Conditional Value-at-Risk Model For Solving Stochastic Second-order-cone Complementarity Problems And Their Convergence Results

The second-order-cone complementarity problems(SOCCP),which is considered as a kind of universal equilibrium optimization problem,becoming one of the most popular problems.In recently years,great progress has been made on SOCCP researches with the utilization of jordan algebra and spectral factorization.The study of deterministic SOCCP are on the upward trend at present,and the main research on theory directions include:various of smoothing methods to solve SOCCP problems,the existence and convergence of solution set and development of some effective algorithms.On practical applications,many problems can be transformed into SOCCP by certain ways.For example,the threedimensional frictional contact problems and robust Nash equilibrium problems.However,in practice,several kinds of uncertainties such as trade,supply,demand etc.may be involved in SOCCP,and if we ignore these factors will lead to serious consequences.Due to the introduction of a random variable,that stochastic second-order-cone complementarity problems(SSOCCP)are more complicated than SOCCP and they are found application in more fields.Therefore,it is meaningful and very necessary to study the general SSOCCP.Based on these reasons,we will introduces Conditional Value-at-Risk(CVaR)model to solve SSOCCP.We will regard the second-order-cone complementarity function as loss function,and present a smoothing approximate CVaR model and Monte Carlo sample average approximate CVaR model by combining smoothing and Monte Carlo sample average approximate method.Furthermore,we will consider the convergence results between approximate problems and original problems,and related numerical example will be given and solving the solution by method we presented.Finally,we will summarize the main contents of the research in this paper and put forward further assumption and consideration of SSOCCP based on CVaR model.