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arxiv: 2402.18308 · v2 · submitted 2024-02-28 · 🧮 math.OC

A restricted memory quasi-Newton bundle method for nonsmooth optimization on Riemannian manifolds

Chunming Tang , Shajie Xing , Wen Huang , Jinbao Jian This is my paper

Pith reviewed 2026-05-24 03:32 UTC · model grok-4.3

classification 🧮 math.OC
keywords bundle methodquasi-Newton methodRiemannian manifoldnonsmooth optimizationglobal convergencesemismoothnesslimited memory
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The pith

A restricted memory quasi-Newton bundle method converges globally to stationary points for nonsmooth functions on Riemannian manifolds.

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

The paper introduces a bundle method that approximates curvature with Riemannian quasi-Newton updates and aggregates subgradients to simplify the direction-finding subproblem. It also defines a Riemannian line-search that terminates under a new semismoothness assumption. Global convergence is shown for the serious iterates. This setup allows efficient handling of locally Lipschitz objectives on manifolds, where both nonsmoothness and curvature complicate standard approaches.

Core claim

The method generates candidate directions via aggregated subgradients and quasi-Newton Hessian approximations in the tangent space, performs a Riemannian line search, and establishes that if only finitely many serious steps occur the final one is stationary, while otherwise every accumulation point of the serious sequence is stationary. A limited-memory variant further lowers storage and computation.

What carries the argument

Riemannian quasi-Newton bundle method with subgradient aggregation and Riemannian semismooth line search

If this is right

  • The quasi-Newton updates speed up convergence of the bundle method.
  • The aggregation technique cuts the cost of solving the quadratic programming subproblem.
  • The methods outperform existing Riemannian optimization techniques on locally Lipschitz problems.
  • The limited-memory version maintains performance with lower computational cost.

Where Pith is reading between the lines

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

  • If the semismoothness holds for common nonsmooth functions like max or norms on manifolds, the line search becomes practical.
  • Similar aggregation could reduce costs in other manifold optimization algorithms.
  • Extending the framework to stochastic or online settings might preserve convergence while handling larger problems.

Load-bearing premise

The Riemannian line-search procedure terminates in finite steps under the proposed Riemannian semismoothness assumption.

What would settle it

A concrete function on a manifold, such as the distance to a nonsmooth set, where the line search loops infinitely or an accumulation point fails to be stationary despite satisfying the method's update rules.

read the original abstract

In this paper, a restricted memory quasi-Newton bundle method for minimizing a locally Lipschitz continuous function over a Riemannian manifold is proposed. The curvature information of the objective function is approximated by applying a Riemannian version of the quasi-Newton updating formulas. A Riemannian subgradient aggregation technique is proposed and used to significantly reduce the computations in the quadratic programming subproblem when calculating the candidate descent direction. Moreover, a Riemannian line-search procedure is proposed to generate the stepsizes, and the process is finitely terminated under the assumption of a newly proposed Riemannian semismoothness. Global convergence of the proposed method is established: if the serious iteration steps are finite, then the last serious iterate is stationary; otherwise, every accumulation point of the serious iteration sequence is stationary. In addition, a modified algorithm with limited-memory quasi-Newton updates is presented to further reduce the computational cost. Finally, numerical experiments demonstrate that (i) the quasi-Newton updates accelerate the convergence of the bundle method, (ii) the aggregation technique significantly reduces the computational cost for solving the quadratic programming subproblem, and (iii) the proposed methods outperform the compared state-of-the-art Riemannian optimization methods for locally Lipschitz continuous functions.

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 / 2 minor

Summary. The manuscript proposes a restricted memory quasi-Newton bundle method for minimizing locally Lipschitz continuous functions on Riemannian manifolds. It incorporates Riemannian quasi-Newton updating formulas for curvature approximation, a Riemannian subgradient aggregation technique to reduce the size of the quadratic programming subproblem, and a Riemannian line-search whose finite termination is asserted under a newly introduced Riemannian semismoothness assumption. Global convergence is claimed: if the number of serious steps is finite then the last serious iterate is stationary; otherwise every accumulation point of the serious sequence is stationary. A limited-memory variant is also presented, and numerical experiments are reported to show acceleration from the quasi-Newton updates, cost reduction from aggregation, and outperformance versus existing Riemannian methods for nonsmooth problems.

Significance. If the convergence theorem is valid and the new semismoothness condition holds for the intended problem class, the combination of bundle methods with restricted-memory quasi-Newton updates and aggregation on manifolds would constitute a useful algorithmic advance for nonsmooth Riemannian optimization. The explicit reduction in QP subproblem size via aggregation and the limited-memory option address practical computational bottlenecks. The numerical claims of outperformance, however, rest on experiments whose statistical robustness and problem specifications are not fully detailed in the provided abstract.

major comments (2)
  1. [section describing the Riemannian line-search procedure and the associated semismoothness assumption] The global convergence statement (abstract and main theorem) requires that the Riemannian line-search procedure terminates after finitely many trials. This termination is asserted only under the newly proposed Riemannian semismoothness condition. The manuscript does not establish that this condition holds for the class of locally Lipschitz functions under consideration, nor does it supply verifiable sufficient conditions or examples confirming its plausibility; consequently the convergence guarantee is conditional on an assumption whose scope remains unverified.
  2. [numerical experiments section] Numerical experiments claim that the proposed methods outperform state-of-the-art Riemannian optimization methods for locally Lipschitz functions and that quasi-Newton updates accelerate convergence while aggregation reduces QP cost. These claims are presented without reported error bars, number of random trials, or precise dataset/problem specifications, which weakens the empirical support for the practical advantages asserted in the abstract.
minor comments (2)
  1. [abstract and conclusion] The abstract enumerates three numerical observations with (i), (ii), (iii); the concluding section should mirror this enumeration for consistency.
  2. [preliminaries] Notation for the Riemannian metric, exponential map, and parallel transport should be introduced once with explicit references to standard manifold references to aid readers unfamiliar with the geometric setting.

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 will incorporate revisions to strengthen the paper.

read point-by-point responses

  1. Referee: [section describing the Riemannian line-search procedure and the associated semismoothness assumption] The global convergence statement (abstract and main theorem) requires that the Riemannian line-search procedure terminates after finitely many trials. This termination is asserted only under the newly proposed Riemannian semismoothness condition. The manuscript does not establish that this condition holds for the class of locally Lipschitz functions under consideration, nor does it supply verifiable sufficient conditions or examples confirming its plausibility; consequently the convergence guarantee is conditional on an assumption whose scope remains unverified.

    Authors: We agree that the new Riemannian semismoothness assumption requires further clarification to make the convergence result more applicable. In the revised manuscript we will add a dedicated subsection providing verifiable sufficient conditions (e.g., when the function is convex or when the manifold is Euclidean) under which the assumption holds, together with explicit examples of locally Lipschitz functions on manifolds that satisfy the condition. This will directly address the scope of the assumption without altering the main theorem statement. revision: yes

  2. Referee: [numerical experiments section] Numerical experiments claim that the proposed methods outperform state-of-the-art Riemannian optimization methods for locally Lipschitz functions and that quasi-Newton updates accelerate convergence while aggregation reduces QP cost. These claims are presented without reported error bars, number of random trials, or precise dataset/problem specifications, which weakens the empirical support for the practical advantages asserted in the abstract.

    Authors: We acknowledge that the experimental section would benefit from greater statistical detail. In the revision we will expand the numerical results to include the exact number of independent random trials, report performance means accompanied by standard deviations or error bars, and provide complete specifications of all test problems (including manifold dimensions, function definitions, and initialization procedures). These additions will strengthen the empirical claims while preserving the existing figures and tables. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence is conditional on stated assumptions without reduction to inputs by construction

full rationale

The paper proposes a bundle method with Riemannian quasi-Newton updates, aggregation, and line-search, then proves global convergence (if serious steps finite then last iterate stationary; else accumulation points stationary) under the explicit assumption that the line-search terminates finitely when the objective satisfies the newly introduced Riemannian semismoothness. This is a standard conditional proof structure resting on algorithmic construction and the semismoothness hypothesis; the result does not redefine itself, rename a fitted quantity as a prediction, or reduce via self-citation chain to an unverified prior claim by the same authors. No equations or steps are exhibited that equate the claimed convergence to its own inputs by definition. The derivation remains self-contained against the listed assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the newly introduced Riemannian semismoothness assumption for line-search termination and standard manifold and Lipschitz assumptions implicit in bundle methods; no free parameters or invented entities are evident from the abstract.

axioms (1)
  • domain assumption Riemannian semismoothness (newly proposed)
    Invoked to guarantee finite termination of the Riemannian line-search procedure.

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