Equilibrium Models and Applications

As already mentioned in the preface, the term “equilibrium” is widespread in science in the study of different phenomena. In this chapter a small selection of equilibrium problems from different areas is given, each leading to a different kind of mathematical model. The equilibrium position of an elastic string in presence of an obstacle, which is depicted in Sect. 1.1, coincides with the solution of a complementarity problem, the Nash equilibrium fits in well to model a power control multi-agent system described in Sect. 1.2, the steady distribution of traffic over a network is represented by a variational inequality in Sect. 1.3, the Markowitz portfolio theory is viewed as a multiobjective problem in Sect. 1.4, the shadow price theory is viewed as a saddle point problem for the nonlinear case in Sect. 1.5, the solution of the input-output model given in Sect. 1.6 is a fixed point and the quality control problem in a production system illustrated in Sect. 1.7 is an inverse optimization problem. Finally, the last section is devoted to show that all these mathematical models, which are apparently different, have a common structure that leads to a unified format: the Ky Fan inequality or the “equilibrium problem” using the “abstract” name introduced by Blum, Muu and Oettli to stress this unifying feature.