Research On Stieltjes Integral Approximation Method For Uncertain Variational Inequalities

The traditional variational inequality is mainly discussed in the deterministic environment,but there are many uncertain factors in the real world.Therefore,some scholars introduced random variables into the variational inequality and established the stochastic variational inequality model.However,random variables are based on a large number of repeatable experiments.In real life,a large number of uncertain events cannot be repeated,such as sudden public crisis events,bridge bearing experiments,etc.In order to deal with the uncertainty of unrepeatable events,many scholars put forward various uncertainty theories,such as fuzzy set theory,rough set theory and so on.Among them,the uncertainty theory proposed by Liu is based on axiomatic system,which is more scientific and practical.This thesis studies variational inequalities based on Liu’s uncertainty theory and considers variational inequalities with uncertain variables.An expected value model for uncertain variational inequality problems is constructed and a discrete approximation method is proposed to solve uncertain variational inequality problems.The research work of this thesis mainly includes the following aspects:1.A discrete approximation method for uncertain variational inequalities based on regularized gap functions is proposed.Firstly,we introduce uncertain variables into variational inequalities and establish the expected value model of uncertain variational inequality problems in this thesis.The uncertain variational inequality problem is transformed into a constrained optimization problem by constructing regularized gap function,.Secondly,the influence of parameters on the boundedness of gap function and the influence of monotonicity of variational inequality on the boundedness of gap function are studied.We construct a discrete approximation method for constrained optimization problems using Stieltjes integral.Under the condition that -bounded,uniform convergence of discrete approximation method and epi convergence to primal optimization problem are proved respectively.Finally,it is proved that the global optimal solution of the discrete approximation method converges to the global optimal solution of the original optimization problem.And because the solution of the expected value model of the uncertain variational inequality problem is equivalent to the solution of the constrained optimization problem,that is,the equivalence of the solution between the approximation problem after the discretization of Stieltjes integral and the original expected value model is proved.2.A discrete approximation method for uncertain variational inequalities based on regularized gap functions is proposed.Firstly,the constraint optimization problem of expected– value model of uncertain variational inequality problem is established by constructing D-gap function,and the equivalence between expected value model and constraint optimization problem is proved.Secondly,because of the uncertain expectation in the constrained optimization problem,the discrete approximation method of the constrained optimization problem is constructed by using Stieltjes integral discrete method,and under the condition of-bounded,the uniform convergence of the discrete approximation method and epi convergence to the constrained optimization problem are proved respectively.Finally,it is proved that the global optimal solution of the discrete approximation method converges to the global optimal solution of the constrained optimization problem,that is,the discrete approximation method based on the D-gap function proposed in this thesis is feasible.