Bregman Forward-Backward Operator Splitting

Minh N. B\`ui; Patrick L. Combettes

REVIEW 2 major objections 4 minor 1 cited by

Bregman Forward-Backward Operator Splitting

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that Bregman forward-backward splitting converges weakly to a zero of A+B in reflexive Banach spaces under a new condition on B.

desk verdict The general convergence theorem is a reduction: the 'focusing' condition essentially assumes the key inclusion, but the corollaries and new minimization rates do the real work. read the letter →

arxiv 1908.03878 v3 pith:JXAEMHR5 submitted 2019-08-11 math.OC

Minh N. Bùi , Patrick L. Combettes This is my paper · ORCID

classification math.OC MSC 47H0547J2590C2565K10

keywords Bregmandistanceforward-backwardsplittingmonotoneinclusionreflexiveBanachspaceweakconvergenceLegendrefunctionratevariablemetric

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that the Bregman forward-backward splitting algorithm converges for monotone inclusions of the form $0\in Ax+Bx$ in reflexive Banach spaces, even when the Bregman kernel changes from iteration to iteration. Convergence of this algorithm was previously known for minimization problems, and the paper's result for general monotone inclusions is new even in Euclidean spaces. The proof rests on a single inequality, condition (1.1), imposed on the pair $(A,B)$ and the Bregman kernel, which is shown to unify older assumptions such as cocoercivity, the descent lemma, strong monotonicity, and angle-boundedness. In the convex minimization case, the method produces monotonically decreasing objective values with summable errors and an $o(1/n)$ rate.

What carries the argument

The load-bearing mechanism is the new inequality (1.1) coupled with a quasi-Fejér monotonicity estimate for the sequence of Bregman energies. For every $x,y\in C$, $z\in S$, and selections $y^*\in Ay$, $z^*\in Az$, condition (1.1) requires $\langle y-x,By-Bz\rangle\leq \kappa D_f(x,y)+\langle y-z,\delta_1(y^*-z^*)+\delta_2(By-Bz)\rangle$; the proof inserts this inequality into a four-point identity for Bregman distances (Proposition 2.3(ii) of [6]) to obtain the telescoping bound $\Delta_{n+1}\leq (1+\eta_n)\Delta_n-\theta_n$, where $\theta_n$ is a sum of nonnegative terms. Summability of $\theta_n$ then forces the desired limits, and the focusing condition converts weak cluster points into zeros of $A+B$.

What would settle it

A concrete test: take $X=\mathbb{R}^2$, $A$ the normal cone of a cone, $B$ a monotone but non-cocoercive operator, and $f=\frac12\|\cdot\|^2$; scan a grid over $x,y\in C$ and $z\in S$ and numerically evaluate the difference between the two sides of (1.1). If any triple gives a positive remainder, the theory's main assumption is violated. In the minimization setting, run Algorithm 3.8 on a convex problem satisfying (3.21) and check whether the objective error decays faster than $1/n$; a plateau at order $1/n$ would contradict the claimed $o(1/n)$ rate.

Watch

Extended reading notes

Core claim

The central finding is that the iteration $x_{n+1}=(\nabla f_n+\gamma_n A)^{-1}(\nabla f_n(x_n)-\gamma_n B x_n)$, with $A,B$ maximally monotone and $f_n$ a sequence of Legendre functions whose Bregman distances are compatible, converges weakly to a point in $S=\operatorname{zer}(A+B)$ whenever the orbit is bounded, its weak cluster points lie in $\operatorname{int}\operatorname{dom} f$, and a focusing condition holds (Theorem 2.8). The argument controls the Bregman energy $\Delta_n=D_{f_n}(z,x_n)+\delta_1\gamma_n\langle x_n-z,x_n^*+Bz\rangle$ through a quasi-Fejér inequality, and condition (1.1) is the place where the coupling between $A$, $B$, and the Bregman kernel enters. The paper shows that condition (1.1) covers most previously used assumptions, so the theorem recovers the Bregman proximal point algorithm, variable-metric forward-backward splitting, and the auxiliary-problem splitting method of [20] as special cases, while also handling examples that none of them can. For convex minimization, the objective error is summable and decays as $o(1/n)$, with a companion summability result for the Bregman distances between successive iterates (Theorem 3.9).

Load-bearing premise

The load-bearing premise is the new inequality (1.1): it must hold for all points in the domain and all solutions, with constants $\delta_1,\delta_2,\kappa$; if it does not, the energy decrease that drives the proof collapses.

Editorial extensions

If this is right

Convergence of Bregman forward-backward splitting now holds for the full monotone inclusion problem $0\in Ax+Bx$ in reflexive Banach spaces, not just for minimization, and the result is new even in Euclidean spaces.
Existing algorithms—the Bregman monotone proximal point method, variable-metric forward-backward splitting, and the auxiliary-problem splitting method of [20]—are recovered as special cases of one theorem, and new instances are constructed that none of those frameworks can handle.
In convex minimization the method yields a monotonically decreasing objective sequence, summable objective errors, and the rate $(\phi+\psi)(x_n)-\min(\phi+\psi)=o(1/n)$, together with $\sum_n n(D_{f_n}(x_{n+1},x_n)+D_{f_n}(x_n,x_{n+1}))<+\infty$.
Variational inequalities with non-cocoercive operators can be solved outside Hilbert spaces by choosing convenient Bregman kernels.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Condition (1.1) is global, but Proposition 2.1 shows it is implied by simpler structural assumptions such as cocoercivity, strong monotonicity plus Lipschitzness, or a Bregman descent inequality; a practical check would be to test these simpler sufficient conditions first, since (1.1) itself is hard to verify directly.
The $o(1/n)$ rate and summability of Bregman displacements suggest the method is first-order optimal in the same sense as gradient descent; whether an accelerated variant with $O(1/n^2)$ objective error exists in Bregman geometry is a natural open question.
The focusing condition is the least transparent assumption; Corollary 3.6 shows that in finite-dimensional spaces it can be replaced by simpler hypotheses, so a plausible conjecture is that focusing is automatic for essentially strictly convex Legendre kernels with open conjugate domain.
Changing the Bregman kernel each iteration opens a design axis: one could choose $f_n$ adaptively to improve conditioning or to make the prox of $A$ easy, and the theorem suggests such adaptivity costs only a mild multiplicative drift condition $D_{f_{n+1}}\le(1+\eta_n)D_{f_n}$.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: the abstract sells Theorem 2.8 as establishing convergence of Bregman forward-backward splitting for inclusions, but the general theorem is weaker than that. The focusing condition (Definition 2.7) is, via Proposition 2.5, logically equivalent to the desired cluster-point inclusion W(xn) ⊂ zer(A+B). So the theorem reduces convergence to exactly the hard analytic step, rather than proving it. That said, the paper is still worth serious attention, because the corollaries genuinely discharge that step case by case, and the minimization results are new even in Euclidean spaces.

What the paper does well: it assembles a clean framework around condition (1.1), which unifies several scattered assumptions (cocoercivity, the Renaud–Cohen condition, the Bregman descent lemma, etc.). Proposition 2.5 is the technical core, and I checked the key inequality chain — it is coherent. The finite-dimensional corollary (3.6) and the minimization rates o(1/n) with summable errors are concrete advances. Example 2.9 gives a real problem outside the reach of all four prior frameworks, which is the right way to justify a new condition.

The soft spots: the focusing condition is the main one. It is not a checkable assumption in the usual sense; verifying it in each application essentially means repeating a convergence proof via Lemma 3.1. If a reader only looks at Theorem 2.8 and the abstract, they will overestimate what is proven for the general inclusion problem. A second, minor concern is that condition (1.1) is a global two-point inequality involving constants that are not constructive. The paper shows it covers known cases, but it does not offer much guidance for genuinely new operators, so practical verification could be a barrier. The paper is also long and technical, but that is justified by the machinery.

Bottom line: this is a real and useful paper for researchers working on Bregman splitting and monotone operator theory in Banach spaces. The corollaries and rate results stand on their own, and the proofs appear sound. I would bring it to a reading group and cite the minimization part.

For peer review: yes, send it to a careful referee who knows the Legendre function literature. The referee should ask the authors to state plainly that Theorem 2.8 is a reduction, and to make the role of the focusing condition explicit in the abstract or introduction. That is a minor revision, not a rejection.

Referee Report

2 major / 4 minor

Summary. The paper studies forward-backward splitting with Bregman distances for finding zeros of the sum of two maximally monotone operators in reflexive Banach spaces. The iteration is (1.3), with Bregman kernels f_n that may vary with n, and the main structural hypothesis is the two-point condition (1.1) on the single-valued operator B. The central result, Theorem 2.8, asserts weak convergence of bounded orbits whose weak cluster points lie in int dom f, provided the algorithm is 'focusing' in the sense of Definition 2.7 and either the solution set is a singleton or the kernels satisfy an asymptotic gradient-strictness condition. The paper then shows that several known frameworks (Bregman proximal point, variable-metric forward-backward splitting, the Renaud-Cohen algorithm, and Nguyen's minimization algorithm) are recovered, and in the minimization setting proves monotone decrease of the objective, summability of the objective error, and the rate o(1/n) for the objective gap.

Significance. The paper contains a genuinely useful unification. The sufficient conditions for condition (1.1) in Proposition 2.1 cover cocoercivity, Lipschitz/angle-bounded operators, strong monotonicity, and descent-type inequalities, and the minimization results in Theorem 3.9 improve known rates from O(1/n) to o(1/n) while also giving summability of the objective error. The concrete corollaries and examples are carefully developed and do recover or extend existing methods, including new Euclidean-space results for monotone inclusions that are not minimization problems. The proofs are long but generally well structured, and the verification of assumptions in each corollary is explicit. The main weakness is that the general convergence theorem is stated in terms of a 'focusing' hypothesis that, because of Proposition 2.5, is logically equivalent to the desired conclusion that all weak cluster points are zeros; the unconditional content therefore sits in the corollaries rather than in Theorem 2.8 itself.

major comments (2)

[Definition 2.7 and Theorem 2.8] Proposition 2.5 proves that, for every sequence generated by Algorithm 2.4 and every z in S, the four antecedent conditions displayed in (2.36) all hold. Consequently, for any admissible orbit, the 'focusing' condition is equivalent to the inclusion W(x_n) subset of zer(A+B). The proof of Theorem 2.8 then obtains the central inclusion W(x_n) subset of S in one line by invoking hypothesis [c], rather than by proving it. This makes Theorem 2.8 a formal reduction rather than an unconditional convergence theorem, and the abstract's claim that the paper establishes the convergence of (1.3) is stronger than what the general theorem actually provides. The corollaries do discharge the focusing condition individually, so the paper is internally coherent, but the main theorem and the abstract should be reframed, for instance by stating explicitly that the new analytic work for the general inclusion problem is the verification of focusing in the corollaries, or by replacing [c] with a sufficient condition that is not equivalent to the conclusion.
[Problem 1.1, condition (1.1), and Theorem 2.8] Condition (1.1) is the main novel assumption, and Proposition 2.1 shows that it captures several useful known conditions. However, the paper does not prove that (1.1), together with the other hypotheses of Algorithm 2.4, implies W(x_n) subset of zer(A+B). The summability results of Proposition 2.5 are derived from (1.1), but none of them yields the cluster-point inclusion without the separate focusing assumption. Since focusing is equivalent to that inclusion, the general convergence result is conditional in a way that is not transparent from the abstract. The authors should either prove a general sufficient condition for focusing from (1.1), or explicitly state in the abstract and in Theorem 2.8 that the general monotone-inclusion convergence statement assumes the cluster-point property and that the unconditional results are those established in the subsequent corollaries and examples.

minor comments (4)

[Definition 2.7, display (2.36)] The four lines in the antecedent of the implication are separated by commas; since the implication concerns their conjunction, please make this explicit, for example by writing 'if all of the following four conditions hold' before the display.
[Proposition 2.5, proof after (2.26)] The step from convergence of (Delta_n) to convergence of (D_{f_n}(z,x_n)) uses the fact that the term delta_1 gamma_n <x_n - z, x*_n + Bz> is summable, which follows from the summability of <x_{n+1}-z, x*_{n+1}+Bz> and the boundedness of (gamma_n). This is implicit; adding one sentence would make the argument fully transparent.
[Example 3.11] The strong convergence claim in Example 3.11 is stated without proof or reference. If it is intended as a new result, please provide the argument or identify the theorem from which it follows; if it is an illustration, make that clear.
[Introduction, Section 1] The sentence 'the convergence of such an iterative process has not yet been established, even in finite-dimensional spaces with a single function f_n = f and constant parameters' should be qualified relative to the Renaud-Cohen algorithm (1.7), which treats a constant strongly convex kernel in Hilbert space; otherwise the novelty claim is easy to misread as stronger than intended.

Circularity Check

1 steps flagged · score 6.0 of 10

The general convergence theorem reduces the hard cluster-point inclusion to the focusing condition, which is logically equivalent to that inclusion once Proposition 2.5 is proved; the corollaries discharge the condition independently.

self definitional [Definition 2.7 (Eq. (2.36)) and Theorem 2.8, proof after Eq. (2.37)]
"Algorithm 2.4 is focusing if, for every z ∈ S, [(Dfn(z,xn))n∈N converges; the three displayed sums are finite] ⇒ W(xn)n∈N ⊂ zer(A+B). ... On the other hand, [c] and items (i)–(iv) in Proposition 2.5 yield W(xn)n∈N ⊂ zer(A+B)."

Proposition 2.5(i)–(iv) proves, for every orbit generated by Algorithm 2.4, exactly the four antecedent clauses appearing in Definition 2.7: convergence of (Dfn(z,xn)), summability of ⟨xn+1−z,γn−1(∇fn(xn)−∇fn(xn+1))−Bxn+Bz⟩, summability of (1−δ2)⟨xn−z,Bxn−Bz⟩, and summability of (1−κγn/α)Dfn(xn+1,xn). Therefore the named property ‘focusing’ is logically equivalent to the consequent W(xn)n∈N⊂zer(A+B). The proof of Theorem 2.8 uses this equivalence to obtain the inclusion, so the central analytic step of the general theorem is not derived but imported through the definition.

full rationale

The paper is an honest conditional analysis rather than a data-fitting exercise: condition (1.1) is an explicit assumption, and Proposition 2.1 shows that familiar hypotheses imply it; the rates in Theorem 3.9 are proved, not assumed. The one significant structural issue is the focusing condition in Definition 2.7. Because Proposition 2.5 establishes all four antecedent clauses of the focusing implication for every admissible orbit, Definition 2.7 collapses to the claim W(xn)n∈N⊂zer(A+B). Theorem 2.8 then cites this assumption to reach exactly that inclusion, so the general convergence theorem is a reduction rather than a proof of the hard inclusion. The applications and corollaries (e.g., Corollaries 3.2, 3.3, 3.5, 3.6 and Theorem 3.9) independently verify the focusing condition via Lemma 3.1 and problem-specific reasoning, which gives the paper genuine external content. Self-citations to [5], [6], [7] and [13] are used for established Bregman, Legendre-function, and monotone-operator lemmas, not as a substitute for the paper's convergence proof. The overall circularity is partial: the general theorem is definitionally close to its conclusion, but the concrete results are separately established.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper does not introduce new entities. The main assumptions are the new condition (1.1) with parameters delta1, delta2, kappa and the focusing condition. These are not fitted to data; they are hypotheses of the theorem. The paper is a theoretical extension of known methods.

free parameters (1)

delta1, delta2, kappa (in condition (1.1)) = not fitted; assumed to exist
These constants parameterize the key assumption (1.1) that the analysis relies on. They are not derived from data, but they are chosen by hand and the entire convergence result depends on their existence. This is more a modeling assumption than a fitted parameter, but it is listed as a free parameter because it is the main tunable condition of the paper.

assumptions (4)

standard math The space X is a reflexive real Banach space.
The entire theory of Legendre functions and monotone operators used here requires reflexivity for weak compactness and many duality arguments.
domain assumption A and B are maximally monotone operators, with B single-valued on int dom B.
Maximal monotonicity is needed to define resolvents and to use the Brézis-Haraux theorem and other tools. The single-valuedness of B is needed to define the explicit step.
ad hoc to paper The sequence of Bregman kernels (fn) lies in C_alpha(f) and satisfies the condition D_{f_{n+1}} <= (1+eta_n) D_{f_n} with summable (eta_n).
This is a technical assumption introduced specifically to handle varying Bregman distances. It is reasonable but not standard in earlier work, and the paper does not show it holds in all applications.
ad hoc to paper The focusing condition in Definition 2.7 holds.
This is a central assumption for the main convergence theorem. It is not a standard condition and requires separate verification in each application. The paper verifies it for several special cases, but it is an ad hoc structural assumption.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Bregman Forward-Backward Operator Splitting." pith.science (2026). https://pith.science/paper/JXAEMHR5

@misc{pith2026190803878,
  author       = {Pith},
  title        = {Pith review of: Bregman Forward-Backward Operator Splitting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JXAEMHR5}},
  note         = {Machine review of arXiv:1908.03878}
}

read the original abstract

We establish the convergence of the forward-backward splitting algorithm based on Bregman distances for the sum of two monotone operators in reflexive Banach spaces. Even in Euclidean spaces, the convergence of this algorithm has so far been proved only in the case of minimization problems. The proposed framework features Bregman distances that vary over the iterations and a novel assumption on the single-valued operator that captures various properties scattered in the literature. In the minimization setting, we obtain rates that are sharper than existing ones.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Warped Proximal Iterations for Monotone Inclusions
math.OC 2019-08 accept novelty 7.0 of 10

Warped resolvents generalize classical resolvents, and new convergence theorems unify and extend operator-splitting algorithms for monotone inclusions.

Reference graph

Works this paper leans on

24 extracted references · 20 canonical work pages · cited by 1 Pith paper

[1]

Baillon and G

J.-B. Baillon and G. Haddad, Quelques propri´ et´ es des op´ erateurs angle-born´ es etn-cycliquement mono- tones, Israel J. Math. , vol. 26, pp. 137–150, 1977

work page 1977
[2]

Banach, Th´ eorie des Op´ erations Lin´ eaires.Seminar

S. Banach, Th´ eorie des Op´ erations Lin´ eaires.Seminar . Matem. Univ . Warszawa, 1932

work page 1932
[3]

H. H. Bauschke, J. Bolte, and M. Teboulle, A descent lemma beyond Lipschitz gradient continuity: First- order methods revisited and applications, Math. Oper . Res., vol. 42, pp. 330–348, 2017

work page 2017
[4]

H. H. Bauschke and J. M. Borwein, Legendre functions and t he method of random Bregman projections, J. Convex Anal. , vol. 4, pp. 27–67, 1997

work page 1997
[5]

H. H. Bauschke, J. M. Borwein, and P. L. Combettes, Essent ial smoothness, essential strict convexity , and Legendre functions in Banach spaces, Commun. Contemp. Math. , vol. 3, pp. 615–647, 2001

work page 2001
[6]

H. H. Bauschke, J. M. Borwein, and P. L. Combettes, Bregma n monotone optimization algorithms, SIAM J. Control Optim. , vol. 42, pp. 596–636, 2003

2003
[7]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spa ces, 2nd ed. Springer , New York, 2017

work page 2017
[8]

H. H. Bauschke, M. N. Dao, and S. B. Lindstrom, Regularizi ng with Bregman–Moreau envelopes, SIAM J. Optim., vol. 28, pp. 3208–3228, 2018

work page 2018

Show all 24 references

[9]

J. M. Borwein and J. D. Vanderwerff, Convex Functions: Constructions, Characterizations and C ounterex- amples. Cambridge University Press, 2010

2010
[10]

Bourbaki, Espaces V ectoriels T opologiques, Chapitres 1 ` a 5

N. Bourbaki, Espaces V ectoriels T opologiques, Chapitres 1 ` a 5. Masson, Paris, 1981. English translation: T opological V ector Spaces, Chapters 1–5. Springer-Verlag, New York, 1987

1981
[11]

Br´ ezis and A

H. Br´ ezis and A. Haraux, Image d’une somme d’op´ erateurs monotones et applications, Israel J. Math. , vol. 23, pp. 165–186, 1976. 20

1976
[12]

Censor and S

Y. Censor and S. A. Zenios, Parallel Optimization – Theory, Algorithms and Applicatio ns. Oxford University Press, New York, 1997

1997
[13]

P. L. Combettes and Q. V. Nguyen, Solving composite mono tone inclusions in reﬂexive Banach spaces by constructing best Bregman approximations from their Kuh n-T ucker set, J. Convex Anal. , vol. 23, pp. 481–510, 2016

2016
[14]

P. L. Combettes and B. C. V˜ u, Variable metric quasi-Fej´ er monotonicity ,Nonlinear Anal., vol. 78, pp. 17–31, 2013

2013
[15]

P. L. Combettes and B. C. V˜ u, Variable metric forward-b ackward splitting with applications to monotone inclusions in duality , Optimization, vol. 63, pp. 1289–1318, 2014

2014
[16]

Frecon, S

J. Frecon, S. Salzo, and M. Pontil, Bilevel learning of t he group lasso structure, Adv. Neural Inform. Process. Syst., vol. 31, pp. 8301–8311, 2018

2018
[17]

Mercier , T opics in Finite Element Solution of Elliptic Problems (Lectures on Mathematics, no

B. Mercier , T opics in Finite Element Solution of Elliptic Problems (Lectures on Mathematics, no. 63). Tata Institute of Fundamental Research, Bombay , 1979

1979
[18]

Q. V. Nguyen, Forward-backward splitting with Bregman distances, Vietnam J. Math. , vol. 45, pp. 519– 539, 2017

2017
[19]

Ortiz-Jim´ enez, M

G. Ortiz-Jim´ enez, M. El Gheche, E. Simou, H. Petric Mar eti´ c, and P. Frossard, Forward-backward splitting for optimal transport based problems, Proc. Intl. Conf. Acoust., Speech, Signal Process. , pp. 5405–5409, 2020

2020
[20]

Renaud and G

A. Renaud and G. Cohen, An extension of the auxiliary pro blem principle to nonsymmetric auxiliary operators, ESAIM Control Optim. Calc. V ar ., vol. 2, pp. 281–306, 1997

1997
[21]

R. T. Rockafellar , Local boundedness of nonlinear , mon otone operators, Michigan Math. J. , vol. 16, pp. 397–407, 1969

1969
[22]

R. T. Rockafellar , On the maximality of sums of nonlinea r monotone operators, Trans. Amer . Math. Soc., vol. 149, no. 1, pp. 75–88, 1970

1970
[23]

Salzo, The variable metric forward-backward splitt ing algorithm under mild differentiability assump- tions, SIAM J

S. Salzo, The variable metric forward-backward splitt ing algorithm under mild differentiability assump- tions, SIAM J. Optim. , vol. 27, pp. 2153–2181, 2017

2017
[24]

Z˘ alinescu, Convex Analysis in General V ector Spaces

C. Z˘ alinescu, Convex Analysis in General V ector Spaces. World Scientiﬁc Publishing, River Edge, NJ, 2002. 21

2002

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Baillon and G

J.-B. Baillon and G. Haddad, Quelques propri´ et´ es des op´ erateurs angle-born´ es etn-cycliquement mono- tones, Israel J. Math. , vol. 26, pp. 137–150, 1977

work page 1977

[2] [2]

Banach, Th´ eorie des Op´ erations Lin´ eaires.Seminar

S. Banach, Th´ eorie des Op´ erations Lin´ eaires.Seminar . Matem. Univ . Warszawa, 1932

work page 1932

[3] [3]

H. H. Bauschke, J. Bolte, and M. Teboulle, A descent lemma beyond Lipschitz gradient continuity: First- order methods revisited and applications, Math. Oper . Res., vol. 42, pp. 330–348, 2017

work page 2017

[4] [4]

H. H. Bauschke and J. M. Borwein, Legendre functions and t he method of random Bregman projections, J. Convex Anal. , vol. 4, pp. 27–67, 1997

work page 1997

[5] [5]

H. H. Bauschke, J. M. Borwein, and P. L. Combettes, Essent ial smoothness, essential strict convexity , and Legendre functions in Banach spaces, Commun. Contemp. Math. , vol. 3, pp. 615–647, 2001

work page 2001

[6] [6]

H. H. Bauschke, J. M. Borwein, and P. L. Combettes, Bregma n monotone optimization algorithms, SIAM J. Control Optim. , vol. 42, pp. 596–636, 2003

2003

[7] [7]

H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spa ces, 2nd ed. Springer , New York, 2017

work page 2017

[8] [8]

H. H. Bauschke, M. N. Dao, and S. B. Lindstrom, Regularizi ng with Bregman–Moreau envelopes, SIAM J. Optim., vol. 28, pp. 3208–3228, 2018

work page 2018