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arxiv: 2605.26909 · v1 · pith:QMKGEHCSnew · submitted 2026-05-26 · 🧮 math.OC

A Nonmonotone Descent Method for Optimization Problems Defined by Upper-mathcal{C}² Functions over Submanifolds

Christian Kanzow , Leo Lehmann This is my paper

Pith reviewed 2026-06-29 15:44 UTC · model grok-4.3

classification 🧮 math.OC
keywords upper-C^2 functionssubmanifoldsprojectional subdifferentialsnonmonotone subgradient methodKurdyka-Lojasiewicz propertydifference of convex functionsmanifold clustering
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The pith

A nonmonotone subgradient method finds stationary points for upper-C^2 optimization on submanifolds.

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

The paper considers minimization of nonsmooth functions that satisfy a nonsmooth descent lemma, a class called upper-C^2 in Euclidean space, but now defined over submanifolds. It establishes that the descent property extends to this setting when subdifferentials are taken in the projectional sense. A nonmonotone subgradient algorithm is introduced for these problems. The method produces sequences whose accumulation points are stationary, and further convergence and rate results hold when the objective satisfies the Kurdyka-Lojasiewicz property. The approach is illustrated on difference-of-convex objectives and on clustering tasks posed directly on manifolds.

Core claim

The descent property of upper-C^2 functions carries over to submanifolds when the recent notion of projectional subdifferentials is used; this permits a nonmonotone subgradient method whose generated sequences have stationary accumulation points, with convergence and rate-of-convergence guarantees under the Kurdyka-Lojasiewicz property.

What carries the argument

Nonmonotone subgradient iteration that employs projectional subdifferentials to enforce the manifold constraint while preserving the upper-C^2 descent inequality.

Load-bearing premise

The descent property of upper-C^2 functions carries over to submanifolds when projectional subdifferentials are used.

What would settle it

An explicit upper-C^2 function on a submanifold together with an initial point for which every sequence generated by the algorithm has a non-stationary accumulation point.

Figures

Figures reproduced from arXiv: 2605.26909 by Christian Kanzow, Leo Lehmann.

Figure 1
Figure 1. Figure 1: Performance profiles of CPU times for the Minimum Sum-of-Squares Clustering [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Performance profiles of CPU times for the Multidimensional Scaling problem, [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Performance profiles of CPU times for clustering on the sphere with artificial [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Performance profiles of CPU times for document clustering on the sphere ( [PITH_FULL_IMAGE:figures/full_fig_p026_4.png] view at source ↗
read the original abstract

We consider the optimization problem of minimizing a nonsmooth function characterized by a nonsmooth formulation of the descent lemma over a manifold. In the unconstrained case over a Euclidean space, this class of functions is called upper-$\mathcal{C}^2$. Using the recent notion of projectional subdifferentials, we show that their descent property carries over to submanifolds. We propose a nonmonotone subgradient method to solve these problems and prove stationarity of accumulation points of the generated sequence as well as convergence and rate-of-convergence results under the Kurdyka-Lojasiewicz property. We also perform numerical experiments and show how our approach can be applied to a certain type of difference of convex functions as well as clustering problems on manifolds.

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 addresses minimization of nonsmooth functions satisfying a nonsmooth descent lemma (upper-C² functions) over submanifolds. It extends the descent property to this setting via projectional subdifferentials, proposes a nonmonotone subgradient algorithm, proves that accumulation points are stationary, and establishes convergence and rate results under the Kurdyka-Łojasiewicz property. Numerical experiments illustrate the method on difference-of-convex problems and manifold clustering.

Significance. If the extension of the descent property and the subsequent convergence analysis hold, the work supplies a new class of tractable nonsmooth problems on manifolds together with a practical algorithm and KL-based guarantees. This could enlarge the scope of nonmonotone methods beyond Euclidean upper-C² functions. The inclusion of concrete numerical tests on clustering is a positive feature.

minor comments (3)
  1. §2: the definition and basic properties of the projectional subdifferential on the manifold should be stated explicitly rather than only referenced, to make the descent-property proof self-contained.
  2. Algorithm 1: the precise rule for choosing the nonmonotone parameter sequence (e.g., the window length or the update of the reference value) is not fully specified; a short pseudocode block would improve reproducibility.
  3. Numerical section: the manifold clustering experiments would benefit from a brief description of the retraction and the projectional-subdifferential oracle used in the implementation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. The referee correctly identifies the core contributions: extension of the nonsmooth descent lemma to submanifolds via projectional subdifferentials, the nonmonotone subgradient algorithm, stationarity of accumulation points, and KL-based convergence and rate results. We are pleased that the numerical experiments on difference-of-convex problems and manifold clustering are viewed favorably. Since the report lists no specific major comments, we interpret the minor_revision recommendation as an invitation to incorporate any editorial or minor technical suggestions that may appear in a subsequent round; we will prepare a revised version accordingly.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's core claims rest on extending the descent property of upper-C² functions to submanifolds via the external notion of projectional subdifferentials, then constructing a nonmonotone subgradient method whose stationarity and KL-convergence results follow from standard arguments in nonsmooth optimization on manifolds. No equations, fitted parameters, or self-citations are shown to reduce any prediction or theorem to a tautology by construction. The derivation chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; full technical assumptions, parameter choices, and any invented constructs cannot be audited.

axioms (2)
  • domain assumption Projectional subdifferentials preserve the nonsmooth descent lemma on submanifolds
    Invoked in abstract paragraph 2 to extend the Euclidean upper-C^2 property.
  • domain assumption Kurdyka-Lojasiewicz property holds for the objective
    Used to obtain convergence rates; standard in nonsmooth optimization literature.

pith-pipeline@v0.9.1-grok · 5657 in / 1287 out tokens · 21857 ms · 2026-06-29T15:44:58.395887+00:00 · methodology

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