Forcing strong convergence of proximal point iterations in a Hilbert space

By M. Solodov, Benar Fux Svaiter. Mathematical programming, 2000

407 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/14e08f784ab8fa758eec060feb6020356cd68996

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Citations in the corpus

A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping (2009, methodology): “It follows that from Theorem 3.2 and 3.3 ,w e can replace by using the CQ-hybrid method (see [16, 17 , 19]) for finding a common element of fixed point problems, equilibrium problems.”
Construction of best Bregman approximations in reflexive Banach spaces (2003, methodology): “We develop a conceptual fixed point method for finding the D-projection of x0 ∈ int dom f onto a closed convex set C ⊂ X in the spirit of a method initially proposed by Haugazeau for metric projections in Hilbert spaces [18] and further studied in this context in [3, 13, 16, 19, 22].”
A New Algorithm for Solving The Multiple-Sets Split Feasibility Problem in Banach Spaces (2014, methodology): “Our modification is mainly based on an idea in the Nakajo-Takahashi’s paper [13] for finding a fixed point of a nonexpansive mapping T in Hilbert spaces (see also [2, 18]).”
Weak and Strong Convergence Theorems for Maximal Monotone Operators in a Banach Space (2004, methodology): “This is a generalization of Solodov and Svaiter [ 22 ].”
A Weak-to-Strong Convergence Principle for Fejé-Monotone Methods in Hilbert Spaces (2001, methodology): “A result closely related to Corollary 6.2(ii) was obtained via a different analysis in Solodov and Svaiter (2000).”
A new proximal point iteration that converges weakly but not in norm (2005, background): “(See [4] and [17] for recent modifications of the Proximal Point Algorithm that produce norm convergent sequences.”
Weak Convergence of a Relaxed and Inertial Hybrid Projection-Proximal Point Algorithm for Maximal Monotone Operators in Hilbert Space (2003, background): “Taking σ > 0, αk ≡ 0, and ρk ≡ 1, one recovers the Hybrid Projection-Proximal Point Algorithm introduced in [23] (see also [24]), whose main feature is the fixed relative error tolerance given by (2.”
Convex Optimization in Normed Spaces: Theory, Methods and Examples (2015, background): “Some variations of the proximal algorithm also converge strongly (see, for instance, [98]).”
Two Strong Convergence Theorems for a Proximal Method in Reflexive Banach Spaces (2010, methodology): “We mention, in particular, the seminal papers [41, 21, 5, 6].”
Solving variational inequalities involving nonexpansive type mappings (2008, unknown): “Recently, Benavides, Acedo and Xu [2], Kamimura and Takahashi [8], Kim and Xu [9], Mainge [13], Nakajo [17] and Solodov and Svaiter [21] modified proximal point algorithm to generate strongly convergent sequences.”