A Smoothing SAA Method For A Stochastic Second-order Cone Linear Complementarity Problem

Complementarity problem is an important model in the field of operations research and have important applicationsin many areas such as mechanics,economics,engineering,etc.In recent years,in order to better reflect the actual uncertainty,stochastic complementarity problems have attracted researchers' interest and attention.This article focuses on a stochastic second-order cone linear complementarity problem.The problem is a special case of stochatic complementarity problem which provides a unified model for the study of stability of SOCP and can be applied in mechanic swith uncertainties.Currently,with the Euclidean Jordan algebr-atechnology,there is significan tprogress on the study of deterministic second-order cone complementarity problems.It is very difficult to find a solution for stochastic second order cone complementarity problems to satisfy all the realizations ofrandom variable.Therefore,how to establish reasonable stochastic order cone linear complementarity problem model and how to solve this model have important theoretical and practical significance.This paper prespresents a method for solving stochastic second-order cone linear complementarity problem problem in the form of expected values,and establishs the convergence theory and its application of the method,in particular.Firstly,based on the Chen-Harker-Kanzow-Smale?CHKS?smooth function,smoothing sample average approximation method has been propoesd for solving stochastic second-order cone linear complementarity problem.The existence of solution of the smoothing sample aveage approximation stochastic second order cone linear complementarity problems is established,which lay the foundation for the convergence theory of this method.Secondly,based on the R₀ and P property,the convergence properties of the smoothing sample average approximation method for two forms of stochastic second order linear complementarity problem?includes a number of second-order cones and a single second-order cone?have been established.Under the error bound theory on constraint system,the convergence rate of the smoothing sample average approximation metod has been established.Finally,based on the convergence theory,the three-dimensional frictional contact problems with themechanics of uncertainty is reformulated to a stochastic second order cone complementarityproblem which is solved by the smoothing sample average approximation method.Numerical results are demonst rated to show the efficiency this method.