Two-stage Stochastic Variational Inequalities:Theory And Applications

The two-stage stochastic variational inequality considers both the decision before the occurrence of uncertainty and the recourse decision after the occurrence of uncertainty.Hence it has wide applications in science,engineering,economics and finance,and becomes one of the hot topics in the optimization field.At present,it is known that the two-stage stochastic variational inequality mainly comes from optimality conditions of the two-stage stochastic optimization and the two-stage stochastic equilibrium.However,the specific conditions and ways of transformation,as well as properties such as monotonicity/nonmonotonicity of the transformed two-stage stochastic variational inequality have not been fully investigated.In this thesis,the transformations of the two-stage stochastic variational inequalities from the two-stage stochastic programming,as well as the two-stage stochastic equilibrium are established under some specific moderate conditions that can be easily verified.The smoothing technique is used to deal with the nonsmooth objective function,the discretization scheme is designed,the monotonicity/nonmonotonicity are discussed,and the two-stage stochastic variational inequalities in real applications are solved by the progressive hedging method.Numerical results show that the transformation is effective,and the progressive hedging method is e cient especially when the number of scenarios is large.First,we consider a general convex two-stage stochastic programming problem,where the random variables have continuous distributions and there are conic constraints in the second-stage.We transform it into a two-stage stochastic variational inequality.The equivalence between the two problems is shown under some moderate conditions,and the monotonicity of the two-stage stochastic variational inequality is discussed under additional conditions.We provide a discretization scheme with convergence results and employ the progressive hedging method with double parameterization to solve the discretized stochastic variational inequality.The water resources allocation under uncertainty and the portfolio selection under uncertainty are given as applications,where the cone of the former is the non-negative orthant,and the cone of the latter is the second-order cone.Second,for a general nonconvex two-stage stochastic programming problem,we transform it into a two-stage stochastic variational inequality under some moderate conditions.The two problems are no longer equivalent,but the stochastic variational inequality is still useful for solving the two-stage stochastic programming problem.As an application,a two-stage shipment planning with pricing is addressed.We show that all the conditions for transformation are satisfied and the transformed two-stage stochastic variational inequality is not monotone.When the product price is regarded as a parameter,the parameterized two-stage stochastic variational inequality is monotone.The progressive hedging method is then employed to solve a sequence of parameterized problems,and find the optimal solution of the two-stage shipment planning with pricing.Finally,for a class of nonsmooth two-stage stochastic equilibrium problem,we use a smooth approximation technique to approximate and transform it into a monotone two-stage stochastic variational inequality.Specially,we propose a new nonsmooth two-stage stochastic equilibrium model of medical supplies in epidemic management.The first stage addresses the storage in the pre-disaster phase,and the second stage focuses on the dynamic distribution by enrolling competitions among multiple hospitals over a period of time in the post-disaster phase.The uncertainties are the numbers of infected people treated in multiple hospitals during the period of time,which are timevarying around a nominal distribution predicted by historical experience.We employ the progressive hedging method to solve a case study in the city of Wuhan in China suffered from the COVID-19 pandemic at the beginning of 2020.Compared with Matlab code “linprog” and Benders decomposition method,numerical results are presented to demonstrate the effectiveness of the proposed model and the progressive hedging method in planning the storage and dynamic distribution of medical supplies in epidemic management.