## Outer approximation algorithms for pseudomonotone equilibrium problems.(English)Zbl 1221.90083

Summary: We propose a new method for solving equilibrium problems on a convex subset $$C$$, where the underlying function is continuous and pseudomonotone which is called an outer approximation algorithm. The algorithm is to define new approximating subproblems on the convex domains $$C_{k}\supseteq C$$, $$k=0,1,\dots$$, which forms a generalized iteration scheme for finding a global equilibrium point. Finally we present some numerical experiments to illustrate the behavior of the proposed algorithms.

### MSC:

 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 49J40 Variational inequalities
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### References:

 [1] Daniele, P.; Giannessi, F.; Maugeri, A., Equilibrium problems and variational models, (2003), Kluwer [2] Konnov, I.V., Combined relaxation methods for variational inequalities, (2000), Springer-Verlag Berlin · Zbl 0986.49004 [3] Anh, P.N., A logarithmic quadratic regularization method for solving pseudomonotone equilibrium problems, Acta Mathematica vietnamica, 34, 183-200, (2009) · Zbl 1200.65044 [4] Anh, P.N., An LQP regularization method for equilibrium problems on polyhedral, Vietnam journal of mathematics, 36, 209-228, (2008) [5] Blum, E.; Oettli, W., From optimization and variational inequality to equilibrium problems, The mathematics student, 63, 127-149, (1994) [6] Mastroeni, G., Gap function for equilibrium problems, Journal of global optimization, 27, 411-426, (2004) · Zbl 1061.90112 [7] Moudafi, A., Proximal point algorithm extended to equilibrium problem, Journal of natural geometry, 15, 91-100, (1999) · Zbl 0974.65066 [8] Noor, M.A., Auxiliary principle technique for equilibrium problems, Journal of optimization theory and applications, 122, 371-386, (2004) · Zbl 1092.49010 [9] Bigi, G.; Castellani, M.; Pappalardo, M., A new solution method for equilibrium problems, Optimization methods & software, 24, 895-911, (2009) · Zbl 1237.90253 [10] Anh, P.N., An interior-quadratic proximal method for solving monotone generalized variational inequalities, East – west journal of mathematics, 10, 81-100, (2008) · Zbl 1197.65076 [11] Auslender, A.; Teboulle, M.; Bentiba, S., A logarithmic-quadratic proximal method for variational inequalities, Journal of computational optimization and applications, 12, 31-40, (1999) · Zbl 1039.90529 [12] P.N. Anh, T. Kuno, A cutting hyperplane method for generalized monotone nonlipschitzian multivalued variational inequalities, in: Proceedings of 4th International Conference on High Performance Scientific Computing, Hanoi, 2010 (in press). [13] Burachik, R.S.; Lopes, J.O.; Svaiter, B.F., An outer approximation method for the variational inequality problem, SIAM journal on control and optimization, 43, 2071-2088, (2005) · Zbl 1076.49003 [14] Browder, F.E., Nonlinear operators an nonlinear equations of evolution in Banach spaces, () · Zbl 0176.45301 [15] Ceng, L.C.; Yao, J.C., Approximate proximal algorithms for generalized variational inequalities with pseudomonone multifunction, Journal of computational and applied mathematics, 213, 423-438, (2008) · Zbl 1144.65045 [16] Schaible, S.; Karamardian, S.; Crouzeix, J.P., Characterizations of generalized monotone maps, Journal of optimization theory and applications, 76, 399-413, (1993) · Zbl 0792.90070 [17] Anh, P.N.; Muu, L.D.; Strodiot, J.J., Generalized projection method for non-Lipschitz multivalued monotone variational inequalities, Acta Mathematica vietnamica, 34, 67-79, (2009) · Zbl 1181.65099
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