×

zbMATH — the first resource for mathematics

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)

MSC:
49J40 Variational inequalities
90C25 Convex programming
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.