A penalty method for generalized Nash equilibrium problems

Abstract

This paper considers a penalty algorithm for solving the generalized Nash equilibrium problem (GNEP). Under the GNEP-MFCQ at a limiting point of the sequence generated by the algorithm, we demonstrate that the limit point is a solution to the GNEP and the parameter becomes a constant after a finite iterations. We formulate the Karush-Kuhn-Tucker conditions for a penalized problem into a system of nonsmooth equations and demonstrate the nonsingularity of its Clarke’s generalized Jacobian at the KKT point under the so-called GNEP-Second Order Sufficient Condition. The nonsingularity facilitates the application of the smoothing Newton method for the global convergence and local quadratic rate. Finally, the numerical results are reported to show the effectiveness of the penalty method in solving generalized Nash equilibrium problem.