O problema de quase-equilíbrio : uma abordagem do tipo-Newton regularizado

Abstract

The quasi-equilibrium problem (QEP) allows, in contrast to standard equilibrium problem (EP), a change in the feasible region together with the considered point. Therefore, QEP provides an unified framework for a wider family of mathematical problems which do not fit within the scope of EP. It includes, for example, generalized Nash equilibrium problems (GNEPs) and quasi-variational inequalities (QVIs). In this work, we propose an implementable Newton-type method to solve EPs as well as its extension to solve QEPs. We analyze some properties of the algorithms proving its local convergence to a solution of the problem under usual assumptions for Newton-type methods as well as its superlinear/quadratic convergence rate. We propose a global version of the algorithm for EP. Applications of the proposed method to GNEP and multiobjective optimization problem (MOP) are considered. For applying our results to the multiobjective optimization problem, we present a smooth approximation of the bifunction which allows a formulation of the problem as an EP. Next, we introduce a formulation that allows us to consider a MOP as a QEP. This formulation has the advantage of providing a QEP with smooth data involving only the original data of the problem. Finally, we illustrate the numerical behavior of the local algorithms through some test-problems involving EP, GNEP and MOP. The proposed algorithm to solve QEPs has been proved to be ecient for solving a QEP which is neither a QVI nor a GNEP, besides the fact that it has been able to solve a nonjointly convex GNEP