Convergence Analysis Of A Smoothing SAA Method For A Stochastic Linear Symmetric Complementarity Problems

Complementarity problem is the crossing field of operations research and computational mathematics,it has wide applications in the field of mathematical programming,game theory,mechanics and supply chain management.In 1964,R.W.Cottle first proposed the "complementary" model,which aroused the majority of scholars' interest on this issue.In practical problems,there are all kinds of uncertain factors.Therefore,according to the needs of theory and practice,more and more people begin to pay attentions to the stochastic complementarity problems.Symmetric cone complementarity problem is a balanced optimization problem,including the nonlinear complementarity problem,the semidefinite complementarity problem,and the two order cone complementarity problem.In recent years,with the application of Euclidean Jordan algebras,the study of standard complementary problem has got very good results,and has wide applications in the economic,transportation,communication,etc.However,in order to be closer to the reality,we need to study symmetric cone complementarity problem with random factors.Firstly,we introduce the determinate complementary problems and the stochastic complementarity problems.Then we propose a new stochastic equilibrium model-random linear symmetric cone complementarity problem,which aims to analyze the existence and convergence of this problem.Secondly,we propose the smoothing sample average approximation?SAA?method to solve the SSCLCP problem,then by the Euclidean Jordan algebra,R₀ property and P property,we get the solvability conditions of the problem and the convergence of the proposed method.Then we use this method to solve the stochastic linear semidefinite cone complementarity problems.Finally,we give a concrete example and verify the validity of the proposed method by a numerical example,which provide a good approach to the practical problems.