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A proximal point method for a class of monotone equilibrium problems with linear constraints

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Abstract

It is well known that the equilibrium problems which often arise in engineering, economics and management applications, provide a unified framework for variational inequality, complementarity problem, optimization problem, saddle point problem and fixed point problem. In this paper, a proximal point method is proposed for solving a class of monotone equilibrium problems with linear constraints (MEP). The updates of all variables of the proximal point method are given in closed form. An auxiliary equilibrium problem is introduced for MEP via its saddle point problem. Further, we present some characterizations for solution of the auxiliary equilibrium problem and fixed point of corresponding resolvent operator. Thirdly, a proximal point method for MEP is suggested by fixed point technique. The asymptotic behavior of the proposed algorithm is established under some mild assumptions. Finally, some numerical examples are reported to show the feasibility of the proposed algorithm.

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References

  • Bauschke HH, Combettes PL (2010) Convex analysis and monotone operator theory in hilbert spaces. Springer, New York

    Google Scholar 

  • Bigi G, Castellani M, Pappalardo M, Passacantando M (2013) Existence and solution methods for equilibria. Eur J Oper Res 227:1–11

    Article  Google Scholar 

  • Blum E, Oettli W (1994) From optimization and variational inequalities to equilibrium problems. Math Stud 63:123–145

    Google Scholar 

  • Cai XJ, Gu GY, He BS, Yuan XM (2013) A proximal point algorithm revisit on the alternating direction method of multipliers. Sci China Math 56:2179–2186

    Article  Google Scholar 

  • Chen JW, Cho YJ, Wan Z (2013) The existence of solutions and well-posedness for bilevel mixed equilibrium problems in Banach spaces. Taiwan J Math 17:725–748

    Google Scholar 

  • Chen CH, He BS, Ye YY, Yuan XM (2015) The direct extension of ADMM for multi-block convex minimization problems is not necessarily convergent. Math Program A. doi:10.1007/s10107-014-0826-5

  • Combettes PL, Hirstoaga SA (2005) Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal 6:117–136

    Google Scholar 

  • Donoho DL, Logan BF (1992) Signal recovery and the large sieve. SIAM J Appl Math 52:577–591

    Article  Google Scholar 

  • Elad M, Bruckstein AM (2002) A generalized uncertainty principle and sparse representations in pairs of bases. IEEE Trans Inf Theory 48:2558–2567

    Article  Google Scholar 

  • Fan K (1972) A minimax inequality and applications. In: Shisha O (ed) Inequalities, vol III. Academic Press, New York, pp 103–113

    Google Scholar 

  • Giannessi F (2000) Vector variational inequalities and vector equilibria. Kluwer Academic Publishers, Dordrecht

    Book  Google Scholar 

  • Han D, Yuan X (2012) A note on the alternating direction method of multipliers. J Optim Theory Appl 155(1):227–238

    Article  Google Scholar 

  • He BS, Yuan XM, Zhang W (2013) A customized proximal point algorithm for convex minimization with linear constraints. Comput Optim Appl 56:559–572

    Article  Google Scholar 

  • Hu R, Fang YP (2013) A characterization of nonemptiness and boundedness of the solution sets for equilibrium problems. Positivity 17:431–441

    Article  Google Scholar 

  • Iusem AN, Nasri M (2007) Inexact proximal point methods for equilibrium problems in Banach spaces. Numer Funct Anal Optim 28:1279–1308

    Article  Google Scholar 

  • Iusem AN, Kassay G, Sosa W (2009) On certain conditions for the existence of solutions of equilibrium problems. Math Program B 116:259–273

    Article  Google Scholar 

  • Konnov IV (2003) Application of the proximal point method to nonmonotone equilibrium problems. J Optim Theory Appl 119:317–333

    Article  Google Scholar 

  • Li XB, Li SJ (2010) Existence of solutions for generalized vector quasi-equilibrium problems. Optim Lett 4:17–28

    Article  Google Scholar 

  • Mordukhovich BS, Panicucci B, Pappalardo M, Passacantando M (2012) Hybrid proximal methods for equilibrium problems. Optim Lett 6:1535–1550

    Article  Google Scholar 

  • Moudafi A (1999) Proximal point methods extended to equilibrium problems. J Nat Geom 15:91–100

    Google Scholar 

  • Moudafi A (2007) On finite and strong convergence of a proximal method for equilibrium problems. Numer Funct Anal Optim 28:1347–1354

    Article  Google Scholar 

  • Rockafellar RT (1976) Monotone operators and the proximal point algorithm. SIAM J Control Optim 14:877–898

    Article  Google Scholar 

  • Takahashi W, Zembayashi K (2008) Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal 70:45–57

    Article  Google Scholar 

Download references

Acknowledgments

This work was partially supported by the Natural Science Foundation of China (No. 11401487, 71471140), the Fundamental Research Funds for the Central Universities (No. SWU113037, XDJK2014C073), and the Grant MOST 103-2923-E-037-001-MY3.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Jiawei Chen

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

    Yeong-Cheng Liou

  3. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Zhongping Wan

  4. Center for General Education, China Medical University, Taichung, 40402, Taiwan

    Jen-Chih Yao

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  1. Jiawei Chen
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  2. Yeong-Cheng Liou
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  3. Zhongping Wan
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  4. Jen-Chih Yao
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Correspondence to Jiawei Chen.

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Chen, J., Liou, YC., Wan, Z. et al. A proximal point method for a class of monotone equilibrium problems with linear constraints. Oper Res Int J 15, 275–288 (2015). https://doi.org/10.1007/s12351-015-0177-x

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  • DOI: https://doi.org/10.1007/s12351-015-0177-x

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