Study On Unconstrained Optimization Reformulation And Expected Residual Minimization Method For Stochastic Complementary Problems

The theories of variational inequality and complementary problem have been playing the important roles in many fields such as engineering design,economy equilibrium,transportation and etc.It is well known that some elements may involve uncertain data in many piratical applications.Ignoring these uncertainties may cause faulty in making decision,so it is necessary to study stochastic variational inequality and stochastic complementary problem.The foremost thought for studying stochastic variational inequality and stochastic complementary problem(SCP)is to establish a deterministic model.In this thesis,we proposed an unconstrained optimization reformulation of SCP.The main results of this thesis can be summarized as follows:(1)For the stochastic linear complementary problem(SLCP),we present an unconstrained expected residual minimization problem(UERM problem)which is to minimize an expected residual defined by D-gap function.On the one hand,we analyze the differentiability of the objective function and the boundedness of the level set.On the other hand,a discrete approximation problem of the UERM problem is obtained by using the quasi-Monte Carlo method,and a sufficient condition for the existence of the optimal solution of the discrete approximation problem is proposed.The convergence of the optimal solution and the stability point of the discrete approximation problem are further analyzed.Furthermore,for a class of SLCP with fixed matrix,a necessary and sufficient condition for the boundedness of the optimal solution sets is discussed.(2)For the SLCP,we further consider the influence caused by variance of random variables.Based the convex combined expectation and variance of D-gap function,another UERM problem is established.Under the slightly stronger conditions,the similar conclusions to part one are obtained.(3)For more general nonlinear SCP,we first study the convergence of optimal solution to approximation problem when the sample space is compact.While the sample space is non-compact,by using the compact approximation approach,we similarly obtain the convergence of optimal solutions under some certain assumptions.