Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization

By Dan Butnariu, Alfredo Noel Iusem. 2000

391 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/fd55aedbab3c0a8fee3efed0901b07b9a8f061bd

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In Computational Mechanics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Bregman Monotone Optimization Algorithms (2003, background): “Extensions to Hilbert [16, 18, 58] and Banach spaces [1, 8, 19, 21, 22, 23, 24, 25, 53, 71] have also been considered more recently.”
How to Compare Different Loss Functions and Their Risks (2007, background): “Although this concept, in an equivalent formulation, has already been introduced in 1966 (see [23], [17]) almost nothing that is useful for us, seems to be known (see however [5, 35] and the references therein for some general information on the modulus).”
Construction of best Bregman approximations in reflexive Banach spaces (2003, neutral): “, [9], (∀t ∈ ]0,+∞[) inf { D(u, x) | u ∈ dom f and ‖u− x‖ = t } > 0.”
On Uniform Convexity, Total Convexity and Convergence of the Proximal Point and Outer Bregman Projection Algorithms in Banach Spaces (2003, background): “The notion termed in [13] sequential consistency occurred first in [20] was also studied in [10] and [11].”
A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators (2007, neutral): “(34) • In the case p ∈ (1, 2], (34) obviously holds with C̄2 := √ ρ/αmax + ‖u0 − u†‖ and (33) follows from the proof of corollary (ii) on page 192 in [10] (see also [11]).”
Inexact Proximal Point Methods for Equilibrium Problems in Banach Spaces (2007, methodology): “First, we introduce the Bregman distance associated with a convex and Gâteaux differentiable function g : B → , together with some related notations, taken from [9].”
Right Bregman Nonexpansive Operators in Banach Spaces (2012, background): “Examples of totally convex functions can be found, for instance, in [11, 18, 19].”
Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces (2010, background): “Examples of totally convex functions can be found, for instance, in [11, 13].”
A Characterization of Bregman Firmly Nonexpansive Operators Using a New Monotonicity Concept (2011, background): “See, for instance, [2, 3, 4, 5, 6, 13, 12, 7, 14, 15, 17, 20, 21, 22, 23, 24] and the references therein.”
Update or wait: How to keep your data fresh (2016, neutral): “Therefore , the Karush-Kuhn-Tucker (KKT) theorem for infinite dimensional space [30], [31] and the calculus of variations are required in the analysis.”
Convex Functions: Constructions, Characterizations and Counterexamples (2010, background): “For studies of Bregman distance and their fundamental importance in optimization and convex feasibility problems, see [129, 140, 147] and the references therein.”
Asymptotic Behavior of Relatively Nonexpansive Operators in Banach Spaces (2001, methodology): “Examples of such operators can be found in [ 5 ].”
INEXACT VERSIONS OF PROXIMAL POINT AND AUGMENTED LAGRANGIAN ALGORITHMS IN BANACH SPACES (2001, methodology): “problems in infinite dimensional spaces with a possibly infinite number of constraints, introduced in ([13], 3.”