Convergence Of Projection Gradient Method And Newton-Steffensen Method

The optimization problem is an important subject in computational mathematics.It has important application prospects in many aspects of social production and de-velopment.Among the methods of optimization problems,the convergence study of projection gradient method and Newton method have attracted wide attention.In this paper,we study the convergence of projection gradient method for solving multiple-sets split feasibility problem and the cubic convergence behavior of New-ton-Steffensen method for solving nonlinear operator equation problem.First,for the projection gradient method,we explore a family of projection gra-dient methods for solving the multiple-sets split feasibility problem.Besides,we in-troduce a family of relaxed projection gradient methods,in which the projections onto the approximated halfspaces are adopted in place of the ones onto the level sets.Global weak convergence theorems are established for these methods.As an applica-tion,we explain the cyclic iteration method and the simultaneous method introduced in[J.Optim.Theory Appl.166,844-860(2015)]are special cases of the projection gradient method given in this paper.Second,Cubic convergence behavior of Newton-Steffensen method is studied when the first derivative of the non-linear operator F satisfies the L-average Lip-schitz condition.Namely,when F satisfies the L-average Lipschitz condition,the sequence generated by Newton-Steffensen iterative has cubic convergence.So cubic convergence criteria of Newton-Steffensen method is established.At the same time,an estimate of the radius of convergence ball is also presented.In particular,as appli-cations,when F satisfies the classical Lipschitz condition or F satisfies the?-condition,cubic convergence criteria of Newton-Steffensen method is established and an estimate of the radius of convergence ball is provided,which extends the re-sults in[Journal of Nonlinear and Convex Analysis,2018,19:433-460].