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A bundle method for solving equilibrium problems. (English) Zbl 1155.49006
Basing on the auxiliary problem principle, the authors study a boundle method for solving the nonsmooth convex equilibrium problem: finding \(x^* \in C\) such that \(f(x^*,y) \geq 0 \,\,{\text{for all}}\,\, y \in C\), and prove the convergence theorems for the general algorithm. Using a bundle strategy an implementable version of this algorithm is proposed together with the convergence results for the bundle algorithm. Some applications to variational inequality problems are also given.
Reviewer: Do Van Luu (Hanoi)

49J40 Variational inequalities
90C25 Convex programming
Full Text: DOI
[1] Anh P.N. and Muu L.D. (2004). Coupling the banach contraction mapping principle and the proximal point algorithm for solving monotone variational inequalities. Acta Math. Vietnam. 29: 119–133 · Zbl 1291.49011
[2] Aubin J.P. and Ekeland I. (1984). Applied Nonlinear Analysis. Wiley, New York · Zbl 0641.47066
[3] Blum E. and Oettli W. (1994). From optimization and variational inequalities to equilibrium problems. Math. Stud. 63: 123–145 · Zbl 0888.49007
[4] Cohen G. (1988). Auxiliary problem principle extended to variational inequalities. J. Optim. Theory Appl. 59: 325–333 · Zbl 0628.90069
[5] Correa R. and Lemaréchal C. (1993). Convergence of some algorithms for convex minimization. Math. Program. 62: 261–275 · Zbl 0805.90083
[6] El Farouq N. (2001). Pseudomonotone variational inequalities: convergence of the auxiliary problem method. J. Optim. Theory Appl. 111: 305–326 · Zbl 1027.49006
[7] Gol’shtein E.G. (2002). A method for solving variational inequalities defined by monotone mappings. Comput. Math. Math. Phys. 42(7): 921–930
[8] Hiriart-Urruty J.B. and Lemaréchal C. (1993). Convex Analysis and Minimization Algorithms. Springer, Berlin · Zbl 0795.49001
[9] Iusem A. and Sosa W. (2003). New existence results for equilibrium problem. Nonlinear Anal. Theory Methods Appl. 52: 621–635 · Zbl 1017.49008
[10] Iusem A. and Sosa W. (2003). Iterative algorithms for equilibrium problems. Optimization 52: 301–316 · Zbl 1176.90640
[11] Kiwiel K.C. (1995). Proximal level bundle methods for convex nondifferentiable optimization, saddle point problems and variational inequalities. Math. Program. 69(1): 89–109 · Zbl 0857.90101
[12] Konnov I.V. (2001). Combined Relaxation Methods for Variational Inequalities. Springer, Berlin · Zbl 0982.49009
[13] Konnov I.V. (1996). The application of a linearization method to solving nonsmooth equilibrium problems. Russ. Math. 40(12): 54–62 · Zbl 1022.49023
[14] Lemaréchal C., Nemirovskii A. and Nesterov Y. (1995). New variants of bundle methods. Math. Program. 69(1): 111–147 · Zbl 0857.90102
[15] Lemaréchal C., Strodiot J.J. and Bihain A. (1981). On a bundle method for nonsmooth optimization. In: Mangasarian, O.L., Meyer, R.R., and Robinson, S.M. (eds) Nonlinear Programming, vol. 4, pp 245–282. Academic, New York · Zbl 0533.49023
[16] Mastroeni G. (2003). On auxiliary principle for equilibrium problems. In: Daniele, P., Giannessi, F. and Maugeri, A. (eds) Equilibrium Problems and Variational Models, pp 289–298. Kluwer, Dordrecht · Zbl 1069.49009
[17] Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton · Zbl 0193.18401
[18] Salmon G., Strodiot J.J. and Nguyen V.H. (2004). A bundle method for solving variational inequalities. SIAM J. Optim. 14(3): 869–893 · Zbl 1064.65051
[19] Zhu D. and Marcotte P. (1996). Co-coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J. Optim. 6: 714–726 · Zbl 0855.47043
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