TY - JOUR

T1 - A Bundle Method for Solving Equilibrium Problems

AU - Nguyen, Thi Thu Van

AU - Nguyen, Van Hien

AU - Strodiot, Jean-Jacques

PY - 2007

Y1 - 2007

N2 - 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 delta-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

AB - 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 delta-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

KW - Equilibrium problems · Variational inequalities · Bundle methods

M3 - Article

JO - Mathematical Programming, DOI 10.1007/s10107-007-0112-x

JF - Mathematical Programming, DOI 10.1007/s10107-007-0112-x

ER -