Research On Inertia Extragradient Algorithms For Variational Inequality

The theory of variational inequality was proposed by Stampacchia in the early 1960s,and has a wide range of applications in the fields of differential equations,economics,optimal control and transportation.With the cross integration and mutual promotion between different disciplines,variational inequality has become one of the core topics of optimization theory,it provides a simple and effective method for solving many linear and nonlinear problems.Therefore,the study of variational inequality theory has certain academic value and application value.Since variational inequality generally does not have analytic solution,a popular research direction is how to design efficient and practical numerical algorithm with the help of various concepts and ideas to find the approximate solution of variational inequality problem,and analyze the convergence of the suggested algorithm.Projected gradient algorithm has played a great role in solving variational inequality problem due to its advantages of small storage and simple calculation.When constructing the iterative algorithm of variational inequality,the selection of the step size is very critical.Some classical algorithms usually use the Lipschitz constant to determine the range of the step size,but in practice,the Lipschitz constant is often difficult to estimate.Based on the existing theory,this paper proposes four modified numerical algorithms for solving variational inequality problems by constructing new step sizes,and proves the convergence of the proposed algorithms without knowing the Lipschitz constant in advance or even assuming the Lipschitz continuity of mapping.The obtained conclusion is a more general extension and improvement of the existing conclusion,and the specific content is expressed as follows:1.The inertial subgradient extragradient projection algorithms of pseudomonotone variational inequality are modified.Based on the subgradient extragradient algorithm proposed by Dong,Jiang and Gibali and other scholars,two iterative algorithms with inertia are given.The modified algorithms combine the merits of projection contraction algorithm and improve the step size of the second step in the subgradient extragradient algorithm.It is worth emphasizing that the monotonic step size strategy adopted by the first algorithm does not need line search,which reduces the amount of calculation at each step,so that the algorithm can work without the prior knowledge of Lipschitz constant.The second algorithm uses the Armijo-like step size method to weaken the Lipschitz continuity of the mapping and make it work under the condition of uniform continuity.Under the condition of mapping F pseudomonotone,it is proved that the two algorithms have weak convergence.Finally,numerical examples are given to illustrate the effectiveness and superiority of the proposed algorithms.2.The inertia extragradient algorithms for the common solution of variational inequality and fixed point problem are modified.On the basis of subgradient extragradient algorithm and Tseng extragradient algorithm,inertia algorithm and viscosity algorithm are introduced,and two simple and effective algorithms are presented.The non-monotonic step size strategy is adopted in the proposed algorithms,so that the step size does not need line search and does not depend on the mapped Lipschitz constant.Under the assumptions of the pseudomonotonicity of the mapping F and the quasi-nonexpansion of the mapping U,it is proved that the two algorithms presented converge strongly to common solutions of variational inequality and fixed point problem,respectively.Finally,numerical experiments in finite and infinite dimensional space show that the proposed algorithms are very effective.