A Weak-to-Strong Convergence Principle for Fejé-Monotone Methods in Hilbert Spaces

@article{Bauschke2001AWC,
  title={A Weak-to-Strong Convergence Principle for Fej{\'e}-Monotone Methods in Hilbert Spaces},
  author={Heinz H. Bauschke and Patrick L. Combettes},
  journal={Math. Oper. Res.},
  year={2001},
  volume={26},
  pages={248-264}
}
We consider a wide class of iterative methods arising in numerical mathematics and optimization that are known to converge only weakly. Exploiting an idea originally proposed by Haugazeau, we present a simple modification of these methods that makes them strongly convergent without additional assumptions. Several applications are discussed. 

A strong convergence result involving an inertial forward–backward algorithm for monotone inclusions

Our interest in this paper is to prove a strong convergence result for finding a zero of the sum of two monotone operators, with one of the two operators being co-coercive using an iterative method

Strong Convergence in Hilbert Spaces via Γ-Duality

A primal-dual pair of problems generated via a duality theory introduced by Svaiter are analyzed and a different viewpoint for the weak-to-strong principle of Bauschke and Combettes is given and many results concerning weak and strong convergence of subgradient type methods are unified.

Strong Convergence Theorems by Shrinking Projection Methods for Class Mappings

We prove a strong convergence theorem by a shrinking projection method for the class of mappings. Using this theorem, we get a new result. We also describe a shrinking projection method for a

Two Strong Convergence Theorems for a Proximal Method in Reflexive Banach Spaces

Two strong convergence theorems for a proximal method for finding common zeroes of maximal monotone operators in reflexive Banach spaces are established. Both theorems take into account possible

A Strongly Convergent Method for Nonsmooth Convex Minimization in Hilbert Spaces

In this article, we propose a strongly convergent variant on the projected subgradient method for constrained convex minimization problems in Hilbert spaces. The advantage of the proposed method is

Strong convergence of a double projection-type method for monotone variational inequalities in Hilbert spaces

We introduce a projection-type algorithm for solving monotone variational inequality problems in real Hilbert spaces without assuming Lipschitz continuity of the corresponding operator. We prove that

Approximating solutions of the sum of a finite family of maximally monotone mappings in Hilbert spaces

The purpose of this paper is to study the method of approximation for a zero of the sum of a finite family of maximally monotone mappings using viscosity type Douglas–Rachford splitting algorithm and

Generalized Mann iterates for constructing fixed points in Hilbert spaces

...

References

SHOWING 1-10 OF 66 REFERENCES

Forcing strong convergence of proximal point iterations in a Hilbert space

This paper proposes a new proximal-type algorithm which does converge strongly, provided the problem has a solution, and solves proximal point subproblems inexactly, with a constructive stopping criterion introduced in [31].

A Survey of Examples of Convex Functions and Classifications of Normed Spaces

This paper represents a slightly extended version of the eponymous talk given at the VII Colloque Franco-Allemand d’Optimisation. My aim is to illustrate the tight connection between the sequential

Hilbertian convex feasibility problem: Convergence of projection methods

The classical problem of finding a point in the intersection of countably many closed and convex sets in a Hilbert space is considered. Extrapolated iterations of convex combinations of approximate

Krasnoselski-Mann Iterations in Normed Spaces

Abstract We provide general results on the behaviour of the Krasnoselski-Mann iteration process for nonexpansive mappings in a variety of normed settings.

An example concerning fixed points

An example is given of a contractionT defined on a bounded closed convex subset of Hilbert space for which ((I+T)/2)n does not converge.

Strong Convergence of Block-Iterative Outer Approximation Methods for Convex Optimization

The strong convergence of a broad class of outer approximation methods for minimizing a convex function over the intersection of an arbitrary number of convex sets in a reflexive Banach space is

Surrogate Projection Methods for Finding Fixed Points of Firmly Nonexpansive Mappings

We present methods for finding common fixed points of finitely many firmly nonexpansive mappings on a Hilbert space. At every iteration, an approximation to each mapping generates a halfspace

A mesh-independence principle for operator equations and their discretizations

A proof of the mesh-independence principle for a general class of operator equations and discretizations covers the earlier results and extends them well beyond the cases that have been considered before.

Monotone Operators and the Proximal Point Algorithm

For the problem of minimizing a lower semicontinuous proper convex function f on a Hilbert space, the proximal point algorithm in exact form generates a sequence $\{ z^k \} $ by taking $z^{k + 1} $
...