Research On Algorithms For Variational Inequalities And Inclusion Problems

Variational inequality problems and monotone inclusion problems are important branches of nonlinear analysis and optimization theory,which have been widely applied in mathematics,physical,engineering technology,pure and other sciences.As the increase of cross subject studies,variational inequality problems and monotone inclusion problems are widely used to characterize problems in diverse fields,such as statistical learning,signal processing,computer science,image processing.Hence,motivated by these practical problems,it is necessary and important to find out some stable and fast numerical methods for solving these problems.In recent decades,several numerical algorithms have been proposed and analyzed after continuous research.Based on the existing theories and algorithms,this dissertation is trying to investigate several effective algorithms and convergence theorems in various spaces,including the existence of the solutions of variational inequalities and monotone inclusions,the construction of algorithms,the convergence behaviors and numerical effects in concrete examples.The results presented in this dissertation improve and supply some known results.The dissertation is divided into four parts,which are described as follows:1.Based on the extragradient algorithm,the Tseng algorithm and the inertial method,an inertial Tseng extragradient algorithm and an inertial Tseng-Halpern extragradient algorithm are proposed for solving the monotone variational inequality problems.The proposed algorithms have the advantage of small computational effort since they only calculate the projection operator once per iteration.By adopting the Armijo type stepsize research rule,the modified algorithms have a simple form and the choice of stepsize does not depend on the Lipschitz constant of the underlying mapping.Under the appropriate conditions,the weak convergence and strong convergence of the algorithms are proved,respectively.Finally,the numerical experiments indicate that the new algorithms are valid and applicable to the monotone variational inequality problems.2.Inspired by the classical subgradient extragradient algorithm,the inertial method and the Mann type iterative method,the dissertation proposes a modified inertial subgradient extragradient algorithm for solving variational inequality problems.Under the mild assumption that the underlying mapping is strong pseudomonotone and Lipschitz continuous,the strong convergence of the sequence generated by the proposed algorithm is proved.It is proved that the modified algorithm is convergent strongly to the solution of variational inequality.The numerical experiments will show the effectiveness of the theoretical results.The modified algorithm has a faster rate of convergence in comparison with some existing algorithms.3.Under the framework of Hadamard manifolds,combining the Tseng iterative algorithm and the extragradient algorithm,a Tseng type extragradient algorithm for pseudomonotone variational inequality problems is constructed.Specifically,Tseng extragradient algorithm for variational inequality problems is extended from Hilbert spaces to Hadamard manifolds.Under certain assumptions,the convergence of the algorithms is proved.Numerical experiments verify the effectiveness and feasibility of the theoretical results.4.For solving the monotone inclusion problems in Hilbert spaces,by combining the inertial method and the Douglas-Rachford splitting algorithm,a two-operator inertial Douglas-Rachford splitting algorithm and a three-operator inertial Douglas-Rachford splitting algorithm are constructed.Under appropriate assumptions,it is proved that the sequences generated by the algorithm is convergent to the solution of monotone inclusion problems,which also analyses that the convergence of the resulting algorithms is weak.Numerical experiments show the effectiveness of the algorithms and the improvement of the numerical performance.It also indicates the new algorithms are applicable to the signal recovery problem.