The Application Of Strong Duality Theorem In Infinite Dimensional Space

As everyone knows,duality theory as an important part of the theory of opti-mization,strong practicability,has been widely used in transportation,military science,economic management,and other aspects.In finite dimensional space,Kuhn,Tuckerhave given dual theorems.The duality theorem plays an important role in the economic problem.But a duality theorem of finite dimensional space cannot directly promote the infinite dimensional space.Reason is that the finite dimensional space may not be satisfied with the regular assumption in finite di-mensional space.AntoninoMaugeri,DanielePuglisi,PatriziaDanielegive infinite dimensional space such as the strong duality theorem and some equivalent form.This paper mainly studies application of strong duality theorem in infinite dimensional space.One studies the application of lubrication problem in physics,it is about the distribution of trunnion bearing pressure under the condition of normal operation,how can let the pressure to achieve minimum.Another is the application to the pollution emissions price problem,this is a bilevel problem.How can let the company and society achieve maximum profits.The government chooses the optimal price of pollution emissions,with consideration to firm’s response to the price.Moreover,The firms choose the optimal quantities of production to maximize their profits,given the price of pollution emissions.What I do is transformed the above problems into forms of variational inequalites,and then the strong duality theorem is applied to variational inequalits.Finally the existence of Lagrange mutipliers associated to above problems are obtained respectively.