Modified Jacobian smoothing method for nonsmooth complementarity problems

This paper is devoted to solving a nonsmooth complementarity problem where the mapping is locally Lipschitz continuous but not continuously differentiable everywhere. We reformulate this nonsmooth complementarity problem as a system of nonsmooth equations with the max function and then propose an approximation to the reformulation by simultaneously smoothing the mapping and the max function. Based on the approximation, we present a modified Jacobian smoothing method for the nonsmooth complementarity problem. We show the Jacobian consistency of the function associated with the approximation, under which we establish the global and fast local convergence for the method under suitable assumptions. Finally, to show the effectiveness of the proposed method, we report our numerical experiments on some examples based on MCPLIB/GAMSLIB libraries or network Nash–Cournot game is proposed.