Minimization of equilibrium problems, variational inequality problems and fixed point problems

AbstractIn this paper, we devote to find the solution of the following quadratic minimization problem $$\min_{x\in \Omega}\|x\|^2,$$where Ω is the intersection set of the solution set of some equilibrium problem, the fixed points set of a nonexpansive mapping and the solution set of some variational inequality. In order to solve the above minimization problem, we first construct an implicit algorithm by using the projection method. Further, we suggest an explicit algorithm by discretizing this implicit algorithm. Finally, we prove that the proposed implicit and explicit algorithms converge strongly to a solution of the above minimization problem.

[1]  Patrick L. Combettes,et al.  Proximal Thresholding Algorithm for Minimization over Orthonormal Bases , 2007, SIAM J. Optim..

[2]  I. Yamada.  A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems , 2002 .

[3]  Tomonari Suzuki,et al.  Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces , 2005 .

[4]  Hong-Kun Xu.  An Iterative Approach to Quadratic Optimization , 2003 .

[5]  P. L. Combettes,et al.  Equilibrium programming in Hilbert spaces , 2005 .

[6]  A. Iusem,et al.  Iterative Algorithms for Equilibrium Problems , 2003 .

[7]  A. Moudafi,et al.  Proximal and Dynamical Approaches to Equilibrium Problems , 1999 .

[8]  Jen-Chih Yao,et al.  Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems , 2009, Math. Comput. Model..

[9]  W. Oettli,et al.  From optimization and variational inequalities to equilibrium problems , 1994 .

[10]  Wataru Takahashi,et al.  Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings , 2006 .

[11]  W. Takahashi,et al.  Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space , 2008 .

[12]  Yair Censor,et al.  Cyclic subgradient projections , 1982, Math. Program..

[13]  I. Yamada,et al.  Hybrid Steepest Descent Method for Variational Inequality Problem over the Fixed Point Set of Certain Quasi-nonexpansive Mappings , 2005 .

[14]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[15]  I. Yamada.  The Hybrid Steepest Descent Method for the Variational Inequality Problem over the Intersection of Fixed Point Sets of Nonexpansive Mappings , 2001 .

[16]  Patrick L. Combettes,et al.  Strong Convergence of Block-Iterative Outer Approximation Methods for Convex Optimization , 2000, SIAM J. Control. Optim..

[17]  Jen-Chih Yao,et al.  On generalized implicit vector equilibrium problems in Banach spaces , 2009, Comput. Math. Appl..

[18]  P. Tseng.  Applications of splitting algorithm to decomposition in convex programming and variational inequalities , 1991 .

[19]  H. Attouch,et al.  A Dynamical Approach to Convex Minimization Coupling Approximation with the Steepest Descent Method , 1996 .

[20]  Hong-Kun Xu,et al.  VISCOSITY METHOD FOR HIERARCHICAL FIXED POINT APPROACH TO VARIATIONAL INEQUALITIES , 2010 .

[21]  Jen-Chih Yao,et al.  An iterative algorithm for approximating convex minimization problem , 2007, Appl. Math. Comput..

[22]  Y. Censor,et al.  Inherently parallel algorithms in feasibility and applications , 1997 .

[23]  Jen-Chih Yao,et al.  On modified iterative method for nonexpansive mappings and monotone mappings , 2007, Appl. Math. Comput..

[24]  Heinz H. Bauschke,et al.  Finding best approximation pairs relative to two closed convex sets in Hilbert spaces , 2004, J. Approx. Theory.

[25]  Patrick L. Combettes,et al.  Inconsistent signal feasibility problems: least-squares solutions in a product space , 1994, IEEE Trans. Signal Process..

[26]  Jen-Chih Yao,et al.  Well-posedness by perturbations of mixed variational inequalities in Banach spaces , 2010, Eur. J. Oper. Res..

[27]  Jen-Chih Yao,et al.  A hybrid iterative scheme for mixed equilibrium problems and fixed point problems , 2008 .

[28]  Xiwen Lu,et al.  Hybrid methods for a class of monotone variational inequalities , 2009 .

[29]  Wataru Takahashi,et al.  Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings , 2003 .

[30]  Jen-Chih Yao,et al.  Implicit Iteration Scheme with Perturbed Mapping for Equilibrium Problems and Fixed Point Problems of Finitely Many Nonexpansive Mappings , 2008 .