Bregman Monotone Optimization Algorithms

By Heinz H. Bauschke, Jonathan Michael Borwein, Patrick L. Combettes. SIAM Journal of Control and Optimization, 2003

330 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/bc65313ebba4da12702ecc3fa3eb3898450fc460

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Citations in the corpus

Optimal Stochastic Approximation Algorithms for Strongly Convex Stochastic Composite Optimization I: A Generic Algorithmic Framework (2012, background): “2) The prox-function V (·, ·) is also called the Bregman’s distance, which was initially studied by Bregman [6] and later by many others (see [1, 2, 17, 39] and references therein).”
A projection method for solving nonlinear problems in reflexive Banach spaces (2011, background): “See [5, 26] for more information on BFNE operators.”
Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization (2013, background): “It is well-known that using a generalized distance generating function, instead of the usual Euclidean distance function, would lead to algorithms that can be adjusted to the geometry of the feasible set and/or efficient solutions of the projection [2,3,5,21,26, 38].”
An iterative regularization method for the solution of the split feasibility problem in Banach spaces (2008, unknown): “[1, 2, 6, 8, 10, 12–15, 18, 23, 26, 31, 32, 38, 43, 44, 46].”
Convex Functions: Constructions, Characterizations and Counterexamples (2010, background): “Essentially strictly convex and essentially smooth functions for infinite-dimensional Banach spaces were introduced in [35] as a precursor to more algorithmic work [36] discussed in Section 7.”
Forward-Backward Splitting with Bregman Distances (2015, unknown): “2 [5, 6] LetX be a reflexive real Banach space, letX ∗ be the topological dual space ofX , let 〈·, ·〉 be the duality pairing between X and X , let f : X → ]−∞,+∞] be a lower semicontinuous convex function that is Gâteaux differentiable on int dom f 6= ∅, let f ∗ : X ∗ → ]−∞,+∞] : x 7→ supx∈X (〈x , x”
Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces (2010, background): “Recall that the resolvent of A, denoted by ResfA : X → 2 , is defined as follows [5]:”
An optimal method for stochastic composite optimization (2011, background): “…the so-called mirror descent algorithm that can be advantageous over the subgradient descent method when X is not an Euclidean ball by using a prox-function (also called Bregman’s distance, which was studied by Bregman [7] and many others, see for example, [1,2,15,49] and references therein).”
Construction of best Bregman approximations in reflexive Banach spaces (2003, methodology): “The following proposition supplies some examples; it also introduces key notation and denitions. Proposition 2.7 ([ 2, Section 3 ]).”
Right Bregman Nonexpansive Operators in Banach Spaces (2012, background): “Some left classes, so to speak, have already been studied quite intensively [4, 8, 29, 35].”
The Complexity of Large-scale Convex Programming under a Linear Optimization Oracle (2013, background): “More speciﬁcally, for a given strongly convex function (3.15) Note that V ( y ) is often referred to as the Bregman distance (from y to c v ) in the literature, which was initially studied by [8] and later by many others (see [2,4,33,37] and references therein).”