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arxiv: 2606.26792 · v1 · pith:CQVGSAITnew · submitted 2026-06-25 · 🧮 math.OC

Second-order Methods for Multiobjective Composite Optimization: Preconditioning Strategies, Subspace Variants and Inexact Solutions

Jian Chen , Xinmin Yang This is my paper

Pith reviewed 2026-06-26 03:22 UTC · model grok-4.3

classification 🧮 math.OC
keywords multiobjective optimizationcomposite optimizationproximal gradient methodsBarzilai-Borwein methodpreconditioningsubspace methodsinexact algorithmsconvergence analysis
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The pith

A preconditioned proximal Barzilai-Borwein method with subspace variants solves ill-conditioned multiobjective composite problems while proving global convergence in the nonconvex case.

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

The paper develops a new optimization algorithm that addresses slow convergence in proximal gradient methods for multiobjective problems where smooth parts are ill-conditioned. It combines objective-wise Barzilai-Borwein scaling to balance different objectives with a shared preconditioner that incorporates curvature information. A subspace variant reduces computations by solving the search direction in a low-dimensional space that decomposes into simple one-dimensional problems. The framework extends to structured nonsmooth terms and includes an inexact version whose convergence properties are analyzed without assuming convexity.

Core claim

We propose a preconditioned proximal Barzilai-Borwein method for multiobjective composite optimization that combines objective-wise Barzilai-Borwein scaling, which reduces imbalance among objectives, with a common preconditioner that captures shared curvature information. To avoid non-diagonal metric proximal mappings, we develop a subspace variant in which the search direction is computed in a two-dimensional subspace generated by a proximal-gradient-type direction and a projected historical direction. By constructing a conjugate basis with respect to the preconditioning metric, the subspace model decomposes into tractable one-dimensional subproblems. The framework is further extended to no

What carries the argument

The preconditioned proximal Barzilai-Borwein method with a two-dimensional subspace variant that uses a conjugate basis with respect to the preconditioning metric to reduce the model to one-dimensional subproblems.

If this is right

  • Global convergence holds for the inexact algorithm without convexity assumptions on the objectives.
  • The subspace construction yields explicit and inexpensive proximal evaluations for problems involving linear operators in the nonsmooth terms.
  • Objective-wise scaling mitigates imbalance among multiple objectives during iteration.
  • The method avoids the full cost of metric proximal mappings required by proximal Newton approaches while still using curvature information.
  • Linear convergence is recovered under the additional error-bound condition for suitable problem instances.

Where Pith is reading between the lines

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

  • The same preconditioning and subspace ideas could be tested on multiobjective problems outside the composite class if analogous shared curvature information is available.
  • Higher-dimensional subspace variants might be examined to trade off per-iteration cost against faster practical progress.
  • The error-bound condition is likely to hold in many sparse-regularization and multi-task learning instances, suggesting the linear rate is often attainable in practice.

Load-bearing premise

The linear convergence result depends on an error-bound condition whose validity is not guaranteed for arbitrary instances of the multiobjective composite problems considered.

What would settle it

A concrete multiobjective composite problem where the error-bound condition fails and the inexact algorithm exhibits sublinear convergence instead of the claimed linear rate.

Figures

Figures reproduced from arXiv: 2606.26792 by Jian Chen, Xinmin Yang.

Figure 1
Figure 1. Figure 1: Performance profiles and purity metric on [PITH_FULL_IMAGE:figures/full_fig_p033_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Numerical results in value space obtained on [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Performance profiles and purity metric on structured [PITH_FULL_IMAGE:figures/full_fig_p035_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical results in value space obtained on structured [PITH_FULL_IMAGE:figures/full_fig_p035_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Performance profiles and purity metric on linear constrained problems. [PITH_FULL_IMAGE:figures/full_fig_p036_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Numerical results in value space obtained on linear constrained problems QPd and QPe. [PITH_FULL_IMAGE:figures/full_fig_p037_6.png] view at source ↗
read the original abstract

Multiobjective composite optimization problems arise in sparse regularization, constrained multiobjective models, and multi-task learning, but their numerical solution remains challenging when the smooth components are ill-conditioned. Proximal gradient methods are inexpensive per iteration but may converge slowly, while proximal Newton and quasi-Newton methods exploit curvature information at the cost of evaluating expensive metric proximal mappings. To address these issues, we propose a preconditioned proximal Barzilai--Borwein method for multiobjective composite optimization. The method combines objective-wise Barzilai--Borwein scaling, which reduces imbalance among objectives, with a common preconditioner that captures shared curvature information. To avoid non-diagonal metric proximal mappings, we develop a subspace variant in which the search direction is computed in a two-dimensional subspace generated by a proximal-gradient-type direction and a projected historical direction. By constructing a conjugate basis with respect to the preconditioning metric, the subspace model decomposes into tractable one-dimensional subproblems. The framework is further extended to nonsmooth terms of the form $g_i(Ax)$ through a linear-operator-aware preconditioner, yielding explicit proximal evaluations via dual subproblems. We also analyze an inexact version based on relaxed descent conditions. We establish the global convergence of the inexact algorithm in the nonconvex setting and prove a linear convergence rate under an error-bound condition. Numerical experiments on ill-conditioned $\ell_1$-regularized, structured $\ell_1$-regularized, and linearly constrained problems demonstrate the effectiveness of the proposed method.

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

0 major / 3 minor

Summary. The manuscript develops a preconditioned proximal Barzilai-Borwein framework for multiobjective composite optimization. It combines objective-wise BB scaling with a shared preconditioner, introduces a two-dimensional subspace variant whose conjugate basis reduces the model to independent one-dimensional subproblems, extends the approach to structured nonsmooth terms g_i(Ax) via a linear-operator-aware preconditioner that yields explicit dual proximal evaluations, and analyzes an inexact variant. The central theoretical claims are global convergence of the inexact algorithm in the nonconvex setting and a linear rate under an error-bound condition. Numerical results are reported on ill-conditioned ℓ1-regularized, structured ℓ1-regularized, and linearly constrained instances.

Significance. If the stated convergence results are established, the work supplies a practical middle ground between cheap proximal-gradient steps and expensive metric proximal mappings for ill-conditioned multiobjective problems. The subspace decomposition via a conjugate basis with respect to the preconditioner and the explicit dual-subproblem construction for the g_i(Ax) case are concrete algorithmic contributions. The paper correctly qualifies the linear-rate result by the error-bound assumption rather than claiming it holds unconditionally.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'constructing a conjugate basis with respect to the preconditioning metric' is stated without an accompanying equation or brief definition; adding a short parenthetical or reference to the relevant subsection would improve immediate readability.
  2. [Section on inexact variant (likely §4 or §5)] The description of the inexact algorithm relies on 'relaxed descent conditions'; a single displayed inequality or numbered assumption in the main text would make the precise relaxation explicit for readers who skip the proofs.
  3. [Numerical Experiments] Numerical experiments section: while effectiveness is asserted on the three problem classes, the abstract supplies no information on the number of random instances, stopping tolerances, or how the preconditioner parameters were chosen; the full manuscript should include these details to support reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our work, the recognition of its practical contributions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claims are global convergence of an inexact algorithm in the nonconvex case and linear convergence under an explicit error-bound condition. These are standard results in composite optimization analysis and are qualified by the assumption rather than asserted unconditionally. No equations, fitted parameters, self-citations, or ansatzes in the abstract reduce any claimed rate to a definition or input by construction. The error-bound condition is a common external device whose validity is not guaranteed for arbitrary instances, consistent with the reader's assessment of score 2.0. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions for composite optimization and an error-bound condition; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption Objective functions possess a composite structure with smooth and nonsmooth components.
    Explicitly stated as the problem class addressed.
  • domain assumption An error-bound condition holds to obtain the linear convergence rate.
    Invoked for the linear-rate result in the abstract.

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discussion (0)

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Reference graph

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