Deterministic Bicriteria Model For Solving Stochastic Mixed Vector Variational Inequalities And Their Convergence Analysis

Mixed Vector Variational Inequality Problems(MVVIP)are a kind of extensive mathematical model,which include the variational inequalities problems,optimization problems and vector variational inequality problems.At present,the research of mixed vector variational inequality is increasingly rich in theory and it has been widely used in electrical engineering,urban transportation,medicine and other fields.However,because the living environment we live is a dynamic environment,we are often affected by many uncertain factors in practical problems.In most cases,the abstract practical problem model can be reduced to the stochastic optimization problem,which is further transformed into the mixed vector variational inequality problem with randomness.In view of this,the stochastic mixed vector variational inequalities problems(SMVVIP)in economic management,transportation and other practical applications of the research more and more attention,is one of the current research hot topics.However,due to the existence of stochastic factors,usually the solution of SMVVIP will not exist.Therefore,this paper focuses on constructing a bi-criteria model for solving SMVVIP,and gives the approximation problem corresponding to the SMVVIP bi-criteria model,and then gives the convergence analysis of the corresponding approximation problem.The research of this thesis is mainly divided into the following three parts:Firstly,this paper starts from the study of mixed variational inequalities,introduces its research background and research status at home and abroad.As the decision problem that needs to satisfy multiple conditions often appears in practical problems,the concept,research significance and development status of deterministic MVVIP are further explained.In addition,the research significance and development status of SMVVIP as well as the research motivation of this paper are introduced.Secondly,it introduces the relevant basic concepts and theorems needed in this paper,including some basic definitions,symbols,double criterion model and Sample Average Approximation method.In addition,it also introduces the Cauchy-Schwarz inequality.Thirdly,this paper uses value function to transform SMVVIP into its equivalent stochastic optimization problem and proposes a bi-criteria model for solving stochastic mixed vector variational inequality problems.Since this model contains non-smooth functions and difficult mathematical expectations,we use smoothing method and Sample Average Approximation method.The smoothed approximation problem of the bi-criteria model and the corresponding smoothed Sample Average Approximation problem are obtained respectively.In addition,theoretically,the convergence of the global optimal solution between the bi-criteria model of solving SMVVIP and the approximation problem is given in this paper when the sample space is a compact set.When the sample space is not a compact set,by using the compact approximation method,the compact approximation problem of the bi-criteria model for solving the stochastic mixed vector variational inequality problem is obtained,and the convergence results of the corresponding compact approximation problem are discussed.Finally,the paper summarizes the full text,and gives two questions about SMVVIP that are worth thinking and studying.