## Abstract and Applied Analysis

### Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets

#### Abstract

Consider the variational inequality $\text{V}\text{I}\left(C,F\right)$ of finding a point ${x}^{\mathrm{*}}\in C$ satisfying the property $〈F{x}^{\mathrm{*}},x-{x}^{\mathrm{*}}〉\ge \mathrm{0}$, for all $x\in C$, where $C$ is the intersection of finite level sets of convex functions defined on a real Hilbert space $H$ and $F:H\to H$ is an $L$-Lipschitzian and $\eta$-strongly monotone operator. Relaxed and self-adaptive iterative algorithms are devised for computing the unique solution of $\text{V}\text{I}\left(C,F\right)$. Since our algorithm avoids calculating the projection ${P}_{C}$ (calculating ${P}_{C}$ by computing several sequences of projections onto half-spaces containing the original domain $C$) directly and has no need to know any information of the constants $L$ and $\eta$, the implementation of our algorithm is very easy. To prove strong convergence of our algorithms, a new lemma is established, which can be used as a fundamental tool for solving some nonlinear problems.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 942315, 8 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511882

Digital Object Identifier
doi:10.1155/2013/942315

Mathematical Reviews number (MathSciNet)
MR3055865

Zentralblatt MATH identifier
1273.47099

#### Citation

He, Songnian; Yang, Caiping. Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets. Abstr. Appl. Anal. 2013 (2013), Article ID 942315, 8 pages. doi:10.1155/2013/942315. https://projecteuclid.org/euclid.aaa/1393511882

#### References

• G. Stampacchia, “Formes bilineaires coercivites sur les ensembles convexes,” Comptes Rendus de l'Académie des Sciences, vol. 258, pp. 4413–4416, 1964.
• C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, John Wiley & Sons, New York, NY, USA, 1984.
• A. Bnouhachem, “A self-adaptive method for solving general mixed variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 309, no. 1, pp. 136–150, 2005.
• H. Brezis, Operateurs Maximaux Monotone et Semigroupes de Contractions dans les Espace d'Hilbert, North-Holland, Amsterdam, The Netherlands, 1973.
• R. W. Cottle, F. Giannessi, and J. L. Lions, Variational Inequalities and Complementarity Problems: Theory and Application, John Wiley & Sons, New York, NY, USA, 1980.
• M. Fukushima, “Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,” Mathematical Programming A, vol. 53, no. 1, pp. 99–110, 1992.
• K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83, Marcel Dekker, New York, NY, USA, 1984.
• F. Giannessi, A. Maugeri, and P. M. Pardalos, Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, Kluwer Academic, Dodrecht, The Netherlands, 2001.
• R. Glowinski, J. L. Lions, and R. Tremolier, Numerical Analysis of Variational Inequalities, vol. 8, North-Holland, The Netherlands, Amsterdam, 1981.
• P. T. Harker and J. S. Pang, “Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications,” Mathematical Programming B, vol. 48, no. 2, pp. 161–220, 1990.
• B. S. He, “A class of implicit methods for monotone variational inequalities,” Reports of the Institute of Mathematics 95-1, Nanjing University, Nanjing, China, 1995.
• B. S. He and L. Z. Liao, “Improvements of some projection methods for monotone nonlinear variational inequalities,” Journal of Optimization Theory and Applications, vol. 112, no. 1, pp. 111–128, 2002.
• B. S. He, Z. H. Yang, and X. M. Yuan, “An approximate proximal-extragradient type method for monotone variational inequalities,” Journal of Mathematical Analysis and Applications, vol. 300, no. 2, pp. 362–374, 2004.
• S. He and H. K. Xu, “Variational inequalities governed by boundedly Lipschitzian and strongly monotone operators,” Fixed Point Theory, vol. 10, no. 2, pp. 245–258, 2009.
• D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and their Applications, SIAM, Philadelphia, Pa, USA, 2000.
• J. L. Lions and G. Stampacchia, “Variational inequalities,” Communications on Pure and Applied Mathematics, vol. 20, pp. 493–519, 1967.
• H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003.
• H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
• H. Yang and M. G. H. Bell, “Traffic restraint, road pricing and network equilibrium,” Transportation Research B, vol. 31, no. 4, pp. 303–314, 1997.
• I. Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.
• M. Fukushima, “A relaxed projection method for variational inequalities,” Mathematical Programming, vol. 35, no. 1, pp. 58–70, 1986.
• L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces,” Numerical Functional Analysis and Optimization, vol. 29, no. 9-10, pp. 987–1033, 2008.
• L. C. Ceng, M. Teboulle, and J. C. Yao, “Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems,” Journal of Optimization Theory and Applications, vol. 146, no. 1, pp. 19–31, 2010.
• K. Goebel and W. A. Kirk, Topics on Metric Fixed Point Theory, Cambridge University Press, Cambridge, UK, 1990.
• H. K. Xu, “Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002.
• P. E. Maingé, “A hybrid extragradient-viscosity method for monotone operators and fixed point problems,” SIAM Journal on Control and Optimization, vol. 47, no. 3, pp. 1499–1515, 2008.
• H. H. Bauschke and J. M. Borwein, “On projection algorithms for solving convex feasibility problems,” SIAM Review, vol. 38, no. 3, pp. 367–426, 1996.
• Q. Yang, “The relaxed CQ algorithm solving the split feasibility problem,” Inverse Problems, vol. 20, no. 4, pp. 1261–1266, 2004.
• G. López, V. Martín-Márquez, F. Wang, and H. K. Xu, “Solving the split feasibility problem without prior knowledge of matrix norms,” Inverse Problems, vol. 28, no. 8, p. 085004, 18, 2012.
• Y. Censor, A. Gibali, and S. Reich, “The subgradient extragradient method for solving variational inequalities in Hilbert space,” Journal of Optimization Theory and Applications, vol. 148, no. 2, pp. 318–335, 2011.