Iterative Algorithm For A Class Of Variational Inclusion Problems

Recently,significant progress has been made in the study of variational inclusion problems,which have become an important research direction in the field of mathematics.Many problems in various disciplines such as mechanics,control theory,and optimization theory can be transformed into variational inclusion problems.Therefore,solving variational inclusion problems has always been a research hotspot in applied mathematics.This thesis proposes two novel algorithms for solving variational inclusion problems involving H-monotone operators in real Hilbert spaces.Firstly,the properties of relevant preconditioning operators are discussed.Then,based on the Preconditioned Projection Algorithm(RPA),Hybrid Halpern-type Proximal Point Algorithm,and Inertial Algorithm,two new algorithms are designed,and their strong convergence is theoretically proven.Algorithm 1 combines the ideas of inertia,preconditioning,and projection.When the single-valued mapping H satisfies strong monotonicity and Lipschitz continuity,the set-valued mapping M is H-monotone and H-monotone with respect to H,and the parameters satisfy appropriate conditions,the algorithm generates a sequence that strongly converges to the solution of the variational inclusion problem involving H-monotone operators,based on the characteristics of projection operators,preconditioning operators,and other related features.Based on Algorithm 1,Algorithm 2 is obtained by constructing the image of the points generated by the projection and preconditioning steps as a Mann iteration point.It is also proven that Algorithm 2 strongly converges to the solution of this type of problem when the combination coefficients satisfy certain conditions.Finally,numerical examples are provided to verify the convergence and efficiency of the two algorithms proposed in this thesis.