The Moreau envelope function and proximal mapping in the sense of the Bregman distance

Kan, Chao; Song, Wen

doi:10.1016/j.na.2011.07.031

Cited by 26 publications

(45 citation statements)

References 16 publications

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“…. We complement the results of [34] by stating equivalent characterizations of relative prox-boundedness. To this end we need the following lemma which is analogue to [54,Exercise 1.14].…”

Section: Definition Properness and Continuitymentioning

confidence: 99%

Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative Prox-Regularity

Laude

Ochs

Cremers

2020

J Optim Theory Appl

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We systematically study the local single-valuedness of the Bregman proximal mapping and local smoothness of the Bregman-Moreau envelope under relative prox-regularity, an extension of prox-regularity for nonconvex functions which has been originally introduced by Poliquin and Rockafellar. Although, we focus on the left Bregman proximal mapping, a translation result yields analogue (and partially sharp) results for the right Bregman proximal mapping. The class of relatively prox-regular functions significantly extends the recently considered class of relatively hypoconvex functions. In particular, relative prox-regularity allows for functions with possibly nonconvex domain. Moreover, as a main source of examples, in analogy to the classical setting, we introduce relatively amenable functions by invoking the recently proposed notion of smooth adaptability or relative smoothness. Exemplarily we apply our theory to interpret joint alternating Bregman minimization with proximal regularization, locally, as a Bregman proximal gradient algorithm.Keywords: Bregman-Moreau envelope · Bregman proximal mapping · prox-regularity · amenable functions 2000 Mathematics Subject Classification: 49J52 · 65K05 · 65K10 · 90C26

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“…. We complement the results of [34] by stating equivalent characterizations of relative prox-boundedness. To this end we need the following lemma which is analogue to [54,Exercise 1.14].…”

Section: Definition Properness and Continuitymentioning

confidence: 99%

Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative Prox-Regularity

Laude

Ochs

Cremers

2020

J Optim Theory Appl

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“…Fix anx ∈ X . Then, by assumption and the fact that Arg min F γ = X , there exist c and > 0 so that (18) holds whenever x −x < . Define…”

Section: Further Properties Of the Forward-backward Envelopementioning

confidence: 98%

“…Hence, F γ is just a generalized Moreau envelope that uses a suitable Bregman distance in place of the square of Euclidean distance. We refer the readers to [6,18] (i) F γ is continuously differentiable for any γ ∈ (0, 1 L ), with its gradient given by ∇F γ (x) = γ −1 (I − γ∇ 2 f (x))(x − prox γP (x − γ∇f (x))).…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Further properties of the forward–backward envelope with applications to difference-of-convex programming

Liu

Pong

2017

Comput Optim Appl

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In this paper, we further study the forward-backward envelope first introduced in [28] and [30] for problems whose objective is the sum of a proper closed convex function and a twice continuously differentiable possibly nonconvex function with Lipschitz continuous gradient. We derive sufficient conditions on the original problem for the corresponding forward-backward envelope to be a level-bounded and Kurdyka-Lojasiewicz function with an exponent of 1 2 ; these results are important for the efficient minimization of the forward-backward envelope by classical optimization algorithms. In addition, we demonstrate how to minimize some difference-of-convex regularized least squares problems by minimizing a suitably constructed forward-backward envelope. Our preliminary numerical results on randomly generated instances of large-scale 1−2 regularized least squares problems [37] illustrate that an implementation of this approach with a limited-memory BFGS scheme usually outperforms standard first-order methods such as the nonmonotone proximal gradient method in [35].

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“…Actually, ← P λTV is a maximal monotone operator. See [30,43] for more details on Bregman proximity operator and the corresponding Bregman-Moreau envelopes.…”

Section: Bregman Proximity Operatorsmentioning

confidence: 99%

A Characterization of the Domain of Beta-Divergence and Its Connection to Bregman Variational Model

Woo

2017

Entropy

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Abstract:In image and signal processing, the beta-divergence is well known as a similarity measure between two positive objects. However, it is unclear whether or not the distance-like structure of beta-divergence is preserved, if we extend the domain of the beta-divergence to the negative region. In this article, we study the domain of the beta-divergence and its connection to the Bregman-divergence associated with the convex function of Legendre type. In fact, we show that the domain of beta-divergence (and the corresponding Bregman-divergence) include negative region under the mild condition on the beta value. Additionally, through the relation between the beta-divergence and the Bregman-divergence, we can reformulate various variational models appearing in image processing problems into a unified framework, namely the Bregman variational model. This model has a strong advantage compared to the beta-divergence-based model due to the dual structure of the Bregman-divergence. As an example, we demonstrate how we can build up a convex reformulated variational model with a negative domain for the classic nonconvex problem, which usually appears in synthetic aperture radar image processing problems.

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The Moreau envelope function and proximal mapping in the sense of the Bregman distance

Cited by 26 publications

References 16 publications

Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative Prox-Regularity

Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative Prox-Regularity

Further properties of the forward–backward envelope with applications to difference-of-convex programming

A Characterization of the Domain of Beta-Divergence and Its Connection to Bregman Variational Model

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