The Chen-Harker-Kanzow-Smale Locally Smoothing Functions And Its Application In Large-scale Mixed Complementarity Problems

In this thesis, we proposed a new class of smoothing projection functions onto the box set:Chen-Harker-Kanzow-Smale locally smoothing functions, and its application in large-scale mixed complementarity problems.Firstly, based on the structure of the projection function onto the box set and the Chen-Harker-Kanzow-Smale (CHKS) smoothing function, a new class of smoothing projection func-tions onto the box set are proposed. The new smoothing projection functions only smooth the projection function in a neighbourhood of nonsmooth points of the projection function and keep consistent with it at other points, and hence they are referred as the Chen-Harker-Kanzow-Smale locally smoothing functions. Compared with other general smoothing projection functions, the calculations of the CHKS locally smoothing function value and its derived function value become cheaper, especially for large-scale mixed complementarity problems. In this thesis, the CHK-S locally smoothing functions are proved to be feasible, continuously different!able, a uniform approximation of the projection function and globally Lipschitz continuous, which are similar to the CHKS smoothing function and important to prove the convergence of the smoothing Newton method. By the definitions and properties of the semismooth functions and strongly semismooth functions, the CHKS locally smoothing functions are proved to be strongly semis-mooth, which is significant to establish the locally superlinear and quadratic convergence of the smoothing Newton method. By some different simple functions, which satisfy some conditions, some special CHKS locally smoothing functions are constructed.Next, based on the Robinson's normal equation and the CHKS locally smoothing functions, a smoothing Newton method is proposed for solving mixed complementarity problems. Accord-ing to the structure of the CHKS locally smoothing functions, at every iteration of the smoothingNewton method, we just solve the equations corresponding to the primal mixed complementarity problem expect in a neighbourhood of the nonsmooth points of the projection function. What's more, compared with general smoothing Newton method, solving a n-dimensional system of linear equations used to find a Newton direction is equivalent to solving a low-dimensional e-quations. It can improve the efficiency of the algorithm. In addition, at every iteration, the calculations of the the Jacobian matrix in the mixed complementarity problems can be reduced. It also can improve the efficiency of the algorithm. Finally, by the MCPLIB test collection and some examples about linear complementarity problems, the smoothing Newton method, which is based on the CHKS locally smoothing functions, is implemented in MATLAB and compared with the PATH solver and the smoothing Newton method based on the CHKS smoothing func-tion, uniform smoothing function and neural networks smoothing function. Primary numerical results show that the proposed smoothing Newton method has promising numerical stability and computational efficiency.