Expected Value And Sample Average Approximation Methods For Solving Stochastic Second-order-cone Complementarity Problems

The second-order-cone complementarity problems (SOCCP) are a kind of equilibri-um optimization problems. The two sets of decision variables satisfy a “complementary relationship" under the conditions of second-order-cone constraints. With the help of Eu-clidean Jordan algebra technique, in recent years, the second-order-cone complementarity problems have rapid development. The second-order-cone complementarity problems have widely applications in economic, engineering and other fields. However, there are some uncertain factors in real life. Ignoring these factors will lead to decision-making errors.Hence, in this paper, we consider stochastic second-order-cone complementarity problems(SSOCCP).Because of the existence of random variables, the stochastic second-order-cone com-plementarity problems may have no solution in general. However, in order to satisfy the urgent need of solving the practice problems with random factors, it is necessary to con-struct a reasonable deterministic model, then to solve the deterministic model and the solution of the deterministic model is regarded as the solution of the stochastic second-order-cone complementarity problems. Hence, in order to obtain the reasonable solution of the stochastic second-order-cone complementarity problems, we use the second-order-cone complementary functions to give a deterministic expected value (EV) model of stochastic second-order-cone complementarity problems.In this paper, we consider to use the second-order-cone complement functions ?T, and?NR to give the EV model, and first give the boundedness of the level set of the EV model. When the second-order-cone complementarity function is ?T, we first discuss the SC1 property of the objective function of the EV model. Since the EV model contains an expectation function, and the expectation is not easy to get. For solving this model, we employ sample average approximation (SAA) method to give approximation problems of the EV model. In theory, the convergence results of global solutions sequence and station?ary points sequence for the corresponding approximation problems are considered. When the second-order-cone complementarity function is ?NR, because the objective function of EV model is non-smooth, we first use the smoothing method to give the smoothing func-tion of the corresponding objective function. We then employ SAA method to give the approximation problems. Similar to EV model of the case that complementarity problem is ?T, in theory, the convergence results of global solutions sequence and stationary points sequence for the approximation problems while the complementarity problem is ?NR are also considered. Finally, in this paper the numerical experiments are given and solved by the proposed method.