Research For Distributionally Robust Stochastic Optimization With Linear Decision Rule

Optimization theory and methods are widely used in decision-making in engineering,military,business and other industries.In these decision-making processes,there are some decision-making problems containing uncertain factors.In mathematics,the distributionally robust stochastic optimization is used to describe a class of stochastic optimization problems with fuzzy distribution information of uncertain parameters.In order to solve this kind of stochastic problem,more and more attention has been paid to the research of distributionally robust stochastic optimization in recent years.In this paper,model and numerical method studies are carried out for the single-stage distributionally robust stochastic optimization and two-stage distributionally robust stochastic bilevel optimization.The main idea is to construct an optimization problem that is easy to calculate and realize,based on the linear decision rule.The main tasks are as follows:The first part studies the single-stage distributionally robust stochastic optimization.Based on two different fuzzy distribution set,we first deal with the inner supremum problem with Lagrange duality theorem,then we use the linear decision rule to the original and dual variables for that handled problem.We finally obtain the upper and lower bounds optimization problems,which are easy to calculate.Numerical experiments verify the effectiveness of the model analysis.In the second part,based on the uncertain distribution set defined by moment information,we study the two-stage distributionally robust stochastic bilevel optimization.We conduct the optimistic and pessimistic formulations of that.In addition,we deduce the optimistic optimization to a lower optimization and the pessimistic optimization to a upper optimization by using linear decision rule,which are easy to calculate.Therefore,we derive the the lower bound and the upper bound for the two-stage distributionally robust stochastic bilevel optimization.We prove the conclusion in theory.