This paper is dedicated to solving a nonsmooth second-order cone complementarity problem, in which the mapping is assumed to be locally Lipschitz continuous, but not necessarily to be continuously differentiable everywhere. With the help of the vector-valued Fischer-Burmeister function associated with second-order cones, the nonsmooth second-order cone complementarity problem can be equivalently transformed into a system of nonsmooth equations. To deal with this reformulated nonsmooth system, we present an approximation function by smoothing the inner mapping and the outer Fischer-Burmeister function simultaneously. Different from traditional smoothing methods, the smoothing parameter introduced is treated as an independent variable. We give some conditions under which the Jacobian of the smoothing approximation function is guaranteed to be nonsingular. Based on these results, we propose a smoothing Newton method for solving the nonsmooth second-order cone complementarity problem and show that the proposed method achieves globally superlinear or quadratic convergence under suitable assumptions. Finally, we apply the smoothing Newton method to a network Nash-Cournot game in oligopolistic electric power markets and report some numerical results to demonstrate its effectiveness.

本文致力于解决非光滑的二阶锥互补问题，在该问题中，假定映射是局部Lipschitz连续的，但不一定到处都是连续可微的。借助于与二阶锥关联的向量值Fischer-Burmeister函数，可以将非光滑的二阶锥互补问题等效地转化为一个非光滑方程组。为了处理这种重新构造的非光滑系统，我们通过同时平滑内部映射和外部Fischer-Burmeister函数来提供近似函数。与传统的平滑方法不同，引入的平滑参数被视为独立变量。我们给出了保证平滑逼近函数的雅可比行列为非奇异的一些条件。基于这些结果，我们提出了一种光滑的牛顿法来解决非光滑的二阶锥互补问题，并表明该方法在适当的假设下实现了全局超线性或二次收敛。最后，我们将平滑牛顿法应用于寡头电力市场中的网络Nash-Cournot博弈，并报告一些数值结果以证明其有效性。