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

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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 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.

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Authors and Affiliations

  1. Department of Mathematics, Tianjin Polytechnic University, 300160, Tianjin, China

    Yonghong Yao

  2. Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan

    Yeong-Cheng Liou

  3. Department of Mathematics and the RINS, Gyeongsang National University, Jinju, 660-701, Republic of Korea

    Shin Min Kang

Authors
  1. Yonghong Yao
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  2. Yeong-Cheng Liou
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  3. Shin Min Kang
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Corresponding author

Correspondence to Shin Min Kang.

Additional information

The first author was partially supported by National Natural Science Foundation of China Grant 10771050. The second author was partially supported by the grant NSC 98-2622-E-230-006-CC3 and NSC 98-2923- E-110-003-MY3.

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Yao, Y., Liou, YC. & Kang, S.M. Minimization of equilibrium problems, variational inequality problems and fixed point problems. J Glob Optim 48, 643–656 (2010). https://doi.org/10.1007/s10898-009-9512-1

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  • DOI: https://doi.org/10.1007/s10898-009-9512-1

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