Equilibria, Supernetworks, and Evolutionary Variational Inequalities

In this paper, we consider multitiered network equilibrium problems with optimizing decision-makers associated at the nodes with applications ranging from supply chain networks and electric power networks to financial networks with intermediation. We assume fixed demands associated with the demand markets and prove that the governing equilibrium conditions satisfy a finite-dimensional variational inequality. We establish that the static network model with its governing equilibrium conditions can be reformulated as a transportation network equilibrium model over an appropriately constructed abstract network or supernetwork. This identification provides a new interpretation of equilibrium in multitiered networks in terms of path flows as well as new computational procedures. The equivalence is then further exploited to construct a dynamic network model with time-varying demands (and flows) using an evolutionary (time-dependent) variational inequality formulation. Recent theoretical results in the unification of projected dynamical systems and evolutionary variational inequalities, along with double-layered dynamics theory, are presented. The theory is then applied to formulate and solve dynamic numerical network examples drawn from several distinct applications in which the curves of equilibria are computed. An example with step-wise time-dependent demand is also given for illustration purposes as well as examples in which the time-dependent trajectories are available in closed form, due to the special network structure of the particular problem.