The Random Kaczmarz Algorithm for Solving Nonlinear Complementarity Problems

非线性互补问题(NCP)在数学规划和工程优化中占据重要地位，在本文中，首先分析了非线性互补问题的数学模型，并介绍了传统求解方法的局限性。然后，探讨了将随机Kaczmarz算法应用于非线性互补问题的框架与实现，提出了改进策略，以加速收敛并提高计算效率。通过数值实验验证，结果显示随机Kaczmarz算法在高维和大规模问题中具有显著的优势，尤其在处理复杂非线性约束问题时，收敛速度和计算效率远超传统方法。实验结果进一步证明了该算法在实际应用中的有效性和潜力。Nonlinear Complementarity Problems (NCPs) play a significant role in mathematical programming and engineering optimization. In this paper, the mathematical model of nonlinear complementarity problems is analyzed first, and the limitations of traditional solution methods are introduced. Then, the framework and implementation of applying the stochastic Kaczmarz algorithm to nonlinear complementarity problems are explored, and improved strategies are proposed to accelerate convergence and enhance computational efficiency. Through numerical experiments, the results show that the stochastic Kaczmarz algorithm has significant advantages in high-dimensional and large-scale problems, especially in dealing with complex nonlinear constraint problems, where its convergence speed and computational efficiency far exceed those of traditional methods. The experimental results further demonstrate the effectiveness and potential of this algorithm in practical applications.