Optimality Conditions For A Class Of Cone-constrained Variational Inequality Problem And Its Application In Eco-industrial Chain

In recent years,variational inequalities have been used as a powerful unified method to study the equilibrium problem,and academic research in related fields has also been increasing.In particular,the image space analysis method is used to study variational inequality problems.In this article,by using the image space analysis,nonlinear separation method for a class of variational inequality with a cone constraint are proposed.By virtue of the nonlinear scalarization function commonly known as the Gerstewitz function,three nonlinear weak separation functions,two nonlinear regular weak separation functions and a nonlinear strong separation function are introduced.Then,by these nonlinear separation functions,some weak or strong alternative theorems are established.Based on this theorems,the optimality conditions for variational inequality with a cone constraint are derived.Because the gap function and the error bound play an important role in the optimization method,two gap functions for a class of variational inequality with a cone constraint are proposed.Using this gap functions and some assumptions,it is proved that the solution set of a class of variational inequality with a cone constraint satisfies the error bound.As a complex industrial ecosystem,eco-industrial chain has a wide range of applications.Variational inequality theory has been used to construct an ecological industry equilibrium model or to analyze the equilibrium conditions of the ecological industry.However,it is rare to use the inverse variational inequality theory to explore the research related to the ecological industry chain.Based on this,this paper uses the ecological industry equilibrium structure and labeling system characterization established by the document [34],against various complex network structures in the eco-industrial chain,discusses the inequality of each company's control of profitability through cost under certain assumptions,and shows that their optimal change in cost is the solution to the corresponding inverse variational inequality,and gives some properties of the solution.