New hybrid inertial CQ projection algorithms with line-search process for the split feasibility problem

In this paper, we propose two hybrid inertial CQ projection algorithms with linesearch process for the split feasibility problem. Based on the hybrid CQ projection algorithm, we firstly add the inertial term into the iteration to accelerate the convergence of the algorithm, and adopt flexible rules for selecting the stepsize and the shrinking projection region, which makes an optimal stepsize available at each iteration. The shrinking projection region is the intersection of three sets, which are the set C and two hyperplanes. Furthermore, we modify the Armijo-type line-search step in the presented algorithm to get a new algorithm.The algorithms are shown to be convergent under certain mild assumptions. Besides, numerical examples are given to show that the proposed algorithms have better performance than the general CQ algorithm.

GENERAL SPLIT FEASIBILITY PROBLEMS FOR TWO FAMILIES OF NONEXPANSIVE MAPPINGS IN HILBERT SPACES

The purpose of this article is to introduce a general split feasibility problems for two families of nonexpansive mappings in Hilbert spaces. We prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the general split feasibility problem. Our results extend and improve some recent known results.

A New Inertial Self-adaptive Gradient Algorithm for the Split Feasibility Problem and an Application to the Sparse Recovery Problem

In this paper,by combining the inertial technique and the gradient descent method with Polyak's stepsizes,we propose a novel inertial self-adaptive gradient algorithm to solve the split feasi-bility problem in Hilbert spaces and prove some strong and weak convergence theorems of our method under standard assumptions.We examine the performance of our method on the sparse recovery prob-lem beside an example in an infinite dimensional Hilbert space with synthetic data and give some numerical results to show the potential applicability of the proposed method and comparisons with related methods emphasize it further.

A Levenberg–Marquardt Method for Solving the Tensor Split Feasibility Problem

This paper considers the tensor split feasibility problem.Let C and Q be non-empty closed convex set and A be a semi-symmetric tensor.The tensor split feasibility problem is to find x∈C such that Axm−1∈Q.If we simply take this problem as a special case of the nonlinear split feasibility problem,then we can directly get a projection method to solve it.However,applying this kind of projection method to solve the tensor split feasibility problem is not so efficient.So we propose a Levenberg–Marquardt method to achieve higher efficiency.Theoretical analyses are conducted,and some preliminary numerical results show that the Levenberg–Marquardt method has advantage over the common projection method.