An Inexact Three-Operator Splitting Algorithm And Its Application In Convex Optimization

Many problems in signal and image processing,medical image reconstruction,and machine learning can be attributed to the optimization problem of solving the sum of two or more convex functions.Since these optimization models are usually not only non-smooth,but also the dimension of the problem is large,the traditional optimization methods will encounter difficulties.How to design efficient,fast and optimization algorithms guaranteed theoretically is a real and important problem.For the optimization problem of the sum of multiple convex functions,operator splitting algorithms have been received much attention in recent years.It not only has the simple structure,but also divide the complex problem into a series of simple sub-problems,which provides a convenient way for the solution of convex optimization problems.In this paper,we propose an inexact three-operator splitting algorithm and study the convergence and convergence rates of this algorithm under mild conditions on the parameters.Compared with the three-operator splitting algorithm,the inexact three-operator splitting algorithm allows errors in the computation of iterative sequences.Further,we apply the proposed algorithm to solve a class of convex optimization problem with the sum of three convex function,which includes a differentiable convex function and a convex function composed of a linear operator.The full paper is divided into four chapters,the specific contents are as follows:In the first chapter,the background of operator splitting algorithms and the research status of the optimization problem of convex combination function are introduced.Then some symbols,definitions and theorems involved in this paper are given.Finally,the main research contents of this paper are expounded.In the second chapter,an inexact three-operator splitting algorithm is proposed to solve the monotone inclusion problem of the sum of three maximal monotone operators,including a cocoercive operator.The convergence of the proposed algorithm is proved under mild condition on the parameters.Furthermore,the global convergence rate of the inexact three-operator algorithm is studied from the perspective of the fixed point residual.Moreover,for the convex optimization problem,the convergence rate of the function values is given in ergodic and nonergodic sense.In the third chapter,we propose several inner-outer iterative algorithms for solving a class of convex optimization problem of the sum of three convex functions.Based on the inexact three-operator splitting algorithm proposed and the algorithm that solves the fixed point of resolvent of composite operator L~*BL,the inner-outer iterative algorithms are proposed to solve the optimization problem of the sum of three convex functions.The convergence of the proposed iterative algorithm is proved in infinite dimensional Hilbert space.At the same time,the proposed algorithm is applied to CT image reconstruction and the experimental results show the effectiveness of our algorithm.The fourth chapter summarizes the full text and gives a prospect for future work.