Research On Proximal Point Algorithms For Monotone Inclusion Problem In Hadamard Space

Mathematical problems encountered in optimization and related fields can be converted into monotone inclusion problems,for instance minimization problems,equilibrium problems and saddle point problems can be transformed into monotone inclusion problems.Monotone inclusion problems find extensive applications in engineering technology,game theory,signal and image processing and machine learning.Monotone inclusion problems involve fundamental theories such as convex analysis,Hadamard space theory,fixed point theory,etc.Recently,researchers have shown increased interest in studying monotone inclusion problems in Hadamard spaces.Therefore,considering these practical problems,it is imperative to develop stable and efficient numerical methods to solve them.The proximal point algorithm is a classic method for solving monotone inclusion problems.Based on the original proximal point algorithm,various improved proximal point algorithms have been proposed by scholars all around the world.This paper studies monotone inclusion problems in the framework of Hadamard spaces.Using Mann iteration,Halpern iteration and generalized viscosity iteration methods,we design different modified proximal point algorithms for various problems and prove that the relevant sequences have certain properties.The main work is summarized as follows:1.The Mann-Halpern-type hybrid proximal point algorithm is proposed for solving monotone inclusion problem.Strong convergence of the proposed algorithm is proven under certain conditions.The proposed algorithm is used to solve linear inverse problems and minimization problems in Hilbert space.The numerical results demonstrate the feasibility and effectiveness of the algorithm.2.We propose a modified proximal point algorithm using Mann-type to solve the zero point of monotone vector fields and the fixed points of non-expansive mappings.Under different assumptions,we rigorously establish Δ-convergence of the proposed algorithm.Furthermore,we use the conclusion of Δ-convergence to propose a modified proximal point algorithm to solve mixed equilibrium problems.In the numerical experiments,we apply the two algorithms to solve minimization problems and mixed equilibrium problems in Hilbert space,and the numerical results demonstrate the feasibility and effectiveness of the proposed algorithms.3.We propose a modified proximal point algorithm utilizing generalized viscosity iteration to solve the zero point of monotone vector fields and the fixed points of quasipseudo non-expansive mappings.We prove that the proposed algorithm has strong convergence under certain assumptions.Furthermore,we use the conclusion of strong convergence to propose a modified proximal point algorithm to solve mixed equilibrium problems.In the numerical experiments,we apply the proposed algorithms to solve minimization problems and mixed equilibrium problems in Hilbert space,and the numerical results demonstrate the feasibility and effectiveness of the proposed algorithms.