Levitin-polyak Well-posedness In Generalized Quasivariational-like Inequality Problems

As we know, it is the source of Variational Inequality which can be led from Engineering project, Mechanics, Mathematical Physics, Cybernetics, Optimization Theory, Economic Mathematics and Differential Engineering while the development of Variational Inequality play an important role in promoting the development of these subjects. The elements of Variational Inequality Theory is to study all kinds of Variational Inequality, including existence and uniqueness of solution, character of solution, approximation of solution and it involves in the use of all types of prac-tice. The well-posedness of Variational Inequality is to study the represent of the solution. Specially speaking, it considers if there has a subsequence of the approx-imating solution sequence converging to one of the best solutions for the problem Many Optimizations which produce appxiomate solution sequence are gradual min-imization sequence, such as Function method, Augmented Lagrangian approach. Therefore, adaptability plays an important part in the mentioned questions.This article studies properties and applications of Variational Inequality from the following aspects:Chapter 1 summarize academic significance, applied significance and researching current situation of studying Variational Inequality.Chapter 2 introduce the basic knowledge which are involved in this article.Chapter 3 In the literature of No.22, author makes use of gap function, Audlen-der to research one kind of Generalized Vector Variational Inequality and discuss the LP well-posedness of a class of minimization problem. Inspired by this, the inter-related result is popularized to a class of Generalized Variational-like Inequality in this chapter. At the same time, author continues to research this kind of Variational Inequality and the LP well-posedness of Optimization. The results that are referred to in this chapter are popularization and application for the corresponding results in the literature of No.22.Chapter 4 In the literature of No.21, author has given a class of quasi-Variational inequality problem and a class of optimization problems which has four kinds of forms about LP well- posedness. At first, author discusses the relationship among these types of LP well-posedness. On this basis, author makes a conclusion on character of a series of Quasi-Variational Inequality and use Audlender gap function to set up the connection between Quasi-Variational Inequality and Optimization. Then the equivalence between LP well-posedness of Quasi-Variational Inequality and LP well-posedness of Optimization is established. The results in this chapter are established in the promotion of Quasi-Variational Inequality, which extends the applications in the literature of No.21.