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REVIEW 4 major objections 7 minor 32 references

rDSM -- A robust Downhill Simplex Method software package for optimization problems in high dimensions

T0 review · 4 major / 7 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The rDSM package claims that reshaping degenerated simplices and re-averaging persistent vertices lets the downhill simplex method reach global minima on noisy and high-dimensional problems where the classic method stalls.

desk verdict Central 2D test is irreproducible as written; the package itself is plausible and deserves a corrected round. read the letter →

arxiv 2509.05917 v1 pith:CMPZXRGS submitted 2025-09-07 math.OC

Tianyu Wang , Xiaozhou He , Bernd R. Noack This is my paper · ORCID
classification math.OC MSC 90C5665K05
keywords downhillsimplexmethodderivative-freeoptimizationdegeneracyvolumemaximizationvertexreevaluationnoisyvalley-shapedtestfunctionhigh-dimensional
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper introduces rDSM, a software package that extends the classic downhill simplex method with two fixes: it detects a simplex that has collapsed toward a line or plane and reshapes it by maximizing volume while keeping perimeter fixed, and it re-evaluates vertices that linger in the simplex for more than 1.5n iterations, replacing their objective value with the average of repeated evaluations. The authors' claim is that these two additions let the method land on the global minimum in cases where plain DSM gets stuck: a two-dimensional gradient with an obstacle, noisy variants of that problem, and a five-dimensional valley function. In their tests, rDSM improves the final objective value in all three settings, at the cost of more function evaluations and substantially longer run time on the 5D case. If the claim holds, rDSM would make derivative-free optimization practical for experimental and computational problems where gradients are unavailable and measurements are noisy.

What carries the argument

The central object is the simplex itself, treated as a geometric figure that can lose dimensionality and as a set of vertices whose recorded costs can be corrupted by noise. Two mechanisms carry the argument. Degeneracy correction compares the shortest edge to the longest (edge test) and the n-th root of the determinant-based volume to the edge lengths (volume test); when either ratio falls below a threshold (default 0.1), the worst vertex is moved by solving a constrained maximization: maximize simplex volume subject to fixed perimeter, using a Newton-type solver, iterating over vertices from worst to best until the simplex is non-degenerate. Reevaluation attaches a counter to each vertex,

What would settle it

Run the 2D obstacle test and the 5D valley test with each enhancement disabled in turn: if turning off degeneracy correction does not degrade rDSM's result, the volume-maximization step is not carrying the claimed gain, and the same test applies to reevaluation on the noisy problems. A sharper check: add a constant positive bias to every evaluation (for instance, U[0.01,0.03] noise); because averaging cannot remove a nonzero mean, rDSM should fail to approach the true optimum. If it converges anyway, the reevaluation mechanism is working differently than the paper's averaging argument suggests

Watch

Extended reading notes

Core claim

The central claim is that the two failure modes of the classic downhill simplex method—geometric degeneration and noise-induced spurious minima—can be corrected without leaving the derivative-free framework. A simplex is declared degenerated when its shortest edge is far shorter than its longest, or when its volume is too small relative to its edge lengths; rDSM then moves the worst-scoring vertex to a position that maximizes the simplex volume while keeping the perimeter unchanged. For noise, any vertex that remains in the simplex for more than 1.5n iterations is re-evaluated and its objective value is replaced by the mean of all its past evaluations. On the 2D obstacle problem, DSM stops a

Load-bearing premise

The reevaluation step's claimed benefit depends on repeated evaluations of a fixed vertex averaging out to the true objective value, which requires zero-mean, independent noise; under biased or state-dependent noise, including the positive-mean uniform noise used in some of the paper's own tests, the averaged value is systematically offset.

Editorial extensions

If this is right

  • For the 2D obstacle test, rDSM reaches the global-minimum neighborhood (J≈0.0047) while plain DSM stalls at J≈0.2269 on a domain boundary.
  • Across 20 noisy runs with uniform and Gaussian perturbations, rDSM converges closer to the known optimum with smaller spread in both endpoint and final cost.
  • On the 5D valley function, rDSM reaches objective 2.92e-10 versus DSM's 1.66e-5, but requires about 97 more evaluations and roughly 19x the run time.
  • The mechanisms preserve the derivative-free character of DSM, so the package can be applied when gradients are unavailable, including CFD-based or experimental objective functions.
  • The paper's own outlook is that rDSM improves applicability to higher dimensions and noise but does not by itself resolve very-high-dimensional scaling; the authors point to dimensionality reduction and hybrid solvers as next steps.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The degeneracy-correction step is formulated as a constrained maximization of the worst vertex, so it could be detached from rDSM and applied to other simplex-based optimizers, including those with adaptive coefficients.
  • The reported run-time jump (64.6 s vs 3.4 s in 5D) suggests the volume-maximization subproblem becomes the bottleneck as dimension grows; a testable extension would be to approximate the correction or trigger it less often in higher dimensions.
  • The reevaluation trigger (1.5n) is a fixed heuristic; making it depend on the estimated noise variance, or using a weighted average that discounts stale evaluations, could improve the trade-off between evaluation budget and accuracy.
  • A direct prediction is that on smooth, unimodal, noiseless problems where the simplex never degenerates, rDSM behaves like classic DSM; deviations from that baseline would indicate the enhancements are doing more than the paper describes.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. The paper introduces rDSM, a MATLAB software package that extends the classical downhill simplex method (DSM) with two mechanisms: (i) detection and correction of degenerated simplices via constrained volume maximization at fixed perimeter, and (ii) reevaluation of persistent vertices by averaging historical objective evaluations to counter noise. The authors claim improved convergence, better robustness to noisy evaluations, and increased applicability to higher dimensions. They support this with a 2D analytical objective with and without an obstacle, a noisy 2D comparison, and one 5D Rosenbrock run. The code is available in a public repository with metadata and output traceability files.

Significance. If the results are reproducible, the package would be a useful, practical contribution for derivative-free optimization in experimental settings, where gradients are unavailable and evaluations are noisy. The manuscript is clearly written, the software is packaged with useful visualization and logging, and the two enhancements are easy to understand. The degeneracy correction is not entirely novel, since the authors themselves list Luersen and Le Riche [13] as prior work on correcting degenerated simplices, but the explicit implementation and the reevaluation mechanism are of interest. However, the current numerical evidence is substantially weakened by a direct inconsistency in the printed test function, by the absence of a comparison with [13], and by the very limited statistical support for the high-dimensional claim.

major comments (4)
  1. [§3.1, Eq. (6)] The printed objective function contradicts every reported result. For J(x1,x2)=-(x1-x2)^4+0.5 on [-1,1]^2, evaluation at the claimed optimum (1,-1) gives -15.5, not 0, and the actual minimum over the domain is -15.5 (at (1,-1) and (-1,1)), while 0.5 is the maximum attained along x1=x2. The reported DSM endpoint (0.9999,-0.9998) with J=9.2146×10^-5, the rDSM endpoint with J=0.0047, and all entries in Table 5 are therefore impossible under Eq. (6). Figures 4, 5 and Table 5 appear to have been generated with a different objective function than the one printed, making the central illustration for the claim 'rDSM can identify the global optimum in the scenarios where DSM fails' irreproducible as written. Please correct Eq. (6) or the reported values and regenerate the entire section consistently.
  2. [§2.4.3 and Table 5] The reevaluation step is justified by the statement that averaging repeated evaluations 'brings the result closer to the true value.' This is only valid for zero-mean noise. The paper's uniform-noise tests U[0,0.01] and U[0,0.02] have positive means 0.005 and 0.01, so the averaged estimate converges to J+0.005 or J+0.01, not to J. Thus the paper's own noise model does not support the unbiased-estimation rationale, and the observed improvement is attributable only to variance reduction. The authors should either use symmetric zero-mean noise (e.g., U[-a,a]) to match the stated rationale, or explicitly acknowledge that averaging reduces variance but does not remove a systematic bias, which can mis-rank vertices when the noise bias is state-dependent.
  3. [§3.2, Table 6] The high-dimensional claim rests on a single run of the 5D Rosenbrock function from one initial point. The text concludes that 'rDSM consistently identifies the global minimum,' but with n=1 there is no statistical basis for 'consistently.' Moreover, no comparison is made with the prior degeneracy-correcting method of Luersen and Le Riche [13], although Table 2 classifies [13] as already addressing degenerated simplices and the thresholds are said to be 'suggested in [13].' Without a comparison against [13] and without multiple random restarts, the claimed advantage over the closest prior method is not established. Please add repeated trials and a comparative benchmark, or soften the claims accordingly.
  4. [§3.1 and §2.4.2] The threshold values are adjusted to the specific test cases: θe and θv are said to be 'determined based on prior tests on 2D problems' and set to 0.1, while in §3.2 they are set to 1×10^-5. The reevaluation trigger 1.5n is calibrated on the same U[0,0.02] case that is then reported as the improvement. Since the reported gains are partly a consequence of thresholds tuned on the same examples, the paper should provide evidence that the thresholds transfer across problems (e.g., use fixed thresholds on a suite of benchmark functions) or present a sensitivity analysis showing the results do not depend critically on these choices.
minor comments (7)
  1. [§3.1] The iteration limit is inconsistent: the text says 'The iterative process is limited to 50 iterations, with a maximum of 100 evaluations,' but later states that rDSM 'ends at point ... after 100 iterations.' Please clarify whether the intended limit is in iterations or evaluations and use consistent wording.
  2. [Eq. (5)] The constraint writes P(x_s1,...,x_sn, y_s^{n+1}) = P(x_s1,...,x_sn, x_s^{n+1}), but y_s^{n+1} is the variable being optimized. Use separate symbols for the candidate point and the fixed old vertex to avoid confusion.
  3. [Eqs. (1) and (2)] The notation 'i=1,n' for edges is not defined for an n-dimensional simplex, which has n(n+1)/2 edges. Please define the edge indexing and explain why the ratio uses only n edges in the denominator of Eq. (2).
  4. [§3.2] The sentence 'For the convergence stability, the cost is divided by 10000 during the optimization process' is ambiguous: are the reported Jr values 1.66×10^-5 and 2.92×10^-10 the divided values or the original ones? Please state the convention explicitly.
  5. [§2.2] There is a typo: 'The “objection function” module' should be 'objective function'.
  6. [Table 5] The table caption says 'sample means and variances,' but the entries are presented as 'mean ± value.' Specify whether the ± term is a standard deviation, standard error, or confidence interval.
  7. [§3.2] The running times 3.4 s (DSM) versus 64.6 s (rDSM) imply a 19-fold slowdown while only 97 additional evaluations are performed. The source of this overhead (e.g., Newton–Raphson solves) should be explained, especially because the Impact section stresses efficiency.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor in-sample calibration of the reevaluation threshold; no load-bearing self-citation or definitional circularity.

  1. fitted input called prediction [Sec. 3.1, final paragraph (calibration of csi threshold) and Table 5 (U[0,0.02] row)]
    "To calibrate the reevaluation trigger, we compare the thresholds csi ≥1.5n and csi ≥2n by the uniform noise case U[0,0.02]. ... In contrast, when the coefficient is set as 1.5, rDSM can converge to (0.9621±0.0057, -0.9629±8×10−4) ... Table 5: JU[0,0.02] 0.1111±0.0014 0.0239±0.0003"

    The reevaluation trigger threshold 1.5n is explicitly selected by comparing rDSM's performance on the U[0,0.02] noise case. The same U[0,0.02] case, with the same endpoint (0.9621, -0.9629) and J=0.0239, is then presented in Table 5 as evidence that rDSM outperforms DSM under noise. That row is therefore the calibration target rather than an independent prediction, so the reported gain partly reflects the chosen constant. Other noise cases and the obstacle/Rosenbrock tests are not calibration targets, so the circularity is partial and does not invalidate the core method.

full rationale

The rDSM derivation does not presuppose its headline result: degeneracy correction and reevaluation are defined from geometric and averaging principles, and the main claims are tested on external benchmark functions (Eq. (6), Eq. (7)) against a re-implemented DSM. No load-bearing uniqueness theorem or ansatz is imported from the authors' prior work; citations [1,12,13,20-28] are external and not used to forbid alternatives. The only circularity-adjacent issue is the in-sample calibration of the reevaluation threshold (1.5n) on the U[0,0.02] case, which is then reused as evidence in Table 5; this row is fitted rather than predicted. The remaining noise configurations and the obstacle/Rosenbrock demonstrations are independent, so the central claim has substantial independent content. Separately, the printed test function Eq. (6) is inconsistent with the stated optimum (J(1,-1)=-15.5, not 0) and with the reported positive J values; this is a correctness/reproducibility defect, not circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on three hand-set detection and trigger parameters, a heuristic constrained volume-maximization step, and an averaging assumption that is violated by the paper's own uniform-noise tests. No new physical entities are introduced.

free parameters (3)
  • Edge degeneracy threshold theta_e = 0.1 (2D), 1e-5 (5D Rosenbrock)
    Chosen from 'prior tests on 2D problems' in Section 3.1 and changed to 1e-5 for the 5D Rosenbrock test; rDSM results depend on this detection threshold.
  • Volume degeneracy threshold theta_v = 0.1 (2D), 1e-5 (5D Rosenbrock)
    Set per test based on prior runs; used to trigger volume-based degeneracy correction in Eq. (2).
  • Reevaluation trigger coefficient 1.5n = 1.5
    Selected in Section 3.1 by comparing c_si >= 1.5n with c_si >= 2n on the U[0,0.02] noisy 2D test; the claimed noise robustness is partly a consequence of this choice.
assumptions (4)
  • standard math The volume of an n-simplex is given by n!^{-1} times the absolute determinant in Eq. (4).
    Used in the degeneracy correction objective; drawn from reference [20] and standard linear algebra.
  • ad hoc to paper Constrained volume maximization with fixed perimeter, Eq. (5), solved by Newton-Raphson, produces a useful corrected simplex.
    No convergence, uniqueness, or benefit analysis is supplied; this is the core heuristic of the degeneracy correction.
  • domain assumption Averaging repeated evaluations of a vertex approximates its true objective value.
    Invoked in Section 2.4.3; only valid for zero-mean noise, which the paper's uniform-noise tests do not satisfy.
  • ad hoc to paper The threshold values theta_e=0.1, theta_v=0.1, and the 1.5n trigger transfer across optimization problems.
    The paper sets different thresholds per test based on prior runs, so transferability is assumed rather than demonstrated.

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Cite this review

Pith. "Pith review of rDSM -- A robust Downhill Simplex Method software package for optimization problems in high dimensions." pith.science (2026). https://pith.science/paper/CMPZXRGS

@misc{pith2026250905917,
  author       = {Pith},
  title        = {Pith review of: rDSM -- A robust Downhill Simplex Method software package for optimization problems in high dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CMPZXRGS}},
  note         = {Machine review of arXiv:2509.05917}
}
read the original abstract

The Downhill Simplex Method (DSM) is a fast-converging derivative-free optimization technique for nonlinear systems. However, the optimization process is often subject to premature convergence due to degenerated simplices or noise-induced spurious minima. This study introduces a software package for the robust Downhill Simplex Method (rDSM), which incorporates two key enhancements. First, simplex degeneracy is detected and corrected by volume maximization under constraints. Second, the real objective value of noisy problems is estimated by reevaluating the long-standing points. Thus, rDSM improves the convergence of DSM, and may increase the applicability of DSM to higher dimensions, even in the presence of noise. The rDSM software package thus provides a robust and efficient solution for both analytical and experimental optimization scenarios. This methodological advancement extends the applicability of simplex-based optimization to complex experimental systems where gradient information remains inaccessible and measurement noise proves non-negligible.

Figures

Figures reproduced from arXiv: 2509.05917 by the authors.

Figure 1
Figure 1. Flowchart of rDSM. (a) shows the overview of rDSM, (b) and (c) give the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Module structure of rDSM software package. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. The two degenerated simplex types: (a) edge-degenerated simplex, (b) volume [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Comparing the optimization process and corresponding learning curve of linear [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Comparing the optimization process and corresponding learning curve of noisy [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

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    https://doi.org/10.2514/1.J063114

    doi

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.