Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A PROXIMAL POINT-TYPE ALGORITHM FOR PSEUDOMONOTONE EQUILIBRIUM PROBLEMS

  • Kim, Jong-Kyu (Department of Mathematics Education Kyungnam University) ;
  • Anh, Pham Ngoc (Department of Mathematics Education Kyungnam University) ;
  • Hyun, Ho-Geun (Department of Mathematics Education Kyungnam University)
  • Received : 2011.03.28
  • Published : 2012.07.31
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Abstract

A globally convergent algorithm for solving equilibrium problems is proposed. The algorithm is based on a proximal point algorithm (shortly (PPA)) with a positive definite matrix M which is not necessarily symmetric. The proximal function in existing (PPA) usually is the gradient of a quadratic function, namely, ${\nabla}({\parallel}x{\parallel}^2_M)$. This leads to a proximal point-type algorithm. We first solve pseudomonotone equilibrium problems without Lipschitzian assumption and prove the convergence of algorithms. Next, we couple this technique with the Banach contraction method for multivalued variational inequalities. Finally some computational results are given.

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References

  1. P. N. Anh, An interior-quadratic proximal method for solving monotone generalized variational inequalities, East-West Journal of Mathematics 10 (2008), 81-100.
  2. P. N. Anh, A logarithmic quadratic regularization method for solving pseudomonotone equilibrium problems, Acta Mathematica Vietnamica 34 (2009), 183-200.
  3. P. N. Anh, An LQP regularization method for equilibrium problems on polyhedral, Vietnam Journal of Mathematics 36 (2008), 209-228.
  4. P. N. Anh and J. K. Kim, A cutting hyperplane method for solving pseudomonotone nonlipschitzian equilibrium problems, Submitted, 2011.
  5. P. N. Anh, L. D. Muu, V. H. Nguyen, and J. J. Strodiot, Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities, J. Optim. Theory Appl. 124 (2005), no. 2, 285-306. https://doi.org/10.1007/s10957-004-0926-0
  6. P. N. Anh, L. D. Muu, and J. J. Strodiot, Generalized projection method for non- Lipschitz multivalued monotone variational inequalities, Acta Mathematica Vietnamica 34 (2009), 67-79.
  7. A. Auslender, M. Teboulle, and S. Bentiba, A logarithmic-quadratic proximal method for variational inequalities, Comput. Optim. Appl. 12 (1999), no. 1-3, 31-40. https://doi.org/10.1023/A:1008607511915
  8. G. Bigi, M. Castellani, and M. Pappalardo, A new solution method for equilibrium problems, Optim. Methods Softw. 24 (2009), no. 6, 895-911. https://doi.org/10.1080/10556780902855620
  9. E. Blum and W. Oettli, From optimization and variational inequality to equilibrium problems, Math. Student 63 (1994), no. 1-4, 123-145.
  10. P. Daniele, F. Giannessi, and A. Maugeri, Equilibrium Problems and Variational Models, Kluwer, 2003.
  11. F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementary Problems Vol. I, II, Springer-Verlag, New York, 2003.
  12. B. S. He, X. L. Fu, and Z. K. Jiang, Proximal-point algorithm using a linear proximal term, J. Optim. Theory Appl. 141 (2009), no. 2, 299-319. https://doi.org/10.1007/s10957-008-9493-0
  13. I. V. Konnov, Combined Relaxation Methods for Variational Inequalities, Springer-Verlag, Berlin, 2001.
  14. O. L. Mangasarian and M. V. Solodov, A linearly convergent derivative-free descent method for strongly monotone complementarity problem, Comput. Optim. Appl. 14 (1999), no. 1, 5-16. https://doi.org/10.1023/A:1008752626695
  15. G. Mastroeni, Gap function for equilibrium problems, J. Global Optim. 27 (2003), no. 4, 411-426. https://doi.org/10.1023/A:1026050425030
  16. A. Moudafi, Proximal point algorithm extended to equilibrium problem, J. Nat. Geom. 15 (1999), no. 1-2, 91-100.
  17. M. A. Noor, Auxiliary principle technique for equilibrium problems, J. Optim. Theory Appl. 122 (2004), no. 2, 371-386. https://doi.org/10.1023/B:JOTA.0000042526.24671.b2
  18. S. Schaible, S. Karamardian, and J. P. Crouzeix, Characterizations of generalized monotone maps, J. Optim. Theory Appl. 76 (1993), no. 3, 399-413. https://doi.org/10.1007/BF00939374

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