In this paper we consider the weighted nonlinear complementarity problem (denoted by wNCP) which contains a wide class of optimization problems. We introduce a family of new weighted complementarity functions and show that it is continuously differentiable everywhere and has several favorable properties. Based on this function, we reformulate the wNCP as a smooth nonlinear equation and propose a nonmonotone Levenberg–Marquardt type method to solve it. We show that the proposed method is well-defined and it is globally convergent without any additional condition. Moreover, we prove that the whole iteration sequence converges to a solution of the wNCP locally superlinearly or quadratically under the nonsingularity condition. In addition, we establish the local quadratic convergence of the proposed method under the local error bound condition. Some numerical results are also reported.

在本文中，我们考虑加权非线性互补问题（由 wNCP 表示），它包含多种优化问题。我们引入了一系列新的加权互补函数，并表明它在任何地方都是连续可微的，并且具有几个有利的特性。基于这个函数，我们将 wNCP 重新表述为一个光滑的非线性方程，并提出了一种非单调的 Levenberg-Marquardt 型方法来解决它。我们表明，所提出的方法是明确定义的，并且在没有任何附加条件的情况下全局收敛。此外，我们证明了整个迭代序列在非奇异性条件下局部超线性或二次收敛到wNCP的解。此外，我们在局部误差边界条件下建立了所提出方法的局部二次收敛。