The locally Chen-Harker-Kanzow-Smale smoothing functions for mixed complementarity problems
The locally Chen-Harker-Kanzow-Smale smoothing functions for mixed complementarity problems
复制标题
混合互补问题的局部 Chen-Harker-Kanzow-Smale 平滑函数
DOI：
10.1007/s10898-019-00739-4
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发表时间：
2019
影响因子：
1.8
通讯作者：
Peng Yunchan
中科院分区：
文献类型：
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作者：
Zhou Zhengyong;Peng Yunchan
根据盒上投影函数的结构和Chen-Harker-Kanzow-Smer(CHKS)光滑函数，提出了一类新的盒上平滑投影函数。新的光滑投影函数只在非光滑点的邻域内光滑，与其他点保持不变，因此称为局部Chen-Harker-Kanzow-Smer(LCHKS)光滑函数。基于Robinson法方程和LCHKS光滑函数，给出了求解混合互补问题的一种光滑牛顿法及其收敛结果。与基于各种平滑投影函数的平滑牛顿法相比，LCHKS平滑函数的计算量和牛顿方程的函数值及其雅可比矩阵的计算量更小，而且牛顿方向可以通过求解低维线性方程来求解，因此基于LCHKS平滑函数的平滑牛顿法在求解大规模混合互补问题时表现出更高的效率。证明了LCHKS光滑函数对全局Lipschitz连续和强半光滑是可行的、连续可微的、一致逼近的，这对于建立光滑牛顿法的超线性和二次收敛是重要的。将提出的平滑牛顿法在MatLab中实现，并在MCPLIB测试集上进行了数值测试。数值结果表明，基于LCHKS光滑化函数的光滑化牛顿法在求解混合互补问题时具有很好的应用前景。
According to the structure of the projection function onto the box setand the Chen–Harker–Kanzow–Smale (CHKS) smoothing function, a new class of smoothing projection functions onto the box set are proposed in this paper. The new smoothing projection functions only smoothin neighborhoods of nonsmooth points of, and keep unchanged withat other points, hence they are referred as the locally Chen–Harker–Kanzow–Smale (LCHKS) smoothing functions. Based on the Robinson’s normal equation and the LCHKS smoothing functions, a smoothing Newton method with its convergence results is proposed for solving mixed complementarity problems. Compared with smoothing Newton methods based on various smoothing projection functions, the computations of the LCHKS smoothing function, the function value and its Jacobian matrix of the Newton equation become cheaper, and the Newton direction can be found by solving a low dimensional linear equation, hence the smoothing Newton method based on the LCHKS smoothing functions shows more efficient for large-scale mixed complementarity problems. The LCHKS smoothing functions are proved to be feasible, continuously differentiable, uniform approximations of, globally Lipschitz continuous and strongly semismooth, which are important to establish the superlinear and quadratic convergence of the smoothing Newton method. The proposed smoothing Newton method is implemented in MATLAB and numerical tests are done on the MCPLIB test collection. Numerical results show that the smoothing Newton method based on the LCHKS smoothing functions is promising for mixed complementarity problems.