On Accelerated Algorithms For Variational Inequalities And Inclusion Problems

Variational inequalities and inclusion problems are two fundamental classes of problems in mathematical optimization that construct a unified model and framework for many practical problems in the applied sciences.Over the past decades,scholars have proposed a large number of projection-based algorithms to solve these two types of problems.The development of fast numerical algorithms is particularly important in the information age where the volume of data and computation is increasing.The aim of this dissertation is to design some accelerated iterative algorithms for solving variational inequalities,inclusion problems,and their generalization problems in infinite-dimensional real Hilbert spaces.The dissertation is divided into eight chapters and the main research work and conclusions are summarised below.1.It is well known that the extragradient algorithm introduced by G.M.Korpelevich requires the calculation of the projection on the feasible set twice in each iteration.Combining the Tseng’s extragradient algorithm,the subgradient extragradient algorithm,and the projection and contraction algorithm,this dissertation introduces some iterative algorithms that require the computation of the projection on the feasible set only once in each iteration,thus improving the computational efficiency of the extragradient-type algorithms in the literature.In addition,two concise modifications to the subgradient extragradient methods are introduced so that they use different step sizes in each iteration to compute the values of the iterative sequences,and numerical experiments show that this modification improves the computational efficiency of the original algorithms when suitable parameter values are chosen.On the other hand,this dissertation also extends these modified subgradient extragradient methods to solve equilibrium problems governed by fixed point problems.2.The algorithms proposed in this dissertation combine two types of step size criteria allowing them to work adaptively without any prior information,which overcomes the fixed step size type of algorithms presented in the literature,where the prerequisite for using such algorithms is the knowledge of the Lipschitz constant(or operator norm)of the operator involved.The algorithms proposed in this dissertation that embed the Armijotype line search step size criterion can be used to solve variational inequality problems with non-Lipschitz continuous operators.However,the use of an Armijo-type step size criterion may increase the computational effort,as the projection on the feasible set needs to be computed several times in each iteration in order to find the appropriate step size.To overcome this drawback,the algorithms proposed in this dissertation for solving the Lipschitz continuous type variational inequality problems use the simple step size criterion introduced by Yang et al.This criterion updates the step size with a simple calculation using only some known information from the current iteration.3.Inertial techniques are added to the suggested algorithms in order to speed up the convergence of the algorithms.However,inertial-type algorithms do not enjoy Fejer monotonicity of the generated sequence with respect to the solution set of the problem.For this reason,three adaptive alternating inertial extragradient algorithms are presented for solving variational inequality problems by combining the subgradient extragradient algorithm,the projection and contraction algorithm,and the alternating inertial method.The advantage of these algorithms is twofold.One is that they only need to compute the projection on the feasible set once in each iteration.The second is that their weak convergence is proved under the condition that the involved operators satisfy pseudo-monotonicity,Lipschitz continuity,and weak sequential continuity.Moreover,the R-linear convergence rate of the proposed alternating inertial algorithms is proved in the case where the operators involved are strongly pseudo-monotone.4.Examples appearing in infinite-dimensional real Hilbert spaces show that strong convergence results are preferable to weak convergence results.The accelerated inertialtype algorithms introduced in this dissertation obtain strong convergence in real Hilbert spaces by combining the Mann-type method with the viscosity method without involving any additional projection calculations.This improves on the corresponding weak convergence results in the literature,as well as the use of projection-based methods to obtain strong convergence results.On the other hand,all the algorithms presented in this dissertation for solving variational inequalities can solve pseudo-monotone type variational inequalities,which enriches and improves upon many algorithms used in the literature for solving monotone type variational inequalities.5.By combining the inertial method,the projection and contraction algorithm,the Tseng’s extragradient method,the Mann-type method,and the viscosity method,four accelerated adaptive algorithms are presented in this dissertation to solve monotone inclusion problems.The strong convergence of the proposed algorithms is proved in the case where the operators involved are Lipschitz continuous and monotone,which weakens the assumption in the literature that requires the operator to be inverse strongly monotone.Furthermore,applications of these algorithms to split feasibility problems,variational inequality problems,and image processing problems are given.On the other hand,these results are also extended to solve a wider range of split variational inclusion problems than monotone inclusion problems and obtain strong convergence results for the presented adaptive algorithms without the prior knowledge of the norm of the operators involved.6.The experimental results offered in this dissertation for a large number of numerical examples occurring in finite-and infinite-dimensional real Hilbert spaces demonstrate the computational efficiency and advantage of the presented algorithms over iterative schemes known in the literature,i.e.,the proposed algorithms require fewer iterations and execution time than the comparison algorithms and achieve higher error accuracy.All the algorithms presented in this dissertation that use inertial techniques and adaptive step size criterion can speed up the convergence of inertial-free algorithms.Furthermore,the theoretical results of this dissertation are verified in practical applications of optimal control problems and signal recovery.