# Regular Sequences of Quasi-Nonexpansive Operators and Their Applications

@article{Cegielski2017RegularSO, title={Regular Sequences of Quasi-Nonexpansive Operators and Their Applications}, author={Andrzej Cegielski and Simeon Reich and Rafał Zalas}, journal={SIAM J. Optim.}, year={2017}, volume={28}, pages={1508-1532} }

In this paper we present a systematic study of regular sequences of quasi-nonexpansive operators in Hilbert space. We are interested, in particular, in weakly, boundedly and linearly regular sequences of operators. We show that the type of the regularity is preserved under relaxations, convex combinations and products of operators. Moreover, in this connection, we show that weak, bounded and linear regularity lead to weak, strong and linear convergence, respectively, of various iterative…

## 25 Citations

### Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings

- MathematicsJournal of Industrial & Management Optimization
- 2021

In this paper, for solving the variational inequality problem over the set of common fixed points of a finite family of demiclosed quasi-nonexpansive mappings in Hilbert spaces, we propose two new…

### Moduli of Regularity and Rates of Convergence for Fejér Monotone Sequences

- MathematicsIsrael Journal of Mathematics
- 2019

In this paper we introduce the concept of modulus of regularity as a tool to analyze the speed of convergence, including the linear convergence and finite termination, for classes of Fejér monotone…

### Convergence rates for boundedly regular systems

- MathematicsAdvances in Computational Mathematics
- 2021

This work establishes convergence rates for the system's trajectories when the nonexpansive operator satisfies an additional regularity property, the natural continuous-time analogue to discrete-time results obtained in Bauschke, Noll & Phan (2015) and Borwein, Li & Tam (2017) by using the same regularity properties.

### Asymptotic behaviour of a nonautonomous evolution equation governed by a quasi-nonexpansive operator

- MathematicsOptimization
- 2021

We study the asymptotic behaviour of the trajectory of a nonautonomous evolution equation governed by a quasi-nonexpansive operator in Hilbert spaces. We prove the weak convergence of the trajectory…

### Weak, strong and linear convergence of the CQ-method via the regularity of Landweber operators

- MathematicsOptimization
- 2019

ABSTRACT We consider the split convex feasibility problem in a fixed point setting. Motivated by the well-known CQ-method of Byrne [Iterative oblique projection onto convex sets and the split…

### Error Bounds for the Method of Simultaneous Projections with Infinitely Many Subspaces

- Mathematics
- 2020

### On Componental Operators in Hilbert Space

- MathematicsNumerical Functional Analysis and Optimization
- 2021

Abstract We consider a Hilbert space that is a product of a finite number of Hilbert spaces and operators that are represented by “componental operators” acting on the Hilbert spaces that form the…

### Finitely convergent iterative methods with overrelaxations revisited

- MathematicsJournal of Fixed Point Theory and Applications
- 2021

We study the finite convergence of iterative methods for solving convex feasibility problems. Our key assumptions are that the interior of the solution set is nonempty and that certain overrelaxation…

### Finitely Convergent Deterministic and Stochastic Methods for Solving Convex Feasibility Problems

- Mathematics, Computer Science
- 2019

This work combines finitely convergent methods for solving convex feasibility problems defined over a possibly infinite pool of constraints with a very general class of deterministic control sequences where it is required that sooner or later the authors encounter a violated constraint if one exists.

### Finitely convergent deterministic and stochastic iterative methods for solving convex feasibility problems

- Mathematics, Computer ScienceMathematical Programming
- 2021

This work combines finitely convergent methods for solving convex feasibility problems defined over a possibly infinite pool of constraints with a very general class of deterministic control sequences where it is required that sooner or later the authors encounter a violated constraint if one exists.

## References

SHOWING 1-10 OF 41 REFERENCES

### Application of Quasi-Nonexpansive Operators to an Iterative Method for Variational Inequality

- MathematicsSIAM J. Optim.
- 2015

The convergence in a norm of sequences generated by an iterative process for solving a variational inequality over the subset of fixed points of a quasi-nonexpansive operator $T$ defined on a Hilbert space is proved.

### Linear and strong convergence of algorithms involving averaged nonexpansive operators

- MathematicsJournal of Mathematical Analysis and Applications
- 2015

### Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings

- Mathematics
- 1973

### Moduli of Regularity and Rates of Convergence for Fejér Monotone Sequences

- MathematicsIsrael Journal of Mathematics
- 2019

In this paper we introduce the concept of modulus of regularity as a tool to analyze the speed of convergence, including the linear convergence and finite termination, for classes of Fejér monotone…

### PROPERTIES OF A CLASS OF APPROXIMATELY SHRINKING OPERATORS AND THEIR APPLICATIONS

- Mathematics
- 2014

In this paper we present an application of a class of quasi-nonexpansive operators to iterative methods for solving the following variational inequality problem VIP(F,C): Find ū ∈ C such that 〈F ū,…

### Weak convergence of the sequence of successive approximations for nonexpansive mappings

- Mathematics
- 1967

In a recent paper [4] F. E. Browder and W. V. Petryshyn have shown that if a nonexpansive mapping T: X—>X of a Hubert space X into itself is asymptotically regular and has a t least one fixed point…

### Iterative methods for solving variational inequalities in Euclidean space

- Mathematics
- 2015

In this paper, we investigate the convergence properties of an iterative method for solving variational inequalities in Euclidean space. We show that under certain assumptions the method can be…

### Methods for Variational Inequality Problem Over the Intersection of Fixed Point Sets of Quasi-Nonexpansive Operators

- Mathematics
- 2013

Many convex optimization problems in a Hilbert space ℋ can be written as the following variational inequality problem VIP (ℱ, C): Find such that for all z ∈ C, where C ⊂ ℋ is closed convex and ℱ: ℋ →…

### Outer approximation methods for solving variational inequalities in Hilbert space

- Mathematics
- 2017

Abstract In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set C. We…

### Weak convergence of infinite products of operators in Hadamard spaces

- Mathematics
- 2016

We first prove the weak convergence of iterates of both products and convex combinations of strongly nonexpansive operators in complete CAT(0) spaces. Then we establish the weak convergence of…