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Composite Algorithms for Minimization over the Solutions of Equilibrium Problems and Fixed Point Problems

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  • Yonghong Yao
  • Yeong-Cheng Liou

Abstract

The purpose of this paper is to solve the minimization problem of finding x∗ such that x∗ = argminx∈Γ∥x∥2, where Γ stands for the intersection set of the solution set of the equilibrium problem and the fixed points set of a nonexpansive mapping. We first present two new composite algorithms (one implicit and one explicit). Further, we prove that the proposed composite algorithms converge strongly to x∗.

Suggested Citation

  • Yonghong Yao & Yeong-Cheng Liou, 2010. "Composite Algorithms for Minimization over the Solutions of Equilibrium Problems and Fixed Point Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    • Handle: RePEc:wly:jnlaaa:v:2010:y:2010:i:1:n:763506
    DOI: 10.1155/2010/763506
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    References listed on IDEAS

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    1. Jian-Wen Peng & Soon-Yi Wu & Jen-Chih Yao, 2010. "A New Iterative Method for Finding Common Solutions of a System of Equilibrium Problems, Fixed‐Point Problems, and Variational Inequalities," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    2. Chakkrid Klin-eam & Suthep Suantai & Wataru Takahashi, 2009. "Strong Convergence of Generalized Projection Algorithms for Nonlinear Operators," Abstract and Applied Analysis, John Wiley & Sons, vol. 2009(1).
    3. Wanpen Chantarangsi & Chaichana Jaiboon & Poom Kumam, 2010. "A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    4. Jian-Wen Peng & Soon-Yi Wu & Jen-Chih Yao, 2010. "A New Iterative Method for Finding Common Solutions of a System of Equilibrium Problems, Fixed-Point Problems, and Variational Inequalities," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-27, August.
    5. Siwaporn Saewan & Poom Kumam & Kriengsak Wattanawitoon, 2010. "Convergence Theorem Based on a New Hybrid Projection Method for Finding a Common Solution of Generalized Equilibrium and Variational Inequality Problems in Banach Spaces," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-25, January.
    6. Siwaporn Saewan & Poom Kumam & Kriengsak Wattanawitoon, 2010. "Convergence Theorem Based on a New Hybrid Projection Method for Finding a Common Solution of Generalized Equilibrium and Variational Inequality Problems in Banach Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2010(1).
    7. Wanpen Chantarangsi & Chaichana Jaiboon & Poom Kumam, 2010. "A Viscosity Hybrid Steepest Descent Method for Generalized Mixed Equilibrium Problems and Variational Inequalities for Relaxed Cocoercive Mapping in Hilbert Spaces," Abstract and Applied Analysis, Hindawi, vol. 2010, pages 1-39, July.
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    Cited by:

    1. Yonghong Yao & Yeong-Cheng Liou & Ngai-Ching Wong, 2013. "Superimposed optimization methods for the mixed equilibrium problem and variational inclusion," Journal of Global Optimization, Springer, vol. 57(3), pages 935-950, November.
    2. Rattanaporn Wangkeeree & Uthai Kamraksa & Rabian Wangkeeree, 2011. "A General Composite Algorithms for Solving General Equilibrium Problems and Fixed Point Problems in Hilbert Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    3. Wei Xu & Yuanheng Wang, 2012. "Strong Convergence of the Iterative Methods for Hierarchical Fixed Point Problems of an Infinite Family of Strictly Nonself Pseudocontractions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).

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