Estudo de um caso de equilíbrio de Nash usando técnicas de inequação variacional, problemas de complementaridade e teoria de otimização

Abstract

In this work, we present a theoretical study of the variational inequalities (VI), showing necessary and sufficient conditions for the existence of solutions. We also present some classi fi cations for several special cases of LV, explaining the interconnection between LV and the complementarity problem. As well as its relationship with a nonlinear optimization program and the strategy game theory that is related to Nash equilibrium problems. We have established some equivalence results between a VI and a CP, for example we have shown that solving a VI associated with a function defined in a set having a given structure is equivalent to solving a mixed complementarity problem (MiCP); As well as, we have seen that under certain hypotheses, a Nash equilibrium is the solution of a given VI. Finally, we apply the theory to the study of a Nash equilibrium problem case, modeled via the KKT conditions as an MiCP, or creating a family of convex programs, where used some algorithms to numerically determine the equilibrium point solution of Nash.