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Schemes for finding minimum-norm solutions of variational inequalities. (English) Zbl 1183.49012
Summary: Consider the Variational Inequality (VI) of finding a point $x^*$ such that $$x^*\in\text{Fix}(T)\text{ and }\langle(I-S)x^*,x-x^*\rangle\ge 0,\quad x\in \text{Fix}(T)\tag*$$ where $T,S$ are nonexpansive self-mappings of a closed convex subset $C$ of a Hilbert space, and $\text{Fix}(T)$ is the set of fixed points of $T$. Assume that the solution set $\Omega$ of this VI is nonempty. This paper introduces two schemes, one implicit and one explicit, that can be used to find the minimum-norm solution of VI $(*)$; namely, the unique solution $x^*$ to the quadratic minimization problem: $x^*=\text{argmin}_{x\in \Omega}\|x\|^2$.

49J40Variational methods including variational inequalities
47J20Inequalities involving nonlinear operators
47H09Mappings defined by “shrinking” properties
65J15Equations with nonlinear operators (numerical methods)
Full Text: DOI
[1] Yamada, I.: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, Stud. comput. Math. 8, 473-504 (2001) · Zbl 1013.49005
[2] 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 · doi:10.1081/NFA-200045815
[3] Yamada, I.; Ogura, N.; Shirakawa, N.: A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems, Contemp. math. 313, 269-305 (2002) · Zbl 1039.47051
[4] Mainge, P. -E.; Moudafi, A.: Strong convergence of an iterative method for hierarchical fixed-point problems, Pacific J. Optim. 3, No. 3, 529-538 (2007) · Zbl 1158.47057 · http://www.ybook.co.jp/online/pjoe/vol3/pjov3n3p529.html
[5] Lu, X.; Xu, H. K.; Yin, X.: Hybrid methods for a class of monotone variational inequalities, Nonlinear anal. (2008)
[6] Chen, R.; Fu, S.; Xu, H. K.: Regularization and iteration methods for a class of monotone variational inequalities, Taiwanese J. Math. 13, No. 2B, 739-752 (2009) · Zbl 1179.58008
[7] Cianciaruso, F.; Colao, V.; Muglia, L.; Xu, H. K.: On an implicit hierarchical fixed point approach to variational inequalities, Bull. austral. Math. soc. 80, No. 1, 117-124 (2009) · Zbl 1168.49005 · doi:10.1017/S0004972709000082
[8] Moudafi, A.; Mainge, P. -E.: Towards viscosity approximations of hierarchical fixed-point problems, Fixed point theory appl., 1-10 (2006) · Zbl 1143.47305 · doi:10.1155/FPTA/2006/95453
[9] G. Marino, H.K. Xu, Explicit hierarchical fixed Point approach to variational inequalities (preprint) · Zbl 1221.49012 · doi:10.1007/s10957-010-9775-1
[10] H.K. Xu, Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math. 14 (2) (2010) (in press) · Zbl 1215.47099
[11] 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
[12] Moudafi, A.: Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse problems 23, 1635-1640 (2007) · Zbl 1128.47060 · doi:10.1088/0266-5611/23/4/015
[13] Xu, H. K.; Kim, T. H.: Convergence of hybrid steepest-descent methods for variational inequalities, J. optim. Theory appl. 119, 185-201 (2003) · Zbl 1045.49018 · doi:10.1023/B:JOTA.0000005048.79379.b6
[14] Xu, H. K.: Iterative algorithms for nonlinear operators, J. London math. Soc. 66, 240-256 (2002) · Zbl 1013.47032
[15] 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
[16] 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
[17] Xu, H. K.: A variable krasnoselski-Mann algorithm and the multiple-set split feasibility problem, Inverse problems 22, 2021-2034 (2006) · Zbl 1126.47057 · doi:10.1088/0266-5611/22/6/007
[18] Yao, Y.; Liou, Y. -C.: Weak and strong convergence of krasnoselski-Mann iteration for hierarchical fixed point problems, Inverse problems 24, No. 1, 015015 (2008) · Zbl 1154.47055 · doi:10.1088/0266-5611/24/1/015015
[19] Geobel, K.; Kirk, W. A.: Topics in metric fixed point theory, Cambridge studies in advanced mathematics 28 (1990) · Zbl 0708.47031