REGULARIZED EQUILIBRIUM PROBLEMS WITH EQUILIBRIUM 3 CONSTRAINTS WITH APPLICATION TO ENERGY MARKETS

Equilibrium problems with equilibrium constraints are challenging both theoretically and compu7 tationally. However, they are appropriate modeling formulations in a number of important areas, such as energy 8 markets, transportation planning, and logistics. Typically, these problems are characterized as bilevel Nash-Cournot 9 games. For instance, the equilibrium price in an energy market involves top-level decisions of the generators that 10 feed the bottom-level problem of an independent system operator. From a bilevel programming point of view, of 11 particular concern is the fact that the outcome of the equilibrium problem corresponds to computing the optimistic 12 solution of the associated bilevel problems. By contrast, the dual-primal regularization proposed in this work yields 13 equilibrium prices with minimal norm, a feature that corresponds to the pessimistic paradigm. Our approach pro14 vides a continuous selection of price signals that proves useful in guiding the solution process when solving such 15 problems computationally, via the mixed complementarity formulations. We prove existence theorems for a specific 16 equilibrium, and convergence of the proposed regularization scheme. Our numerical results using the PATH solver 17 illustrate the proposal. In particular, we exhibit that, in the given context, PATH with the regularization approach 18 computes a genuine equilibrium almost always, while without regularization the outcome is quite the opposite. 19