Regularized equilibrium problems with equilibrium constraints with application to energy markets

Equilibrium problems with equilibrium constraints are appropriate modeling formulations in a number of important areas, such as energy markets, transportation planning, and logistics. These models often correspond to bilevel games, in which certain dual variables, representing the equilibrium price, play a fundamental role. We consider multileader single-follower equilibrium problems having a linear program in the lower-level. Because in this setting the lowerlevel response to the leaders' decisions may not be unique, the game formulation becomes ill-posed. We resolve possible ambiguities by considering a sequence of bilevel equilibrium problems, endowed with a special regularization term. We prove convergence of the approximating scheme. Our technique proves useful numerically over several instances related to energy markets. When using PATH to solve the corresponding mixed-complementarity formulations, we exhibit that, in the given context, the regularization approach computes a genuine equilibrium price almost always, while without regularization the outcome is quite the opposite.