Applications of accelerated computational methods for quasi-nonexpansive operators to optimization problems

Let , and be real Hilbert spaces, let and be two bounded linear operators. Moudafi introduced simultaneous iterative algorithms with weak convergence for the following split common fixed-point problem:Section.Display where and are two firmly quasi-nonexpansive operators with nonempty fixed-point sets and . Note that, by taking and , we recover the split common fixed-point problem originally introduced by Cesnor and Segal. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. In this paper, we will continue to consider the split common fixed-point problem (1) governed by the general class of quasi-nonexpansive operators. We introduce a simultaneous iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. The weak convergence result of algorithm is obtained and some applied nonlinear analysis examples are stated.

In this paper, we first introduce a new algorithm with a viscosity iteration method for solving the split common fixed point problem (SCFP) for a finite family of nonexpansive semigroups. We also present a new algorithm for solving the SCFP for an infinite family of quasi-nonexpansive mappings. We establish strong convergence of these algorithms in an infinite-dimensional Hilbert spaces. As application, we obtain strong convergence theorems for split variational inequality problems and split common null point problems. Our results improve and extend the related results in the literature.

In this paper, we adopt an iterative approach to solve the class of optimization problem for the sum of finite functions over split equality optimization problems for the sum of two functions. This type of problem contains many optimization problems, and bilevel problems, as well as split equality problems, and split feasibility problems as special cases. Here, we are able to establish a strong convergence theorem for an iterative method for solving this problem. As consequences of this convergence theorem, we study the following problems: optimization for the sum of finite functions over the common solution set of optimization problems for the sum of two functions; optimization for the sum of finite functions; optimization for the sum of finite functions with split equality inconsistent feasibility constraints; optimization for the sum of finite functions over the solution set for split equality constrained quadratic signal recovery problem; optimization for the sum of finite functions over the solution set of generalized split equality multiple set feasibility problem, and optimization for the sum of finite functions over the solution set of split equality linear equations problem. We use simultaneous iteration to establish strong convergence theorems for these problems. Our results generalize and improve many existing theorems for these types of problems in the literature and will have applications in nonlinear analysis, optimization problems and signal processing problems.

<abstract><p>The theory of variational inequalities is an important tool in physics, engineering, finance, and optimization theory. The projection algorithm and its variants are useful tools for determining the approximate solution to the variational inequality problem. This paper introduces three distinct extragradient algorithms for dealing with variational inequality problems involving quasi-monotone and semistrictly quasi-monotone operators in infinite-dimensional real Hilbert spaces. This problem is a general mathematical model that incorporates a set of applied mathematical models as an example, such as equilibrium models, optimization problems, fixed point problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithms employ both fixed and variable stepsize rules that are iteratively transformed based on previous iterations. These algorithms are based on the fact that no prior knowledge of the Lipschitz constant or any line-search framework is required. To demonstrate the convergence of the proposed algorithms, some simple conditions are used. Numerous experiments have been conducted to highlight the numerical capabilities of algorithms.</p></abstract>

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.

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 present a conditional gradient type (CGT) method for solving a class of composite optimization problems where the objective function consists of a (weakly) smooth term and a (strongly) convex regularization term. While including a strongly convex term in the subproblems of the classical conditional gradient method improves its rate of convergence, it does not cost per iteration as much as general proximal type algorithms. More specifically, we present a unified analysis for the CGT method in the sense that it achieves the best known rate of convergence when the weakly smooth term is nonconvex and possesses (nearly) optimal complexity if it turns out to be convex. While implementation of the CGT method requires explicitly estimating problem parameters like the level of smoothness of the first term in the objective function, we also present a few variants of this method which relax such estimation. Unlike general proximal type parameter free methods, these variants of the CGT method do not require any additional effort for computing (sub)gradients of the objective function and/or solving extra subproblems at each iteration. We then generalize these methods under stochastic setting and present a few new complexity results. To the best of our knowledge, this is the first time that such complexity results are presented for solving stochastic weakly smooth nonconvex and (strongly) convex optimization problems.

This paper studies a strongly convergent of a modified tri-inertial viscosity self-adaptive technique for common solution to the split generalized equilibrium problem together with the split common null point problem with multiple output sets. In our convergent analysis, no assumption was made of the sum condition on the iteration parameters, which is not the case for several results that involve strongly convergent algorithm with inertial extrapolation step proposed for split generalized equilibrium problem and split common null point problem with multiple output sets. The method uses variable step-sizes that are modified at each iteration and dependent on prior iterations. This method has the advantage of not requiring a prior knowledge of the Lipschitz constant and the linesearch process. A series of numerical experiments are performed to validate the efficacy and superiority of the presented iterative technique over the existing methods in this direction. The experiment serves to affirm the effectiveness of the proposed algorithm in split generalized equilibrium problem and split common null point problem with multiple output sets in real Hilbert spaces.

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.

New inertial modification of regularized algorithms for solving split variational inclusion problem

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.

In this paper, we construct a novel algorithm for the split common fixed point problem for two demicontractive operators in Hilbert spaces. By using inertial self-adaptive algorithms, we obtain strong convergence results for finding a solution of the split common fixed point problems. Applications to solving the split minimization problem and the split feasibility problem are included. Our results extend and generalize many previously known results in this research area. Moreover, numerical experiments are supplied to demonstrate the convergence behavior and efficiency of the proposed algorithm.

The purpose of this paper is to introduce and study the general viscosity approximation methods for quasi-nonexpansive mappings in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequences generated by the proposed new algorithm converge strongly to a fixed point of quasi-nonexpansive mappings in Hilbert spaces, which is also the unique solution of some variational inequality. Then this result is used to study the split equality fixed point problems, the split equality common fixed point problems, the split equality null point problems, etc. Our results improve and generalize many results in the literature and they should have many applications in nonlinear science.

In this paper, we study the generalized multiple-set split feasibility problem including the common fixed-point problem for a finite family of generalized demimetric mappings and the monotone inclusion problem in 2-uniformly convex and uniformly smooth Banach spaces. We propose an inertial Halpern-type iterative algorithm for obtaining a solution of the problem and derive a strong convergence theorem for the algorithm. Then, we apply our convergence results to the convex minimization problem, the variational inequality problem, the multiple-set split feasibility problem and the split common null-point problem in Banach spaces.

In the paper, we introduce several accelerated iterative algorithms for solving the multiple-set split common fixed-point problem of quasi-nonexpansive operators.Based on the primal-dual method, we construct several iterative algorithms in a way that combines inertial technology and self-adaptive stepsizes such that the implementation of the algorithms does not need any prior information about bounded linear operator norm.Under suitable assumptions, we obtain the weak convergence of the proposed algorithms in real Hilbert spaces.As applications, we obtain relative iterative algorithms to solve the multiple-set split feasibility problem.Finally, the performance of the proposed algorithms is illustrated by numerical experiments.