Simeon Reich’s research while affiliated with Technion – Israel Institute of Technology and other places

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Publications (501)


Trajectories {xk}k=030\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{x^k\}_{k=0}^{30}$$\end{document} generated by methods (77), (78) and (79)
Absolute errors {log‖xk‖}k=030\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\log \Vert x_k\Vert \}_{k=0}^{30}$$\end{document} obtained by methods (77), (78) and (79)
Regularity of the product of two relaxed cutters with relaxation parameters beyond two
  • Article
  • Publisher preview available

April 2025

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7 Reads

Numerical Algorithms

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Simeon Reich

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We study the product of two relaxed cutters having a common fixed point. We assume that one of the relaxation parameters is greater than two so that the corresponding relaxed cutter is no longer quasi-nonexpansive, but rather demicontractive. We show that if both of the operators are (weakly/linearly) regular, then under certain conditions, the resulting product inherits the same type of regularity. We then apply these results to proving convergence in the weak, norm and linear sense of algorithms that employ such products.

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Convergence of Infinite Products of Uniformly Locally Nonexpansive Mappings

February 2025

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6 Reads

The generic convergence of infinite products of nonexpansive mappings was established in a 1999 paper of ours. In the present paper, such results are extended to infinite products of uniformly locally nonexpansive mappings. In particular, the convergence of infinite products of uniformly locally contractive mappings, as well as its stability, are proved. Moreover, the Baire category approach and the porosity notion are used to show that most sequences of uniformly locally nonexpansive mappings are, in fact, uniformly locally contractive.


Figure 1: Trajectories {x k } 30 k=0 generated by methods (77), (78) and (79).
Figure 2: Absolute errors {log ∥x k ∥} 30 k=0 obtained by methods (77), (78) and (79).
Regularity of the Product of Two Relaxed Cutters with Relaxation Parameters Beyond Two

February 2025

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26 Reads

We study the product of two relaxed cutters having a common fixed point. We assume that one of the relaxation parameters is greater than two so that the corresponding relaxed cutter is no longer quasi-nonexpansive, but rather demicontractive. We show that if both of the operators are (weakly/linearly) regular, then under certain conditions, the resulting product inherits the same type of regularity. We then apply these results to proving convergence in the weak, norm and linear sense of algorithms that employ such products.


The behavior of σn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _n$$\end{document} with the stopping rule σn<10-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _n<10^{-4}$$\end{document}
The behavior of σn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _n$$\end{document} with the stopping rule σn<10-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _n<10^{-3}$$\end{document}
A Self-Adaptive Explicit Iterative Algorithm for Solving the Split Common Solution Problem

January 2025

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18 Reads

Mediterranean Journal of Mathematics

This paper aims to introduce a novel self-adaptive explicit iterative algorithm for solving the split common solution problem with monotone operator equations in real Hilbert spaces. This new approach does not employ resolvent operators nor does it use the norms of the bounded linear operators (transfer mappings) from the source space to the image spaces.


An efficient algorithm with double inertial steps for solving split common fixed point problems and an application to signal processing

January 2025

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11 Reads

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3 Citations

Computational and Applied Mathematics

The split common fixed point problem is an optimization challenge that involves finding an element within one fixed point set such that when transformed by a bounded linear operator, it belongs to another fixed point set. This problem falls under the category of inverse problems in mathematics. We present a novel self-adaptive algorithm based on double inertial steps for solving the split common fixed point problem for demicontractive mappings. We also establish a weak convergence theorem for our method. Furthermore, we also present some numerical experiments illustrating the convergence behavior and the efficiency of our proposed algorithm.


An efficient algorithm for solving the split mixed variational inequality problem in real Hilbert spaces

December 2024

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22 Reads

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas

The paper concerns the split mixed variational inequality problem with multiple output sets. In order to find a solution to this problem, we introduce an efficient algorithm with multiple inertial steps. Using simple techniques of variational analysis and imposing some mild conditions on the control parameters, we establish strong convergence of the sequences generated by our proposed algorithm.



Convergence of an inertial reflected-forward–backward splitting algorithm for solving monotone inclusion problems with application to image recovery

December 2024

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64 Reads

Journal of Computational and Applied Mathematics

We first propose a reflected-forward–backward splitting algorithm with two inertial effects for solving monotone inclusions and then establish that the sequence of iterates it generates converges weakly in a real Hilbert space to a zero of the sum of a set-valued maximal monotone operator and a single-valued monotone Lipschitz continuous operator. The proposed algorithm involves only one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration. One inertial parameter is non-negative while the other is non-positive. These features are absent in many other available inertial splitting algorithms in the literature. Finally, we discuss some problems in image restoration in connection with the implementation of our algorithm and compare it with some known related algorithms in the literature.


Three shrinking projection methods with multiple inertial effects for solving a class of split feasibility problems

November 2024

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18 Reads

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3 Citations

Journal of Fixed Point Theory and Applications

We introduce three new inertial shrinking projection algorithms with multiple inertial effects for solving split common solution problems with multiple output sets. We establish the convergence of the sequences generated by our proposed algorithms under some mild conditions on the control parameters. More precisely, we only require the boundedness of the coefficients of the inertial components. Moreover, our algorithms do not depend on the norms of the transfer mappings.


Top left: Case Ia; Top right: Case Ib; Bottom left: Case Ic; Bottom right: Case Id
Top left: Case IIa; Top right: Case IIb; Bottom left: Case IIc; Bottom right: Case IId
Numerical results
New iterative algorithms for solving split variational inclusions

November 2024

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97 Reads

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1 Citation

Journal of Global Optimization

In this paper we study a class of split variational inclusion (SVI) and regularized split variational inclusion (RSVI) problems in real Hilbert spaces. We discuss various analytical properties of the net generated by the RSVI and establish the existence and uniqueness of the solution to the RSVI. Using analytical properties of this net and under certain assumptions on the parameters and mappings associated with the SVI, we establish the strong convergence of the sequence generated by our proposed iterative algorithm. We also deduce another iterative algorithm by taking the regularization parameters to be zero in our proposed algorithm. We establish the weak convergence of the sequence generated by our new algorithm under certain assumptions. Moreover, we discuss two special cases of the SVI, namely the split convex minimization and the split variational inequality problems, and give several numerical examples.


Citations (55)


... Iyiola and Shehu (2022) [24] proposed a two-step inertial proximal point algorithm for convex minimization, establishing a non-asymptotic O(1/n) convergence rate. Recently, Thong et al. (2025) [25] introduced a double-inertial-step algorithm for split common fixed-point problems, demonstrating strong convergence with an application to signal processing. Motivated by the literature discussed above, we propose a new two-step inertial algorithm that incorporates two inertial parameters and exhibits enhanced convergence. ...

Reference:

A novel fixed-point based two-step inertial algorithm for convex minimization in deep learning data classification
An efficient algorithm with double inertial steps for solving split common fixed point problems and an application to signal processing
  • Citing Article
  • January 2025

Computational and Applied Mathematics

... Proximal split feasibility problems [11,24], split variational inequality problems [2,3], and split null point problems [4,30,37] are some special cases of the SVIP. These studied problems have been applied to some important fields such as intensitymodulated radiation therapy treatment planning [7,25,34] and data compression [10,18,26]. These applications have been deeply solved by many numerical algorithms, for example proximal algorithms [15,16,37,38], inertial proximal methods [22,28] and conjugate gradient methods [15,17,33,40] and the references therein. ...

New iterative algorithms for solving split variational inclusions

Journal of Global Optimization

... There are many variations in projection methods, and it is one of the most critical schemes among them. For the recent works, see [12,13], for instance. This method Symmetry 2024, 16, 483 2 of 10 has also been studied in the setting of complete geodesic space, and several convergence theorems were proved. ...

Cyclic projection methods for solving the split common zero point problem in Banach spaces
  • Citing Article
  • February 2024

Fixed Point Theory

... In 2016, Bose and et al. [7] introduced the notions of the x-distance in G-metric spaces and, also, some fixed point results by using the concept of the generalized weak contraction in complete G-metric spaces via the x-distance. For some related results with the generalized distance, see Kata et al. [20] and others [9], [8,16,21,25,26,31,32]. ...

Two existence results regarding strict contractions on metric spaces with graphs

The Journal of Analysis

... The final outcome of the game often depends on the choices made by each rational actor, and their choices are usually considered strategic because of the need to guess how the others would make decisions in the same scenario. In non-cooperative games, the participating actors tend to consider only how to obtain their own maximum benefit, but this does not mean that they are in opposition to each other [12]. ...

Multidimensional Evolution Effects on Non-Cooperative Strategic Games

... By using algorithms specifically designed for the SCSPMOS, one can effectively tackle many related problems. Some recent results for the SCSPMOS can be found in [8][9][10]. Some other related problems and their applications can be seen in [11][12][13][14][15]. ...

Two inertial hybrid projection algorithms for solving a class of split common solution problems
  • Citing Article
  • July 2024

Rendiconti del Circolo Matematico di Palermo

... where ‖ ⋅ ‖ 1 = ∑ | | and ‖ ‖ is the discrete TV regularization term. Several efforts have been made to improve the TV output, [4], [5], [6], [7], [8], [9]. ...

Iterative methods for solving monotone variational inclusions without prior knowledge of the Lipschitz constant of the single-valued operator

Numerical Algorithms

... In this section, our goal is to demonstrate the efficiency and robustness of the proposed strategy, denoted by (Algo3), in comparison with the Double Inertial Steps Into the Single Projection Method with Non-Monotonic Step Sizes for Solving Pseudomonotone Variational Inequalities [26], the Two Subgradient Extragradient Methods Based on the Golden Ratio Technique for Solving Variational Inequality Problems [19], and the Versions of the Subgradient Extragradient Method for Pseudomonotone Variational Inequalities [12], respectively denoted by (Algo1), (Algo2), and (Algo4). We used Matlab 2021a on a Dell Core i7 computer to conduct the numerical simulation. ...

Two subgradient extragradient methods based on the golden ratio technique for solving variational inequality problems

Numerical Algorithms

... By using algorithms specifically designed for the SCSPMOS, one can effectively tackle many related problems. Some recent results for the SCSPMOS can be found in [8][9][10]. Some other related problems and their applications can be seen in [11][12][13][14][15]. ...

Inertial Proximal Point Algorithms for Solving a Class of Split Feasibility Problems

Journal of Optimization Theory and Applications

... The inertial approach has the advantage of increasing the speed of convergence of iterative techniques. There has recently been a surge in interest in researching inertial type algorithms for solving optimization problems (see [4,5,11,21,27,32,38,39,51] and other references therein). Motivated by the above mentioned works in literature, we propose an inertial iterative method in a reflexive Banach space for approximating the common solution to a finite family of monotone inclusion problems and fixed point problems of a finite family of relatively nonexpansive mappings. ...

Two regularized inertial Tseng methods for solving inclusion problems with applications to convex bilevel programming
  • Citing Article
  • December 2023

Optimization