Optimization for the Sum of Finite Functions Over the Solution Set of Split Equality Optimization Problems with Applications

In this paper, we consider split equality generalized mixed equilibrium problem, which is more general than many problems such as split feasibility problem, split equality problem, split equilibrium problem, and so on. We propose a new modified algorithm to obtain strong and weak convergence theorems for split equality generalized mixed equilibrium problem for nonexpansive mappings in Hilbert spaces. Also, we give some applications to other problems. Our results extend some results in the literature.

The split feasibility problem is an inverse problem which arises in signal processing and medical image reconstruction. So there is practical value in studying it. While both the split equality problem and the split variational inclusion problem are generalized form of the split feasibility problem which are more meaningful than the split feasibility problem. In this paper, fusing the two problems, we research a split inclusion problem and propose relevant methods for solving it. What counts is that not only the proposed algorithms have strong convergence, but also the limit points of the algorithms are the minimal norm solution of the split inclusion problem.

In this paper, we consider the split equality fixed point problem for quasi-pseudo-contractive mappings without a priori knowledge of operator norms in Hilbert spaces, which includes split feasibility problem, split equality problem, split fixed point problem, etc., as special cases. A unified framework for the study of this kind of problems and operators is provided. The results presented in the paper extend and improve many recent results.

In this paper, we introduce a new iterative algorithm for solving the split equality generalized mixed equilibrium problems. The weak and strong convergence theorems are proved for demi-contractive mappings in real Hilbert spaces. Several special cases are also discussed. As applications, we employ our results to get the convergence results for the split equality convex differentiable optimization problem, the split equality convex minimization problem, and the split equality mixed equilibrium problem. The results in this paper generalize, extend, and unify some recent results in the literature.

In this paper, we investigate the problem of solving strongly monotone variational inequalities over the solution set of the split variational inequality problem with multiple output sets in real Hilbert spaces and establish a new iterative algorithm for it. Our algorithms are accelerated by the inertial technique and eliminate the dependence on the norm of the transformation operators and the strongly monotone and Lipschitz continuous constants of the involved operator by employing a self-adaptive step size criterion. The strong convergence result is given under some mild conditions widely used in the convergence analysis. Two corollaries for the solutions of the split variational inequality problem and the split feasibility problem with multiple output sets are also obtained using our main result. Finally, some numerical experiments have been conducted to illustrate the effectiveness of the proposed algorithms and compare them with the related ones.

ABSTRACTIn this paper, we suggest two new iterative methods for finding an element of the solution set of split variational inclusion problem in real Hilbert spaces. Under suitable conditions, we present weak and strong convergence theorems for these methods. We also apply the proposed algorithms to study the split feasibility problem. Finally, we give some numerical results which show that our proposed algorithms are efficient and implementable from the numerical point of view.

The inverse problem is one of the four major problems in computational mathematics. There is an inverse problem in medical image reconstruction and radiotherapy that is called the multiple-sets split equality problem. The multiple-sets split equality problem is a unified form of the split feasibility problem, split equality problem, and split common fixed point problem. In this paper, we present two iterative algorithms for solving it. The suggested algorithms are based on the gradient method with a selection technique. Based on this technique, we only need to calculate one projection in each iteration.

This paper first introduces a new problem that is the general split variational inequality problem and invents a mathematical tool for solving our new problem which is Lemma 2.5. Then, we establish and prove a strong convergence theorem aimed at finding an element of the set of the solution of the general split variational inequality problem. Furthermore, we apply our main theorem to demonstrate a strong convergence theorem for finding solutions to the split variational inequality problem, the split feasibility problem, and the minimization problem. Finally, we provide numerical examples to advocate our main result

The purpose of this paper is to introduce a new parallel algorithm for finding solutions to pairs of variational inequalities over the solution sets of systems of split equality null-point problems in Hilbert spaces. More precisely, we propose a new self-adaptive algorithm based on the hybrid steepest descent method and establish a strong convergence theorem for it. Moreover, we also give some applications of our proposed algorithm to solving some related problems, such as systems of split equality minimum point problems and systems of split equality feasibility problems.

The split feasibility problem (SFP) is finding a point xin C such that Axin Q, where C and Q are nonempty closed convex subsets of Hilbert spaces H_{1} and H_{2}, and A:H_{1}rightarrow H_{2} is a bounded linear operator. Byrne’s CQ algorithm is an effective algorithm to solve the SFP, but it needs to compute |A|, and sometimes |A| is difficult to work out. López introduced a choice of stepsize lambda_{n}, lambda_{n}=frac{rho_{n}f(x_{n})}{|nabla f(x_{n})| ^{2}}, 0<rho_{n}<4. However, he only obtained weak convergence theorems. In order to overcome the drawbacks, in this paper, we first provide a regularized CQ algorithm without computing |A| to find the minimum-norm solution of the SFP and then obtain a strong convergence theorem.

In this paper, two viscosity proximal type algorithms involving the superiorization method are presented for solving the split variational inclusion problems in real Hilbert spaces. Strong convergence theorems and bounded perturbation resilience analysis of the proposed algorithms are obtained under mild conditions. The split feasibility problems, the split minimization problems, and the variational inequality problems are concerned as the applications, and several numerical experiments are performed to show the efficiency and implementation of the proposed algorithms.

In this paper, four self-adaptive iterative algorithms with inertial effects are introduced to solve a split variational inclusion problem in real Hilbert spaces. One of the advantages of the suggested algorithms is that they can work without knowing the prior information of the operator norm. Strong convergence theorems of these algorithms are established under mild and standard assumptions. As applications, the split feasibility problem and the split minimization problem in real Hilbert spaces are studied. Finally, several preliminary numerical experiments as well as an example in the field of compressed sensing are proposed to support the advantages and efficiency of the suggested methods over some existing ones.

Three fixed-point equations for solving both split common and feasibility problems of firmly non-expansive operator in a stochastic domain was studied. Stochastic convergence of the iterations based on random-type iterative scheme for solving stochastic split common and feasibility problems was also established. The mapping of the operators in Hilbert space was adopted.

Strong convergence theorem of viscosity approximation methods for nonexpansive mapping have been studied. We also know that CQ algorithm for solving the split feasibility problem (SFP) has a weak convergence result. In this paper, we use viscosity approximation methods and some related knowledge to solve a class of generalized SFP’s with monotone variational inequalities in Hilbert space. We propose some iterative algorithms based on viscosity approximation methods and get strong convergence theorems. As applications, we can use algorithms we proposed for solving split variational inequality problems (SVIP), split constrained convex minimization problems and some related problems in Hilbert space.

Inspired by Moudafi (2011) and Takahashi et al. (2008), we present the shrinking projection method for the split common fixed-point problem in Hilbert spaces, and we obtain the strong convergence theorem. As a special case, the split feasibility problem is also considered.