First-Order Algorithms for Nonlinear Generalized Nash Equilibrium Problems
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We consider the problem of computing an equilibrium in a class of nonlinear generalized Nash equilibrium problems (NGNEPs) in which the strategy sets for each player are defined by equality and inequality constraints that may depend on the choices of rival players. While the asymptotic global convergence and local convergence rate of solution procedures have been studied in this setting, the analysis of iteration complexity is still in its infancy. Our contribution is to provide two simple first-order algorithmic frameworks based on the quadratic penalty method and the augmented Lagrangian method, respectively, with an accelerated mirror-prox algorithm as the inner loop. We provide nonasymptotic theoretical guarantees for these algorithms. More specifically, we establish the global convergence rate of our algorithms for solving (strongly) monotone NGNEPs and we provide iteration complexity bounds expressed in terms of the number of gradient evaluations. Experimental results demonstrate the efficiency of our algorithms.
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