A bundle method for solving equilibrium problems

Authors:
T. T. V. Nguyen_aff1n2;J. J. Strodiot;V. H. Nguyen
Affiliations:
af2 University of Natural Sciences, Vietnam National University, Faculty of Mathematics and Informatics, Ho Chi Minh City, Vietnam;af1 University of Namur (FUNDP), Department of Mathematics, Namur, Belgium;af1 University of Namur (FUNDP), Department of Mathematics, Namur, Belgium
Venue:
Mathematical Programming: Series A and B - Nonlinear convex optimization and variational inequalities
Year:
2008

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Abstract

We present a bundle method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle. First, we consider a general algorithm that we prove to be convergent. Then we explain how to make this algorithm implementable. The strategy is to approximate the nonsmooth convex functions by piecewise linear convex functions in such a way that the subproblems are easy to solve and the convergence is preserved. In particular, we introduce a stopping criterion which is satisfied after finitely many iterations and which gives rise to Δ-stationary points. Finally, we apply our implementable algorithm for solving the particular case of singlevalued and multivalued variational inequalities and we find again the results obtained recently by Salmon et al. [18].