Research On Algorithms For Solving Variational Inclusion Problems And Their Applications

The variational inclusion problem is an important branch of nonlinear analysis and optimization theory.It is a subject that has found applications in a wide array of disciplines,including optimization control,engineering technology,game theory,economics,transportation,computer science,signal and image processing,machine learning and so on.The research of the variational inclusion problem involves modern analysis theory of convex analysis,variational method,geometric theory of Banach spaces,nonlinear approximation theory,critical point theory and fixed point theory and so on.The objective of this dissertation is to investigate the effective iterative approximation algorithms for solving variational inclusion problems in Hilbert spaces and Banach spaces base on projection algorithm,forward-backward splitting algorithm,Tseng algorithm,Mann iterative algorithm,Halpern type algorithm,viscosity iterative algorithm and so on.The main contents are described as follows.1.Variational inequality problems involving pseudomonotone operators in real Hilbert spaces will be investigated.The contents split into two parts.Firstly,the pseudomonotone variational inequality problem will be investigated.It is known that the step size of the classical projection algorithm for the variational inequality problem is associate with the Lipschitz constant of involved operators.For avoiding the use of Lipschitz constant,the usual way is to use an Armijo-like search rule to get a self-adaptive step size.In this dissertation,a self-adaptive iterative algorithm which combines the inertial technique and the Tseng extragradient idea with a Armijo-like step size rule will be proposed.The construction of the proposed algorithm is without the prior knowledge of the Lipschitz constant of cost operators.Secondly,the bilevel pseudomonotone variational inequality problem will be investigated.A self-adaptive algorithm involving inertial technique will be introduced to solve the bilevel pseudomonotone variational inequality problem.The main advantage of the proposed algorithm is that the strong convergence theorem is proved without Lipschitz continuity condition of the associated operators.The effectiveness and superiority of the proposed algorithms are proposed by numerical experiments.2.The variational inclusion problem of the sum of two monotone operators will be investigated.Two self-adaptive inertial forward-backward splitting algorithms are proposed for solving variational inclusion problems in real Hilbert spaces.The proposed algorithms base on inertial extrapolation technique,forward-backward splitting algorithm,Tseng extragradient algorithm and Armijo-like search method.Strong convergence theorems are obtained under some weak assumptions imposed on the sequence of parameters.Moreover,applications to signal processing are also considered.Some numerical experiments of proposed algorithms and comparisons with existing algorithms are given to demonstrate the efficiency of the proposed algorithms.The numerical results show that the proposed algorithms are superior to some related algorithms.3.Inertial splitting algorithms for solving the common solution of accretive operators and ?-strictly pseudocontractive operators will be proposed.Weak and strong convergence algorithms are established in uniformly convex and q-uniformly smooth Banach spaces base on inertial extrapolation technique and forward-backward splitting algorithms.The other highlight of the proposed algorithms is that the algorithms work for the class of ?-strictly pseudocontractive mappings.The main results extend and complement the recent results obtained in the literatures.Numerical examples are given to illustrate the effectiveness of the proposed algorithms.4.The common solution of accretive operators and nonexpansive operators in the framework of uniformly convex and uniformly smooth Banach spaces will be investigated.A weak convergence Mann type iterative algorithm and a strong convergence Halpern type iterative algorithm will be proposed.The main results extend and complement the recent results obtained in the literatures.A numerical example is given to illustrate the convergence of the proposed iterative algorithms.