Appl. 300, 362–374 (2004) · Zbl 1068.65087 · doi:10.1016/j.jmaa.2004.04.068
[14]Wang, Y., Xiu, N., Wang, C.: A new version of extragradient method for variational inequality problems. Comput. Math. Appl. 42, 969–979 (2001) · Zbl 0993.49005 · doi:10.1016/S0898-1221(01)00213-9
[15]Xiu, N., Wang, C., Zhang, J.: Convergence properties of projection and contraction methods for variational inequality problems. Appl. Math. Optim. 43, 147–168 (2001) · Zbl 0980.90093 · doi:10.1007/s002450010023
[16]Yang, X.Q.: Vector variational inequality and its duality. Nonlinear Anal. 21, 869–877 (1993) · Zbl 0809.49009 · doi:10.1016/0362-546X(93)90052-T
[17]Giannessi, F., Mastroeni, G., Yang, X.Q.: A survey on vector variational inequalities. Boll. Unione Mat. Ital. (9) 2(1), 225–237 (2009)
[18]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 · doi:10.1016/j.jmaa.2005.05.028
[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]Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002) · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[21]Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004) · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059
[22]Yamada, I.: The hybrid steepest descent for the variational inequality problems over the intersection of fixed point sets of nonexpansive mappings. In: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473–504. Elservier, New York (2001)
[23]Mainge, P.-E., Moudafi, A.: Strong convergence of an iterative method for hierarchical fixed-points problems. Pac. J. Optim. 3, 529–538 (2007)
[24]Moudafi, A., Mainge, P.-E.: Towards viscosity approximations of hierarchical fixed-points problems. Fixed Point Theory Appl. 2006, 1–10 (2006). Article ID 95453 · Zbl 1143.47305 · doi:10.1155/FPTA/2006/95453
[25]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 · doi:10.1007/BF00251595
[26]Ceng, L.C., Ansari, Q.H., Yao, J.C.: 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
[27]Ceng, L.C., Ansari, Q.H., Yao, J.C.: On relaxed viscosity iterative methods for variational inequalities in Banach spaces. J. Comput. Appl. Math. 230, 813–822 (2009) · Zbl 1178.65074 · doi:10.1016/j.cam.2009.01.015
[28]Ceng, L.C., Xu, H.K., Yao, J.C.: A hybrid steepest-descent method for variational inequalities in Hilbert spaces. Appl. Anal. 87, 575–589 (2008) · Zbl 1158.47050 · doi:10.1080/00036810802140608
[29]Halpern, B.: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957–961 (1967) · Zbl 0177.19101 · doi:10.1090/S0002-9904-1967-11864-0
[30]Lions, P.L.: Approximation de points fixes de contractions. C.R. Acad. Sci. Sèr. A–B Paris 284, 1357–1359 (1977)
[31]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
[32]Reich, S.: Approximating fixed points of nonexpansive mappings, Panamerican. Math. J. 4, 23–28 (1994)
[33]Shioji, N., Takahashi, W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 125, 3641–3645 (1997) · Zbl 0888.47034 · doi:10.1090/S0002-9939-97-04033-1
[34]Tam, N.N., Yao, J.C., Yen, N.D.: On some solution methods for pseudomonotone variational inequalities. J. Optim. Theory Appl. 138, 253–273 (2008) · Zbl 05314923 · doi:10.1007/s10957-008-9376-4
[35]Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992) · Zbl 0797.47036 · doi:10.1007/BF01190119
[36]Wong, N.C., Sahu, D.R., Yao, J.C.: Solving variational inequalities involving nonexpansive type mappings. Nonlinear Anal., Theory Methods Appl. 69, 4732–4753 (2008) · Zbl 1182.47050 · doi:10.1016/j.na.2007.11.025
[37]Xu, H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 65, 109–113 (2002) · Zbl 1030.47036 · doi:10.1017/S0004972700020116
[38]Xu, H.K.: Remarks on an iterative method for nonexpansive mappings. Commun. Appl. Nonlinear Anal. 10, 67–75 (2003)
[39]Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003) · Zbl 1043.90063 · doi:10.1023/A:1023073621589
[40]Yao, J.C., Chadli, O.: Pseudomonotone complementarity problems and variational inequalities. In: Crouzeix, J.P., Haddjissas, N., Schaible, S. (eds.) Handbook of Generalized Convexity and Monotonicity, pp. 501–558 (2005)
[41]Zeng, L.C., Wong, N.C., Yao, J.C.: On the convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. J. Optim. Theory Appl. 132, 51–69 (2007) · Zbl 1137.47059 · doi:10.1007/s10957-006-9068-x
[42]Xu, H.K.: Viscosity method for hierarchical fixed Point approach to variational inequalities. Taiwan. J. Math. 14, 463–478 (2010)
[43]Geobel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics, vol. 28. Cambridge University Press, Cambridge (1990)
[44]Attouch, H.: Variational Convergence for Functions and Operators. Applicable Math. Series. Pitman, London (1984)
[45]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 · doi:10.1007/BF03007664
[46]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 · doi:10.1137/S105262340343467X