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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. [ABSTRACT FROM AUTHOR]

We present a framework for analyzing convergence and local rates of convergence of a class of descent algorithms, assuming the objective function is weakly convex. The framework is general, in the sense that it combines the possibility of explicit iterations (based on the gradient or a subgradient at the current iterate), implicit iterations (using a subgradient at the next iteration, like in the proximal schemes), as well as iterations when the associated subgradient is specially constructed and does not correspond either to the current or the next point (this is the case of descent steps in bundle methods). Under the subdifferential-based error bound on the distance to critical points, linear rates of convergence are established. Our analysis applies, among other techniques, to prox-descent for decomposable functions, the proximal-gradient method for a sum of functions, redistributed bundle methods, and a class of algorithms that can be cast in the feasible descent framework for constrained optimization. [ABSTRACT FROM AUTHOR]