# The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

@article{Deutsch2008TheRO, title={The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets}, author={Frank Deutsch and Hein Hundal}, journal={J. Approx. Theory}, year={2008}, volume={155}, pages={155-184} }

## 42 Citations

The rate of convergence for the cyclic projections algorithm II: Norms of nonlinear operators

- MathematicsJ. Approx. Theory
- 2006

A Randomized Algorithm for Generalized Accelerated Projection Method

- MathematicsIEEE Control Systems Letters
- 2021

A randomized accelerated projection algorithm is devised and its linear convergence rate is proved when all the convex sets are half-spaces in a finite-dimensional Euclidean space.

Restricted Normal Cones and the Method of Alternating Projections: Applications

- Mathematics
- 2013

The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of…

Nonconvex Notions of Regularity and Convergence of Fundamental Algorithms for Feasibility Problems

- MathematicsSIAM J. Optim.
- 2013

A notion of local subfirm nonexpansiveness with respect to the intersection is introduced for consistent feasibility problems of alternating projections and the Douglas--Rachford algorithm.

The optimal error bound for the method of simultaneous projections

- Mathematics, Computer ScienceJ. Approx. Theory
- 2017

Fixed Point Algorithms for Nonconvex Feasibility with Applications

- Mathematics
- 2014

Projection algorithms for solving (nonconvex) feasibility problems provide powerful and computationally efficient schemes for a wide variety of applications.
Algorithms as Alternating Projections…

Convergence analysis under consistent error bounds

- Computer Science, Mathematics
- 2020

One of the main results is that the convergence rate of several algorithms for feasibility problems can be expressed explicitly in terms of the underlying consistent error bound function.

Random Multi-Constraint Projection: Stochastic Gradient Methods for Convex Optimization with Many Constraints

- Computer Science, MathematicsArXiv
- 2015

The rate analysis and numerical experiments reveal that the algorithm using the polyhedral-set projection scheme is the most efficient one within known algorithms, providing new convergence rate benchmarks for stochastic first-order optimization with many constraints.

## References

SHOWING 1-10 OF 40 REFERENCES

The rate of convergence for the cyclic projections algorithm I: Angles between convex sets

- MathematicsJ. Approx. Theory
- 2006

The rate of convergence for the cyclic projections algorithm II: Norms of nonlinear operators

- MathematicsJ. Approx. Theory
- 2006

A Dual Approach to Constrained Interpolationfrom a Convex Subset of Hilbert Space

- Mathematics
- 1997

Many interesting and important problems of best approximationare included in (or can be reduced to) one of the followingtype: in a Hilbert spaceX, find the best approximationPK(x) to anyx?Xfrom the…

The rate of convergence in the method of alternating projections

- Mathematics
- 2010

A generalization of the cosine of the Friedrichs angle between two subspaces to a parameter associated to several closed subspaces of a Hilbert space is given. This parameter is used to analyze the…

Local Linear Convergence for Alternating and Averaged Nonconvex Projections

- MathematicsFound. Comput. Math.
- 2009

It is proved that von Neumann’s method of “alternating projections” converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity.

The strong conical hull intersection property for convex programming

- MathematicsMath. Program.
- 2006

It is established that the strong CHIP of intersecting sets of constraints is the key characterizing property for optimality and strong duality of convex programming problems.

On the convergence of von Neumann's alternating projection algorithm for two sets

- Mathematics
- 1993

We give several unifying results, interpretations, and examples regarding the convergence of the von Neumann alternating projection algorithm for two arbitrary closed convex nonempty subsets of a…

Error bounds for the method of alternating projections

- MathematicsMath. Control. Signals Syst.
- 1988

The method of alternating projections produces a sequence which converges to the orthogonal projection onto the intersection of the subspaces, and the sharpest known upper bound for more than two subspaced is obtained.

Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization

- MathematicsMath. Program.
- 1999

It is shown that the standard constraint qualification from convex analysis implies bounded linear regularity, which in turn yields the strong conical hull intersection property.

Conical open mapping theorems and regularity

- Mathematics
- 1999

Suppose $T$ is a continuous linear operator between two Hilbert spaces $X$ and $Y$ and let $K$ be a closed convex nonempty cone in $X$. We investigate the possible existence of $\delta > 0$ such that…