Schemes for finding minimum-norm solutions of variational inequalities. (English) Zbl 1183.49012

Yao, Yonghong; Chen, Rudong; Xu, Hong-Kun

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 7-8, 3447-3456 (2010).

Summary: Consider the Variational Inequality (VI) of finding a point \(x^*\) such that
\[ x^*\in\text{Fix}(T)\text{ and }\langle(I-S)x^*,x-x^*\rangle\geq 0,\quad x\in \text{Fix}(T)\tag{*} \]
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: \(x^*=\text{argmin}_{x\in \Omega}\|x\|^2\).

MSC:

49J40	Variational inequalities
47J20	Variational and other types of inequalities involving nonlinear operators (general)
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
65J15	Numerical solutions to equations with nonlinear operators

Keywords:

variational inequality; nonexpansive mapping; iterative algorithm; implicit scheme; explicit scheme; fixed point; minimum norm

References:

[1]	Yamada, I., The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings, (Butnariu, D.; Censor, Y.; Reich, S., Inherently Parallel Algorithm for Feasibility and Optimization. Inherently Parallel Algorithm for Feasibility and Optimization, Stud. Comput. Math., vol. 8 (2001), North-Holland: North-Holland Amsterdam), 473-504 · 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
[3]	Yamada, I.; Ogura, N.; Shirakawa, N., A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems, (Inverse Problems, Image Analysis, and Medical Imaging. Inverse Problems, Image Analysis, and Medical Imaging, Contemp. Math., vol. 313 (2002), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 269-305 · Zbl 1039.47051
[4]	Mainge, P.-E.; Moudafi, A., Strong convergence of an iterative method for hierarchical fixed-point problems, Pacific J. Optim., 3, 3, 529-538 (2007) · Zbl 1158.47057
[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, 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, 1, 117-124 (2009) · Zbl 1168.49005
[8]	Moudafi, A.; Mainge, P.-E., Towards viscosity approximations of hierarchical fixed-point problems, Fixed Point Theory Appl., 1-10 (2006), Article ID 95453 · Zbl 1143.47305
[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
[12]	Moudafi, A., Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Problems, 23, 1635-1640 (2007) · Zbl 1128.47060
[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
[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
[16]	Xu, H. K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298, 279-291 (2004) · Zbl 1061.47060
[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
[18]	Yao, Y.; Liou, Y.-C., Weak and strong convergence of Krasnoselski-Mann iteration for hierarchical fixed point problems, Inverse Problems, 24, 1, 015015 (2008), 8 pp · Zbl 1154.47055
[19]	Geobel, K.; Kirk, W. A., (Topics in Metric Fixed Point Theory. Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, vol. 28 (1990), Cambridge University Press) · Zbl 0708.47031