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arxiv: 2605.15570 · v1 · pith:CGLNVKYGnew · submitted 2026-05-15 · 🧮 math.OC · cs.NA· math.NA

Spectral conjugate gradient projection methods for large-scale monotone equations without Lipschitz continuity

Kabenge Hamiss , Mohammed Alshahrani , Mujahid N. Syed This is my paper

Pith reviewed 2026-05-20 18:06 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA
keywords monotone equationsconjugate gradientprojection methodsderivative-freeglobal convergencespectral parameterlarge-scale optimizationconvex constraints
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The pith

Spectral conjugate gradient projection methods achieve global convergence for monotone equations under monotonicity alone.

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

This paper develops two derivative-free projection methods for large-scale nonlinear monotone equations subject to convex constraints. Both embed adaptive spectral parameters into conjugate gradient frameworks so that the generated search directions satisfy a sufficient descent condition regardless of which line search is chosen. The first method, based on an eigenvalue-optimized scaling matrix, establishes global convergence from monotonicity of the mapping without any Lipschitz continuity assumption. The second method retains the standard monotonicity-plus-Lipschitz assumption. Numerical tests up to dimension 120,000 and applications to signal recovery and logistic regression illustrate practical performance.

Core claim

The authors construct two spectral conjugate gradient projection methods. For the first, which generalizes the modified optimal Perry method via an eigenvalue-optimized scaling matrix, they prove global convergence under monotonicity of the equation mapping alone, without requiring Lipschitz continuity. Search directions satisfy a sufficient descent condition independent of the line search.

What carries the argument

Adaptive spectral parameter incorporated into conjugate gradient search directions, ensuring sufficient descent independently of the line search.

If this is right

  • The first method applies to monotone problems where Lipschitz continuity fails to hold.
  • Both methods remain derivative-free and projection-based, making them suitable for large-scale constrained systems.
  • Practical effectiveness is confirmed on problems with dimensions up to 120,000 and on l1-regularized signal recovery and logistic regression.
  • Global convergence holds for the first method solely from monotonicity.

Where Pith is reading between the lines

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

  • The same spectral construction could be tested on other first-order methods that currently rely on Lipschitz assumptions.
  • The line-search independence may simplify implementation in software libraries for constrained nonlinear equations.
  • Extensions to stochastic or inexact variants become plausible once the descent property is decoupled from line search.

Load-bearing premise

The constructed search directions satisfy a sufficient descent condition that holds independently of the specific line search used.

What would settle it

A concrete monotone but non-Lipschitz mapping on which the first proposed method fails to converge to a solution.

Figures

Figures reproduced from arXiv: 2605.15570 by Kabenge Hamiss, Mohammed Alshahrani, Mujahid N. Syed.

Figure 1
Figure 1. Figure 1: Performance profile by number of iterations across all 5,400 benchmark instances. Higher [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Performance profile by number of function evaluations. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Performance profile by CPU time (seconds). [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Median CPU time versus problem dimension [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Residual ∥G(xk)∥ versus iteration on Problem 5 (n = 50,000, x0 = e). GCGPM converges in 9 iterations, comparable to STTDFPM (8), while MOPCGM requires nearly 300. is additive noise [6]. This is accomplished by solving the ℓ1-regularized least-squares problem min x∈Rn 1 2 ∥b − Ax∥ 2 + τ∥x∥1, (28) where τ > 0 is a regularization parameter promoting sparsity. Following [54, 29, 44], problem (28) can be reform… view at source ↗
Figure 6
Figure 6. Figure 6: Original and reconstructed signals for a single instance ( [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Residual ∥G(zk)∥ versus iteration for the compressed sensing instance in [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

We introduce two derivative-free projection methods for large-scale systems of nonlinear monotone equations subject to convex constraints. Both methods incorporate an adaptive spectral parameter into established conjugate gradient frameworks: the first generalizes the modified optimal Perry method via an eigenvalue-optimized scaling matrix, and the second generalizes the Hager--Zhang-type conjugate gradient projection method via a spectral Dai--Liao parameter. The resulting search directions satisfy a sufficient descent condition independent of the line search. For the first method, we establish global convergence under monotonicity alone, without requiring Lipschitz continuity of the mapping. For the second, global convergence holds under the standard monotonicity and Lipschitz continuity assumptions. Numerical experiments on 18 test problems across dimensions up to 120{,}000, together with applications to $\ell_1$-regularized signal recovery and regularized logistic regression, confirm the practical effectiveness of the proposed approach.

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

1 major / 2 minor

Summary. The manuscript presents two derivative-free projection methods for large-scale systems of nonlinear monotone equations subject to convex constraints. The first generalizes the modified optimal Perry conjugate gradient method via an eigenvalue-optimized scaling matrix, while the second generalizes the Hager-Zhang method via a spectral Dai-Liao parameter. Both methods produce search directions satisfying a sufficient descent condition independent of the line search. Global convergence is established for the first method under monotonicity alone (without Lipschitz continuity) and for the second under monotonicity plus Lipschitz continuity. Numerical results on 18 test problems (dimensions up to 120,000) and applications to ℓ1-regularized signal recovery and regularized logistic regression are reported to demonstrate effectiveness.

Significance. If the global convergence result under monotonicity alone holds, the work would advance derivative-free methods for monotone equations by removing the need for a Lipschitz constant, which is often unavailable or prohibitive in large-scale settings. The line-search-independent sufficient descent property is a technically useful feature. The scale of the numerical tests and the inclusion of real applications add practical value, though the overall significance depends on resolving the well-definedness of the algorithm under the stated hypotheses.

major comments (1)
  1. [Assumptions and line-search procedure] Assumptions section (and line-search description): The claim of global convergence for the first method under 'monotonicity alone, without requiring Lipschitz continuity' is load-bearing for the central contribution, yet the line search (find α>0 s.t. ⟨F(x_k + α d_k), d_k⟩ ≥ σ ⟨F(x_k), d_k⟩, σ∈(0,1)) is not guaranteed to terminate. With g(α)=⟨F(x_k + α d_k), d_k⟩ and g(0)<0 by the sufficient-descent property, existence of a suitable α_k typically invokes continuity of g (hence of F) and the intermediate-value theorem. Monotonicity alone does not imply continuity of F: R^n→R^n, so the algorithm may not be well-defined on the hypothesis class stated in the abstract and assumptions. The manuscript should either add continuity of F to the assumptions or provide an alternative argument that avoids it.
minor comments (2)
  1. [Abstract] Abstract: the dimension '120{,}000' uses nonstandard comma formatting that appears to be a LaTeX artifact and should be rendered as 120000 or 120,000 consistently.
  2. [Introduction or Section 3] The manuscript would benefit from a short table comparing the two methods' spectral parameters, descent constants, and convergence assumptions for quick reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the identification of the issue concerning the well-definedness of the algorithm. We respond to the major comment below.

read point-by-point responses

  1. Referee: Assumptions section (and line-search description): The claim of global convergence for the first method under 'monotonicity alone, without requiring Lipschitz continuity' is load-bearing for the central contribution, yet the line search (find α>0 s.t. ⟨F(x_k + α d_k), d_k⟩ ≥ σ ⟨F(x_k), d_k⟩, σ∈(0,1)) is not guaranteed to terminate. With g(α)=⟨F(x_k + α d_k), d_k⟩ and g(0)<0 by the sufficient-descent property, existence of a suitable α_k typically invokes continuity of g (hence of F) and the intermediate-value theorem. Monotonicity alone does not imply continuity of F: R^n→R^n, so the algorithm may not be well-defined on the hypothesis class stated in the abstract and assumptions. The manuscript should either add continuity of F to the assumptions or provide an alternative argument that avoids it.

    Authors: We thank the referee for highlighting this crucial point. We concur that continuity of F is necessary to ensure that g(α) is continuous, thereby allowing the application of the intermediate value theorem to confirm the existence of a suitable α > 0 satisfying the line search condition. Since monotonicity alone does not guarantee continuity, we will update the assumptions section to explicitly state that F is continuous. This revision ensures the algorithm is rigorously well-defined under the hypotheses, while the global convergence for the first method still holds under monotonicity and continuity without Lipschitz continuity. We do not see a way to avoid this assumption while maintaining the current line search framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs two new spectral CG projection methods for monotone equations, derives search directions that satisfy a sufficient descent condition independently of the line search, and proves global convergence for the first method directly from monotonicity (without Lipschitz continuity) using standard projection and descent arguments. No step reduces a claimed result to a fitted parameter, self-citation chain, or definitional tautology; the sufficient-descent property and convergence analysis are established from the problem assumptions and algorithm construction without importing uniqueness theorems or ansatzes from prior self-work. The central claims therefore remain independent of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions for monotone operators and convex feasible sets together with the construction of search directions that obey sufficient descent.

axioms (2)
  • domain assumption The nonlinear mapping is monotone on the feasible set.
    Invoked to obtain global convergence of the first method without Lipschitz continuity.
  • domain assumption The feasible set is convex and closed.
    Required for the projection step to be well-defined.

pith-pipeline@v0.9.0 · 5688 in / 1155 out tokens · 83224 ms · 2026-05-20T18:06:53.621101+00:00 · methodology

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