Hybrid Steepest Descent Method Research Articles

A parallel algorithm for solving a pair of two variational inequalities in Hilbert spaces

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.

A New Viscosity Implicit Approximation Method for Solving Variational Inequalities over the Common Fixed Points of Nonexpansive Mappings in Symmetric Hilbert Space

In this paper, based on the viscosity approximation method and the hybrid steepest-descent iterative method, a new implicit iterative algorithm is presented for finding the common fixed points set of a finite family of nonexpansive mappings in a reflexive Hilbert space, which is called a symmetric space. We prove that the sequence generated by this new implicit rule strongly converges to the unique solution of a class of variational inequalities under certain appropriate conditions of the parameters. Moreover, we also study the applications to a broader family of strictly pseudo-contractive mappings and generalized equilibrium problems that involve several variational inequality problems, optimization problems, and fixed-point problems. Finally, numerical results are provided to clarify the stability and effectiveness of the algorithm and to compare with some existing iterative algorithms.

On Strengthened Inertial-Type Subgradient Extragradient Rule with Adaptive Step Sizes for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive Mappings

In a real Hilbert space, let the VIP denote a pseudomonotone variational inequality problem with Lipschitz continuity operator, and let the CFPP indicate a common fixed-point problem of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping. On the basis of the Mann iteration method, the viscosity approximation method and the hybrid steepest-descent method, we propose and analyze two strengthened inertial-type subgradient extragradient rules with adaptive step sizes for solving the VIP and CFPP. With the help of suitable restrictions, we show the strong convergence of the suggested rules to a common solution of the VIP and CFPP, which is the unique solution of a hierarchical variational inequality (HVI).

A New Approximation Method for Finding Zeros of Maximal Monotone Operators

One of the major problems in the theory of maximal monotone operators is to find a point in the solution set Zer( ), set of zeros of maximal monotone mapping . The problem of finding a zero of a maximal monotone in real Hilbert space has been investigated by many researchers. Rockafellar considered the proximal point algorithm and proved the weak convergence of this algorithm with the maximal monotone operator. Güler gave an example showing that Rockafellar’s proximal point algorithm does not converge strongly in an infinite-dimensional Hilbert space. In this paper, we consider an explicit method that is strong convergence in an infinite-dimensional Hilbert space and a simple variant of the hybrid steepest-descent method, introduced by Yamada. The strong convergence of this method is proved under some mild conditions. Finally, we give an application for the optimization problem and present some numerical experiments to illustrate the effectiveness of the proposed algorithm.

Convergence Theorems for the Variational Inequality Problems and Split Feasibility Problems in Hilbert Spaces

In this paper, we establish an iterative algorithm by combining Yamada’s hybrid steepest descent method and Wang’s algorithm for finding the common solutions of variational inequality problems and split feasibility problems. The strong convergence of the sequence generated by our suggested iterative algorithm to such a common solution is proved in the setting of Hilbert spaces under some suitable assumptions imposed on the parameters. Moreover, we propose iterative algorithms for finding the common solutions of variational inequality problems and multiple-sets split feasibility problems. Finally, we also give numerical examples for illustrating our algorithms.

Hybrid viscosity implicit scheme for variational inequalities over the fixed point set of an asymptotically nonexpansive mapping in the intermediate sense in Banach spaces

With the help of the generalized viscosity implicit method and hybrid steepest-descent method, we introduce an iterative scheme for approximating the solution of a variational inequality over the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense. Some strong convergence results for our proposed iterative scheme are established in the framework of Banach spaces. Applicability of our proposed method is shown in variational inclusion problem and convex minimization problem. We discuss some examples to demonstrate the numerical implementation and efficiency of our main results in comparison of other related results. Our results improve, extend and unify previously known results given in literature.

New projection methods for equilibrium problems over fixed point sets

In this paper, we introduce some new approximation projection algorithms for solving monotone equilibrium problems over the intersection of fixed point sets of demicontractive mappings. By combining subgradient projection methods and hybrid steepest descent methods, strong convergence of the algorithms to a solution is shown in a real Hilbert space. Some numerical illustrations and comparisons are also reported to show the efficiency and advantage of the proposed algorithms.

Perturbed iterative methods for a general family of operators: convergence theory and applications

ABSTRACT We study perturbations of a hybrid steepest descent method for locating common fixed points of an arbitrary pool of nonexpansive mappings. The difficulty of handling a possibly uncountable family is settled down to dealing with a countable family of auxiliary mappings associated to , in the sense that the approximate fixed points of provide the common fixed points of . Algorithms with strong convergence for solving the associated variational inequality problems are presented. Applications to convex minimization problems and convex feasibility problems are provided, together with numerical examples for comparisons of our algorithms with the existing ones.

Basin Mapping Method for Extracting Comparative Assessments of Protein Phase Behavior from In Vivo Measurements

To uncover sequence-encoded driving forces for and the mechanisms of protein-specific phase transitions, we require knowledge of the underlying phase diagram, the degree of supersaturation, and the distance from critical point. Recent technological developments [1] based on Distributed Amphifluoric FRET (DAmFRET) allow the quantitative tracking of phase transitions as a function of protein expression in yeast. These in vivo experiments afford unprecedented throughput while enabling quantitative comparisons of protein-specific phase behavior. For a given protein, the DAmFRET dataset provides 2-dimensional histograms of AmFRET values and expression levels. Here, we adapt advances from energy landscape theory, specifically the inherent structure formalism of Stillinger and coworkers [2], to uncover protein-specific free energy functionals that underlie the phase behavior observed in DAmFRET measurements. We have developed and deployed a hybrid steepest-descent basin mapping method that identifies and quantifies major expression level dependent basins using the 2-dimensional histograms extracted from DAmFRET measurements. This approach identifies; (i) basins, which are envelopes in the 2-dimensional histograms; (ii) the relative weights of basins, and (iii) the centers of basins as defined by the steepest descent mapping. Results from the basin mapping procedure are used to infer protein-specific free energy functionals and these free energy functionals are used to reconstruct phase diagrams. Our analysis enables quantitative comparisons of phase transitions encoded by different protein sequences in a model cellular system. 1. T. Khan et al., (2018), Molecular Cell, 71: 155-168. 2. F. H. Stillinger, Energy Landscapes, Inherent Structures, and Condensed Matter Phenomena, Princeton University Press, 2016.

On general implicit hybrid iteration method for triple hierarchical variational inequalities with hierarchical variational inequality constraints

In this paper, we study the triple hierarchical variational inequality problem (THVIP, for short) with the hierarchical variational inequality constraint for finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Based on Korpelevich's extragradient method, hybrid steepest-descent method, Mann's implicit iteration method and Halpern's iteration method, a general implicit hybrid iterative algorithm for solving this THVIP is proposed. Under mild conditions, the strong convergence of the iterative sequences generated by the algorithm is established. Our results improve and extend the corresponding results in the earlier and recent literature.

On perturbed hybrid steepest descent method with minimization or superiorization for subdifferentiable functions

For finding the minimum value of differentiable functions over a nonempty closed convex subset of a Hilbert space, the hybrid steepest descent method (HSDM) can be applied. In this work, we study perturbed algorithms in line with a generalized HSDM and discuss how some selections of perturbations enable us to increase the convergence speed. When we specialize these results to constrained minimization then the perturbations become bounded perturbations used in the superiorization methodology (SM). We show usefulness of the SM in studying the constrained convex minimization problem for subdifferentiable functions and proceed with the study of the computational efficiency of the SM compared with the HSDM. In the computational experiment comparing the HSDM with superiorization, the latter seems to be advantageous for the specific experiment.

Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems

In this paper, we introduce and investigate composite inertial gradient-based algorithms with a line-search process for solving a variational inequality problem (VIP) with a pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and a strictly pseudocontractive mapping in the framework infinite-dimensional Hilbert spaces. The proposed algorithms are based on an inertial subgradient–extragradient method with the line-search process, hybrid steepest-descent methods, viscosity approximation methods and Mann iteration methods. Under weak conditions, we prove strong convergence of the proposed algorithms to the element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.

Hybrid inertial subgradient extragradient methods for variational inequalities and fixed point problems involving asymptotically nonexpansive mappings

ABSTRACT In this paper, we introduce hybrid inertial subgradient extragradient algorithms with line-search process for solving a variational inequality problem (VIP) with pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. The proposed algorithms are based on inertial subgradient extragradient method with line-search process, hybrid steepest-descent method, and viscosity approximation method. Under mild conditions, we prove strong convergence of the proposed algorithms to an element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.

Convergence of hybrid viscosity and steepest-descent methods for pseudocontractive mappings and nonlinear Hammerstein equations

In this article, we first introduce an iterative method based on the hybrid viscosity approximation method and the hybrid steepest-descent method for finding a fixed point of a Lipschitz pseudocontractive mapping (assuming existence) and prove that our proposed scheme has strong convergence under some mild conditions imposed on algorithm parameters in real Hilbert spaces. Next, we introduce a new iterative method for a solution of a nonlinear integral equation of Hammerstein type and obtain strong convergence in real Hilbert spaces. Our results presented in this article generalize and extend the corresponding results on Lipschitz pseudocontractive mapping and nonlinear integral equation of Hammerstein type reported by some authors recently. We compare our iterative scheme numerically with other iterative scheme for solving non-linear integral equation of Hammerstein type to verify the efficiency and implementation of our new method.

Accelerated hybrid viscosity and steepest-descent method for proximal split feasibility problems

The purpose of this paper is to present an accelerated hybrid viscosity and steepest-descent method for solving proximal split feasibility problems (if solutions exist) in real Hilbert spaces. We obtain strong convergence of the sequence generated by our scheme under some suitable conditions on the parameters. In all our results in this work, our iterative schemes are proposed with a way of selecting the step sizes such that their implementation does not need any prior information about the bounded linear operator norm. Finally, we give numerical comparisons of our results with other established result to verify the efficiency and implementation of our new method.

Convergence and some control conditions of hybrid steepest-descent methods for systems of variational inequalities and hierarchical variational inequalities

Convergence and some control conditions of hybrid steepest-descent methods for systems of variational inequalities and hierarchical variational inequalities

A modified iterative method for capturing a common solution of split generalized equilibrium problem and fixed point problem

A new iterative method is proposed based on hybrid viscosity approximation method and hybrid steepest-descent method to approximate a common solution of a split generalized equilibrium problem and a fixed point problem of a finite collection of nonexpansive mappings. We prove that the sequences generated by the iterative schemeconverges strongly to a common solution of the above said problems. Applications and numerical results are also given to illustrate the effectiveness and superiority of the proposed algorithm.

Weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space

We know that variational inequality problem is very important in the nonlinear analysis. For a variational inequality problem defined over a nonempty fixed point set of a nonexpansive mapping in Hilbert space, the strong convergence theorem has been proposed by I. Yamada. The algorithm in this theorem is named the hybrid steepest descent method. Based on this method, we propose a new weak convergence theorem for zero points of inverse strongly monotone mapping and fixed points of nonexpansive mapping in Hilbert space. Using this result, we obtain some new weak convergence theorems which are useful in nonlinear analysis and optimization problem.

Hybrid steepest-descent viscosity methods for triple hierarchical variational inequalities with constraints of mixed equilibria and bilevel variational inequalities

Hybrid steepest-descent viscosity methods for triple hierarchical variational inequalities with constraints of mixed equilibria and bilevel variational inequalities

Hybrid steepest-descent methods for systems of variational inequalities with constraints of variational inclusions and convex minimization problems

Hybrid steepest-descent methods for systems of variational inequalities with constraints of variational inclusions and convex minimization problems