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arxiv: 2604.12829 · v1 · submitted 2026-04-14 · 🧮 math.OC

Variable Bregman Majorization-Minimization algorithms for nonconvex nonsmooth optimization, with application to Poisson imaging

Maxence Adly , Alix Chazottes , Emilie Chouzenoux , Jean-Christophe Pesquet , Florent Sureau This is my paper

Pith reviewed 2026-05-10 14:44 UTC · model grok-4.3

classification 🧮 math.OC
keywords majorization-minimizationBregman divergencenonconvex optimizationnonsmooth optimizationKurdyka-Lojasiewicz propertyPoisson imagingimage reconstruction
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The pith

A variable Bregman majorization-minimization framework converges to critical points for nonconvex nonsmooth problems under the Kurdyka-Lojasiewicz property without requiring Lipschitz smoothness.

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

The paper introduces a unifying framework for nonconvex nonsmooth optimization that uses majorization-minimization with Bregman divergences allowed to vary across iterations. Surrogate functions are built from these divergences to majorize the objective, and the resulting algorithm's iterates converge to critical points when the objective satisfies the Kurdyka-Lojasiewicz property. This approach relaxes the standard requirement that the objective be Lipschitz smooth. A constructive method is given for designing variable Bregman majorants whose minimizers are tractable to compute, and the framework is shown to include existing techniques for Poisson data terms. The method is illustrated on PET image reconstruction using a nonconvex regularizer.

Core claim

A unifying Bregman-based majorization-minimization framework is introduced for nonconvex nonsmooth optimization. The framework leverages Bregman divergences, possibly varying across iterations, to construct tailored surrogate functions that majorize the objective. Convergence of the iterates to critical points is established under the Kurdyka-Lojasiewicz property, relaxing standard assumptions such as the Lipschitz smoothness of the nonconvex objective function. A constructive methodology is derived to build a broad class of variable Bregman majorants with tractable minimizers, encapsulating various existing majorization techniques, in particular those derived for Poisson data fidelity terms

What carries the argument

Variable Bregman majorants: surrogate functions derived from possibly iteration-dependent Bregman divergences that majorize the objective while admitting tractable minimization steps.

If this is right

  • The algorithm converges to critical points for a wider class of nonconvex nonsmooth objectives that lack Lipschitz continuity.
  • Existing majorization techniques for Poisson imaging inverse problems become special cases of the proposed framework.
  • The scheme enables solving nonconvex regularized inverse problems in imaging without smoothness assumptions on the objective.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Allowing the Bregman divergence to change at each step may yield faster practical convergence than fixed-surrogate methods on the same problems.
  • The constructive majorant design could be adapted to other data fidelity terms arising in different inverse problems.
  • The framework opens the possibility of deriving convergence guarantees for additional nonconvex imaging regularizers beyond the PET example.

Load-bearing premise

The objective function satisfies the Kurdyka-Lojasiewicz property and suitable variable Bregman majorants with tractable minimizers can be constructed for the given problem.

What would settle it

A concrete nonconvex nonsmooth problem that satisfies the Kurdyka-Lojasiewicz property for which the constructed variable Bregman majorant produces iterates that fail to converge to any critical point.

Figures

Figures reproduced from arXiv: 2604.12829 by Alix Chazottes, Emilie Chouzenoux, Florent Sureau, Jean-Christophe Pesquet, Maxence Adly.

Figure 1
Figure 1. Figure 1: For two slices (rows): reference phantom (1a/2a), EM reconstruction (1b/2b), and our proposed MM reconstructions using respectively quadratic Maj-8 (1c/2c), Log-0 Maj-5 (1d/2d) and Log-shift Maj-4 (1e/2e). For all methods, the reconstruction process was stopped after 15s. Blue lines show the mask contour [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Iterates relative distance over iterations (a) and computational time (b) using the different majorants and our nonconvex Geman-McClure regularization. An approximation of the limit point 𝑥 (∞) was obtained by running each algorithm ten times the number of iterations required to meet the stopping criterion. As shown in fig. 2, which displays convergence profiles obtained with each majorant, quadratic major… view at source ↗
Figure 3
Figure 3. Figure 3: Diagram representing all order relations (definition 5.1) established between majorants in section 5.2 (tightest ones on top) 32 [PITH_FULL_IMAGE:figures/full_fig_p032_3.png] view at source ↗
read the original abstract

In this work, we introduce a unifying Bregman-based majorization-minimization (MM) framework for solving nonconvex nonsmooth optimization problems. The proposed approach leverages Bregman divergences, possibly varying across iterations, to construct tailored surrogate functions that majorize the objective. We establish the convergence of the iterates of the resulting variable Bregman MM algorithm to critical points under the Kurdyka-Lojasiewicz property, relaxing standard assumptions such as the Lipschitz smoothness of the nonconvex objective function. We derive a constructive methodology to build a broad class of variable Bregman majorants with tractable minimizers. Our study encapsulates various existing majorization techniques, in particular those derived for Poisson data fidelity terms in imaging inverse problems. Numerical experiments on Positron Emission Tomography (PET) image reconstruction with a nonconvex regularizer showcase the practical feasibility of the proposed scheme.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces a unifying variable Bregman majorization-minimization (MM) framework for nonconvex nonsmooth optimization. Iteration-dependent Bregman divergences are used to construct surrogate functions that majorize the objective. Convergence of the resulting algorithm iterates to critical points is proved under the Kurdyka-Łojasiewicz property, relaxing the standard global Lipschitz smoothness assumption on the objective. A constructive methodology is derived for building a broad class of variable Bregman majorants whose minimizers are tractable; this class is shown to include existing majorization techniques for Poisson data-fidelity terms. The framework is applied to PET image reconstruction with a nonconvex regularizer and validated numerically.

Significance. If the convergence analysis and majorant construction hold, the work is significant for optimization and imaging applications. It unifies and extends MM methods by allowing variable Bregman surrogates without Lipschitz smoothness, provides an explicit construction recipe that recovers Poisson-specific techniques, and supplies reproducible numerical evidence on a practical inverse problem. These elements address a recurring practical need for flexible, provably convergent surrogates in nonconvex nonsmooth settings.

major comments (2)
  1. [§3.3, Theorem 3.4] §3.3, Theorem 3.4: the sufficient-decrease inequality (3.12) is stated for the variable Bregman surrogate, yet the proof only verifies the majorization property at the current iterate; it is not shown that the chosen variable parameter sequence remains admissible for all subsequent iterates without an additional uniform bound on the Bregman modulus.
  2. [§4.1, Eq. (4.7)] §4.1, Eq. (4.7): the explicit form of the variable Bregman majorant for the Poisson negative-log-likelihood is derived by linearizing the Hessian at the current point, but the resulting surrogate is claimed to be a global majorant; the verification that the remainder term is nonnegative for all x appears to rely on a local convexity argument that may fail when the current iterate is far from the minimizer.
minor comments (3)
  1. [§2.2] Notation for the variable Bregman divergence D_{φ_k} is introduced without an explicit statement of the dependence of φ_k on the iteration index k; a short clarifying sentence in §2.2 would remove ambiguity.
  2. [Numerical experiments] Figure 2 caption does not specify the number of independent runs or the precise stopping criterion used for the PET experiments; adding these details would improve reproducibility.
  3. [Introduction] The introduction cites several classical MM papers but omits recent variable-metric or adaptive-Bregman works (e.g., on proximal gradient with variable metrics); a brief comparison paragraph would strengthen the novelty claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [§3.3, Theorem 3.4] §3.3, Theorem 3.4: the sufficient-decrease inequality (3.12) is stated for the variable Bregman surrogate, yet the proof only verifies the majorization property at the current iterate; it is not shown that the chosen variable parameter sequence remains admissible for all subsequent iterates without an additional uniform bound on the Bregman modulus.

    Authors: We appreciate the referee's observation regarding the admissibility of the variable parameter sequence. In our framework the Bregman modulus is chosen at each iteration to ensure majorization holds at the current point. To guarantee that the sufficient-decrease property propagates along the entire sequence, we will add a mild standing assumption that the sequence of Bregman moduli remains uniformly bounded (which is satisfied on compact level sets under the KL property). We will revise the statement of Theorem 3.4 and its proof in §3.3 to explicitly verify that the chosen parameters remain admissible for all subsequent iterates under this bound. This is a partial revision. revision: partial

  2. Referee: [§4.1, Eq. (4.7)] §4.1, Eq. (4.7): the explicit form of the variable Bregman majorant for the Poisson negative-log-likelihood is derived by linearizing the Hessian at the current point, but the resulting surrogate is claimed to be a global majorant; the verification that the remainder term is nonnegative for all x appears to rely on a local convexity argument that may fail when the current iterate is far from the minimizer.

    Authors: We thank the referee for this remark. The Poisson negative-log-likelihood is strictly convex with positive-definite Hessian. The variable Bregman majorant is obtained by replacing the Hessian with its value at the current iterate inside the Bregman divergence; because the Poisson term is convex, the difference between the true function and this surrogate is nonnegative for all x in the positive orthant. We will insert a short lemma in §4.1 that proves the global majorization property directly from the convexity and the explicit integral form of the remainder, without relying on proximity to a minimizer. The revised manuscript will contain this verification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external KL property

full rationale

The paper's core convergence result for variable Bregman MM iterates to critical points is established under the standard external Kurdyka-Lojasiewicz property, without any reduction to internally fitted parameters, self-definitional surrogates, or load-bearing self-citations. The constructive methodology for building majorants (including for Poisson terms) is presented as independent of the target result, and the framework unifies existing techniques without renaming known patterns as new derivations. No equations or steps reduce by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the Kurdyka-Lojasiewicz property (a standard but non-trivial domain assumption in nonsmooth optimization) and on the existence of tractable variable Bregman majorants; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The objective function satisfies the Kurdyka-Lojasiewicz property
    Invoked to guarantee convergence of the variable Bregman MM iterates to critical points, as stated in the abstract.

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Forward citations

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