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

By Heinz H. Bauschke, Patrick L. Combettes. Mathematics of Operations Research, 2001

479 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/fb8a34027e7e99064fd71f46a2ebbca6b5a7afa2

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Citations in the corpus

A strong convergence result involving an inertial forward–backward algorithm for monotone inclusions (2017, background): “(see, for example, [5,33,39,40,42,43,57] and the references contained therein for more details.”
Construction of best Bregman approximations in reflexive Banach spaces (2003, methodology): “The method presented in this paper extends a fixed point algorithmic framework proposed in [3] from metric best approximation in Hilbert spaces to Bregman best approximation in reflexive Banach spaces.”
The Adaptive Projected Subgradient Method over the Fixed Point Set of Strongly Attracting Nonexpansive Mappings (2006, unknown): “A mapping T :ℋ → ℋ is called quasi-nonexpansive [ 4 36 37 ] if Fix( T ) ≠ ∅ and”
Iterative selection methods for common fixed point problems (2006, background): “8) goes back at least to [13] (see also [3,4] for examples of quasi-nonexpansive operators).”
Gradient methods with selection technique for the multiple-sets split feasibility problem (2020, background): “2 ([44]): Let H be a real Hilbert space and {xn} a sequence in H such that there exists a nonempty closed set ∈ H satisfying (i) For every z ∈ , limn→∞ ‖xn − z‖ exists; (ii) ωw(xn) ⊂ .”
Equilibrium programming in Hilbert spaces (2005, background): “Then T (i) belongs to the class T [3] if (∀(x, y) ∈ H × FixT ) 〈y − Tx | x− Tx〉 ≤ 0; (ii) is nonexpansive if (∀(x, y) ∈ H2) ‖Tx− Ty‖ ≤ ‖x− y‖; (iii) is firmly nonexpansive if (∀(x, y) ∈ H2) ‖Tx− Ty‖2 ≤ 〈Tx− Ty | x− y 〉; (iv) is α-averaged for some α ∈ ]0, 1[ if T = (1 − α) Id+αR for some nonexpansiv”
Bregman Monotone Optimization Algorithms (2003, background): “If X is Hilbertian, f = w, and only single-valued operators are considered, then B reverts to the class T of operators introduced in [9] and further investigated in this context in [39, 40].”
Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space (2011, background): “These modications originate in the work of Haugazeau [14], which was successfully generalized and extended in recent papers by Combettes [9], Solodov and Svaiter [24], Bauschke and Combettes [2, 3], and by Burachik, Lopes and Svaiter [5].”