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

Authors:
Yonghong Yao;Yeong-Cheng Liou;Shin Min Kang
Affiliations:
Department of Mathematics, Tianjin Polytechnic University, Tianjin, China 300160;Department of Information Management, Cheng Shiu University, Kaohsiung, Taiwan 833;Department of Mathematics and the RINS, Gyeongsang National University, Jinju, Republic of Korea 660-701
Venue:
Journal of Global Optimization
Year:
2010

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Abstract

In this paper, we devote to find the solution of the following quadratic minimization problem $$\min_{x\in \Omega}\|x\|^2,$$ where 驴 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.