Spatial equilibrium problems for auction type systems

At the present time rather a lot of spatial equilibrium models exist which are utilized for description of behavior and for management of complex systems containing many interacting elements, with taking into account their spatial distribution. Such models are greatly developed in economics where they usually describe systems of markets joined by transport communications within either perfect (in the sense of Walras) or imperfect (in the sense on Cournot–Bertrand) competition conditions (see, e.g., [1–3]). However, these kinds of competition are not sufficient for an adequate description of modern management mechanisms in economic systems. In particular, this is the case for systems related to natural monopolies in energy sectors where a state permits their privatization, but keeps certain tools for influence in this field, for instance, via the price policy (see, e.g., [4–6]). Since the optimal control of such a system requires consideration of both separate markets (participants) reactions and capacity and topology restrictions, the hierarchic (multilevel) optimization and game theory problems are usually utilized as suitable mathematical models. As a result, one must solve either very complicated global optimization problems having equilibrium constraints, or mixed integer programming problems. It is very difficult to find then efficient computational solution methods, especially for high-dimensional problems, which arise typically in applications.