Research On The Theories And Algorithms Of Variational Inequality Generalization Models

Variational inequality is a generalization and development of the classical variational problems,which is also a variational method that relaxes the constraints of the classical variational problem to some one-sided constraints(that is,the inequality is used to replace the equation).Moreover,it is an important tool for studying partial differential equations,optimal control and others.At the same time,there are a large number of convex optimization problems in management science and statistical calculations.There are also many problems in information technology such as signal processing,image restoration,matrix integrity theory and machine learning that can be reduced to(or relaxed into)a convex optimization problem.First-order necessary conditions for convex optimization can be transformed into a monotone variational inequality,and then the methods of solving convex optimization can be studied in the framework of variational inequality.In addition to the most commonly used optimization problems,complementary problem is a kind of variational inequality that is constrained to be positive hexagrams.Furthermore,many practical problems,such as space price balance in economic activities,regulatory measures in protection of resources and protection of supply,and the use of economic methods to solve traffic diversion,can be described by variational inequality(or its special form complementarity problem).Multi-valued variational inequality,as a further extension of variational inequality,has a wide range of applications in economics,operations research,mathematical physics,nonlinear equations,optimization problems and nonlinear complementary problems.Therefore,it is necessary to design and study algorithms for variational inequality and multi-valued variational inequality.In this paper,Chapter 1 briefly introduces the research background,the research status,and the preliminaries of the algorithms for solving variational inequalities and multi-valued variational inequalities.The Chapter 2 proposes a new Newton algorithm for a special kind of variational inequality — nonlinear complementary problem.Meanwhile,the function involved is a non-Lipschitz function and the coefficient matrix of the linear part of the function is anM-matrix.Under these conditions,the nonlinear complementarity problem has an unique solution.We first use the special properties of the M-matrix to define a minimum function which is equivalent to the nonlinear complementarity problem,and then design a new algorithm based on the relationship between the elements of the constant vector and zeroand the properties of the minimum function,and the algorithm has global convergence.In addition,the convergence and effectiveness of the algorithm are further verified by numerical experiments.In Chapter 3,we propose two new algorithms for solving multi-valued variational inequality based on the extragradient algorithm.In the existing algorithms,they can not be guaranteed that the iterative sequence produced by the algorithm is strictly monotonous decreasing.In this regard,we combine the inertial algorithm with the subgradient extragradient algorithm to ensure that the iterative sequence generated by the algorithm is strictly monotonous decreasing.In order to make the iterative sequence generated by the algorithm has global convergence,the involved function needs to be monotone and Lipschitz continuously,and has a nonempty convex compact value.We propose an improved extragradient algorithm which just needs the function to be continuous and pseudo-monotone.Moreover,the algorithm is monotonically decreasing.The above two algorithms have global convergence,and the convergence and effectiveness of the algorithms are further verified by numerical experiments.In Chapter 4,we summarized the main results of the paper.Then,we put forward the further research on the stochastic variational inequality problems,which is based on the problems of uncertainty in the actual modeling.