Several Smoothing Newton-type Algorithm For Solving Stochastic Nonlinear Complementarity Problems

This paper focuses on stochastic nonlinear complementarity problems’ algorithmsdesign and theoretical analysis. Based on the equivalent transfer between stochastic com-plementary problems and non-smooth constraint equations, the author proposed stochas-tic nonlinear complementarity problems’ smoothing Newton-type algorithms: smoothingprojected Guass-Newton algorithm and smoothing Levenberg-Marquardt algorithm, stud-ied the convergence of these two algorithms, and conducted numerical tests.Complementarity problems are throughout the computational mathematics and oper-ations research. However, complementarity problems often contain studied with stochas-tic factors,and the function of the equation is not defined on the whole space, the con-straint set is also often non-convex. In this way, there is a need for further study ofmethods for these problems. One method is to convert them into non-smooth constraintequations,then we can use Guass-Newton algorithm or Levenberg-Marquardt algorithmto solve these problems caused by transfer. We know that these algorithms require contin-uously diferentiable function, but transformed from these equations may not be able tomeet this condition, so this paper presents smoothing projected Guass-Newton algorithmand smoothing Levenberg-Marquardt algorithm for solving stochastic nonlinear comple-mentarity problems.The main contents are as follows: First, review the development situation comple-mentarity problems and stochastic complementarity problems and the traditional Newton-type algorithm for solving equations. Secondly, the analysis from the stochastic nonlinearcomplementarity problems to non-smooth constraint equations, and introduces the relatedsymbols and basic concepts. Again, this paper presents the important elements: smooth-ing projected Guass-Newton algorithm and smoothing Levenberg-Marquardt algorithm,and analyze the convergence of these two algorithms. Finally, the algorithm mentionedabove will be used to solve the stochastic nonlinear complementarity problems.