On equilibrium problems.
Muhammad Aslam Noor, Khalida Inayat Noor · Applied Mathematics E-Notes [electronic only] · 2004
Some iterative methods for solving equilibrium problems are suggested and analyzed by using the technique of the auxiliary principle. We have shown that the convergence of the proposed methods either requires only pseudomonotonic- ity, which is a weaker condition than monotonicity or partially relaxed strongly monotonicity. Our results represent an improvement and refinement of previously known results. Since the equilibrium problems include variational inequalities and complementarity problems as special cases, results proved in this paper continue to hold for these problems Equilibrium problems theory provides us with a unified, natural, innovative and gen- eral framework to study a wide class of problems arising in finance, economics, network analysis, transportation, elasticity and optimization. This theory has witnessed an ex- plosive growth in theoretical advances and applications across all disciplines of pure and applied sciences. Equilibrium problems include variational inequalities as special cases. In recent years, several numerical techniques including projection, resolvent and auxil- iary principle have been developed and analyzed for solving variational inequalities, see (1-13). It is well-known and projection and resolvent type methods cannot be extended for mixed quasi variational inequalities. To overcome this drawback, one usually uses the auxiliary principle technique. Glowinski et al. (5) used this technique to study the existence of a solution of mixed variational inequalities, whereas Noor (7,8,10) used this technique to suggest and analyze a number of predictor-corrector and proximal methods for solving various classes of variational inequalities. In this paper, we again use the auxiliary principle technique to suggest and analyze some iterative methods for equilibrium problems. We have studied the convergence criteria of these methods under some mild conditions. As a consequence of this approach, we construct the gap (merit) function for equilibrium problems, which can be used to develop descent-type methods for solving equilibrium problems. Our results can be viewed as significant extension and generalization of the previously known results for solving equilibrium problems.