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An algorithm for a class of split feasibility problems: application to a model in electricity production

Abstract

We propose a projection algorithm for solving split feasibility problems involving paramonotone equilibria and convex optimization. The proposed algorithm can be considered as a combination of the projection ones for equilibrium and convex optimization problems. We apply the algorithm for finding an equilibrium point with minimal environmental cost for a model in electricity production. Numerical results for the model are reported.

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References

  1. Anh PN, Muu LD (2014) A hybrid subgradient algorithm for nonexpansive mapping and equilibrium problems. Optim Lett 8:727–738

    MathSciNet  Article  MATH  Google Scholar 

  2. Bauschke HH, Combettes PH (2011) Convex analysis and monotone operator in hilbert spaces. Springer, Berlin

    Book  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  4. Blum E, Oettli W (1994) From optimization and variational inequalities to equilibrium problems. Math Stud 63:127–149

    MathSciNet  MATH  Google Scholar 

  5. Byrne C (2002) Iterative oblique projection onto convex sets and the split feasibility problems. Inverse Prob 18:441–453

    MathSciNet  Article  MATH  Google Scholar 

  6. Byrne C (2004) A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Prob 20:103–120

    MathSciNet  Article  MATH  Google Scholar 

  7. Censor Y, Elving T (1994) A multiprojections algorithm using Bregman projections in a product spaces. Numer Algorithms 8:221–239

    MathSciNet  Article  Google Scholar 

  8. Censor Y, Segal A (2008) Iterative projection methods in biomedical inverse problems. In: Censor Y, Jiang M, Louis AK (eds) Mathematical methods in biomedical imaging and intensity-modulated therapy. IMRT, Edizioni della Norale, Pisa, pp 65–96

    Google Scholar 

  9. Censor Y, Bortfeld T, Martin B, Trofimov A (2006) A unified approach for inversion problems in intensity-modulated radiation therapy. Phys Med Biol 51:2353–2365

    Article  Google Scholar 

  10. Contreras J, Klusch M, Krawczyk JB (2004) Numerical solution to Nash-Cournot equilibria in coupled constraint electricity markets. EEE Trans Power Syst 19:195–206

    Article  Google Scholar 

  11. Iusem AN (1998) On some properties of paramonotone operator. Convex Anal 5:269–278

    MathSciNet  MATH  Google Scholar 

  12. Korpelevich GM (1976) The extragradient method for finding saddle points and other problems. Ekon Mat Metody 12:747–756

    MathSciNet  MATH  Google Scholar 

  13. Kraikaew R, Saejung S (2014) On split common fixed point problems. J Math Anal Appl 415:513–524

    MathSciNet  Article  MATH  Google Scholar 

  14. Lopez G, Martin-Maquez V, Wang F, Xu HK (2012) Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Prob 28:085–094

    MathSciNet  Article  Google Scholar 

  15. Moudafi A (2011) Split monotone variational inclusions. J Optim Theory Appl 150:275–283

    MathSciNet  Article  MATH  Google Scholar 

  16. Moudafi A, Thakur BS (2014) Solving proximal split feasibility problems without prior knowledge of operator norms. Optim Lett 8:2099–2110

    MathSciNet  Article  MATH  Google Scholar 

  17. Nikaido H, Isoda K (1955) Note on noncooperative convex games. Pac J Math 5:807–815

    MathSciNet  Article  MATH  Google Scholar 

  18. Quoc TD, Muu LD (2012) Iterative methods for solving monotone equilibrium problems via dual gap functions. Comput Optim Appl 51:709–728

    MathSciNet  Article  MATH  Google Scholar 

  19. Rockafellar TR, Wets R (1998) Variational analysis. Springer, Berlin

    Book  MATH  Google Scholar 

  20. Santos P, Scheimberg S (2011) An inexact subgradient algorithm for equilibrium problems. Comput Appl Math 30:91–107

    MathSciNet  MATH  Google Scholar 

  21. Tang J, Chang S, Yuan F (2014) A strong convergence theorem for equilibrium problems and split feasibility problems in Hilbert spaces. Fixed Point Theory Appl 36:1687–1812

    MathSciNet  MATH  Google Scholar 

  22. Vuong PT, Strodiot JJ, Nguyen VH (2015) On extragradient-viscosity methods for solving equilibrium and xed point problems in a Hilbert space. Optimization 64:429–451

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

We would like to thank the editor and the referee very much for their useful comments, remarks and suggestions that helped us very much to improve quality of the paper.

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Correspondence to Le Dung Muu.

Additional information

This paper is supported by the NAFOSTED, Grant 101.01-2014.24.

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Yen, L.H., Muu, L.D. & Huyen, N.T.T. An algorithm for a class of split feasibility problems: application to a model in electricity production. Math Meth Oper Res 84, 549–565 (2016). https://doi.org/10.1007/s00186-016-0553-1

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Keywords

  • Split feasibility
  • Equilibria
  • Convex optimization
  • Practical model