Research On Variational Inequality Theory And Algorithms Of Different Operators

Variational inequality is a very important research direction in the field of optimization.It can be regarded as a natural framework for uniformly handling a variety of optimization problems and equilibrium problems.Optimization problems in many fields can be transformed into variational inequality problems to solve.For example,in information technology such as signal processing,image restoration,matrix completion,and machine learning can be reduced to a convex optimization problem.The first-order necessary condition of convex optimization can be transformed into a monotonic variational inequality(in this case,the algorithm is a gradient).Because the constraint conditions of the operators of variational inequalities are not fixed,and the problems that can be solved by the variational inequalities of different operators are also different,it is very necessary to analyze the variational inequalities of different operators to solve many practical problems.This paper mainly studies the inertial super-gradient algorithm for the variational inequality of two different operators,and gives the global convergence theorem of the algorithm,and further verifies the effectiveness of the algorithm through numerical experiments.The article is divided into the following five chapters,The specific chapters are as follows:The first chapter is the introduction.It mainly introduces the research background and significance of solving variational inequality problems and the current research status at home and abroad.It summarizes the main content of current research on variational inequality problems and the advantages and disadvantages of existing solving algorithms.Finally,this paper is summarized main content of the research.The second chapter is preliminary knowledge,which mainly introduces the commonly used symbols and basic definitions used in the article and related lemmas used in the proof process.The third chapter mainly proposes a new projection algorithm for variational inequalities.When the operators involved are Lipschitz continuous and pseudo-monotonic,we combine the inertial algorithm with the exgradient algorithm.The algorithm optimizes the selection of the step size.It is verified that the algorithm is still feasible without any relevant Lipschitz constant.The algorithm converges globally and satisfies Q-linear convergence.Compared with the existing algorithm,the numerical experiment results verify the effectiveness of the algorithm.The fourth chapter is an improved adaptive exgradient projection algorithm based on nonLipschitz continuous and pseudo-monotonic operators.The idea of this algorithm is mainly derived from the constraint conditions of the operators in Chapter 3.If the operator is weakened from Lipschitz continuous to non-Lipschitz continuous,how to construct a suitable algorithm to solve the variational inequality and realize the global convergence.Inspired by the selection of the step size of Trinh algorithm,we added the inertia term to the algorithm.In the case of nonLipschitz continuous,we analyze the convergence of the algorithm when the step size decreases monotonically to zero and when the step size decreases to a constant d(d>0),and gives the global convergence theorem.Finally,calculation examples and simulation experiments confirm that our algorithm has better performance.The fifth chapter is the summary and outlook.Summarizes the research content and research results of this article,and then analyzes the current problems and further puts forward the research prospects.