Smoothing And Sample Average Approximation Methods For Solving Stochastic Symmetric Cone Complementarity Problems

The Symmetric Cone Complementarity Problem(SCCP)refers to a kind of:"com-plementary" relationship between two groups of decision variables under symmetric cone constraints,which is a novel and rich equilibrium optimization problem.Its research is based on the Euclidean Jordan algebra theoretical,in recent years,The Symmetric Cone Complementarity Problem has become a.research hotspot in the field of optimization.Its theoretical research results are commonly used in finance,management,communication,control and other related practical problems.However,it is often influenced by stochastic factors in dealing with daily practical problems,such as:weather,demand,price,etc.If decision makers neglect the existence of these factors in the process of solving practical problems,it will lead to decision making errors,and unable to get effective and reasonable results.Therefore,people gradually considering Symmetric Cone Complementarity Prob-lem containing random variables,that is Stochastic Symmetric Cone Complementarity Problem(SSCCP).In general,due to the existence of random variables,we can not directly solve the Stochastic Symmetric Cone Complementarity Problem(SSCCP).Therefore,on Euclidian Jordan algebra theoretical basis,we use the symmetric cone complementary function ?NR to give a deterministic ERM model of the Stochastic Symmetric Cone Complementary Problem,and the solution of the deterministic ERM model is regarded as the solution of the Stochastic Symmetric Cone Complementarity Problem.In the process of solving the ERM model,firstly,we need to give the boundedness of the level set of the ERM model,because it can ensure the existence of the solution of the optimization model.Secondly,the symmetric cone complementary function ?NR that given in this paper is a non-smooth function,so the objective function of its corre-sponding ERM model is also non-smooth.Therefore,we use the smoothing method to give the corresponding smoothing function of the objective function.Furthermore,due to the ERM model contains an expectation function,the expectation is not easy to solve Therefore,we use the sample average approximation(SAA)method to give the smoothing approximation of the ERM model.Finally,we prove the convergence of the global optimal solutions corresponding the ERM model and the corresponding smoothing and smoothing approximation models.