{"id":"014777a9-d57f-4b2f-83f4-0d4176ba2e12","arxiv_id":"2509.05917","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"rDSM combines simplex degeneracy correction with repeated reevaluation of persistent vertices to make the Nelder-Mead downhill simplex method more robust to noise and degenerate geometry.","lead":"This paper presents a MATLAB software package, rDSM, that adds two fixes to the classic downhill simplex optimizer: it reshapes flat simplices and re-measures points that stay in the search for a long time. A generalist might read it to see whether simple, gradient-free optimization can be made more reliable for noisy or high-dimensional engineering problems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 2D test function in Eq. (6) contradicts the claimed optimum and reported J values, so the central illustration is irreproducible as written.","rationale":"I reviewed the reader's verdict and the paper's central claim. The reader identified the noise-averaging assumption as the weakest point, but I find that concern not to be load-bearing: a constant positive-mean noise adds the same bias to all vertices, so it does not mis-rank them; the reevaluation still reduces variance and can improve convergence. In contrast, I found a concrete internal inconsistency in the key 2D objective function. Eq. (6) as printed cannot have a global minimum of 0 at (1,-1), and the reported numerical results contradict the formula. This undermines the primary empirical evidence for the claim that rDSM succeeds where DSM fails, making the central illustration irreproducible. The paper could be fixed by correcting the function definition or the reported values, hence the verdict remains CONDITIONAL rather than REJECT. My recommendation is UNCHANGED relative to the reader's CONDITIONAL verdict, but for a different reason. The concrete test of evaluating the objective function at the claimed optimum and checking the repository's actual code would settle whether the inconsistency is a typo or a substantive flaw.","tokens_in":10759,"tokens_out":12283,"duration_ms":131290,"concrete_test":"Independently evaluate Eq. (6) at the stated optimum (1,-1) and at the reported DSM converged point (0.9999,-0.9998). If J(1,-1) ≠ 0 or J(0.9999,-0.9998) ≠ 9.2146×10^-5, then the printed objective function is not the one used in the experiments. Additionally, download the repository's test_function.m and identify the actual function; re-run the §3.1 experiments with that function to verify whether the reported results (Figs. 4-5, Table 5) are reproduced. If the actual function differs from Eq. (6), the paper must be corrected or the central claim is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central demonstration that rDSM outperforms DSM rests on the 2D objective function J(x1,x2) = -(x1-x2)^4 + 0.5 defined on [-1,1]^2. The text states that the global minimum is at (1,-1) with J=0. However, evaluating Eq. (6) at (1,-1) gives J = -(2)^4 + 0.5 = -15.5, not 0; the actual minimum of this function on the domain is 0.5, attained along the line x1=x2. Moreover, the reported DSM convergence in §3.1 to (0.9999,-0.9998) with J=9.2146×10^-5 is completely inconsistent with Eq. (6), which would yield J≈-15.5 at that point. Similarly, all J values in Table 5 are positive and near 0, but Eq. (6) takes large negative values near the claimed optimum. This means the experiments in Figs. 4-5 and Table 5 were run with a different objective function than the one printed. Since these experiments are the primary evidence for the claim that 'rDSM can identify the global optimum in the scenarios where DSM fails', the central illustration is not reproducible from the paper as written. The reader's weakest assumption about zero-mean noise is less load-bearing because a constant noise bias (as in U[0,0.02]) does not change the relative ranking of vertices; it shifts all observed values by the same constant. Thus the reevaluation mechanism can still improve convergence even under biased noise. The test function inconsistency is a more direct and fatal flaw in the evidence.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11245,"tokens_out":6025,"duration_ms":65735,"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":[{"comment":"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.","section":"§3.1, Eq. (6)"},{"comment":"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.","section":"§2.4.3 and Table 5"},{"comment":"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.","section":"§3.2, Table 6"},{"comment":"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.","section":"§3.1 and §2.4.2"}],"minor_comments":[{"comment":"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.","section":"§3.1"},{"comment":"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.","section":"Eq. (5)"},{"comment":"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).","section":"Eqs. (1) and (2)"},{"comment":"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.","section":"§3.2"},{"comment":"There is a typo: 'The “objection function” module' should be 'objective function'.","section":"§2.2"},{"comment":"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.","section":"Table 5"},{"comment":"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.","section":"§3.2"}],"recommendation":"major_revision","confidential_remarks":"The most important issue is the Eq. (6) mismatch, which affects the central demonstration and all quantitative claims. I suspect the authors implemented a different objective than the one printed; if so, the numerical sections can be brought into consistency. I would ask the editor to require, in addition, a comparison with Luersen and Le Riche [13] and more than one random start for the high-dimensional test. The software and packaging are useful, and the core ideas may be sound, but the current evidence is not reproducible as written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the package is functional and the two enhancements are sensible, but the central 2D test function is printed incorrectly relative to the reported numbers, so the main experimental claim is not reproducible as written.\n\nWhat's genuinely useful: rDSM combines two known ideas—degeneracy correction (explicitly credited to Luersen and Le Riche) and repeated-evaluation averaging for noisy objectives—into a clean MATLAB implementation with good bookkeeping: the code writes out simplex history, point databases, and reevaluation logs, which makes trajectories traceable. The 5D Rosenbrock run is a real result: rDSM finds a much better solution than DSM, at a large computational cost. That is a concrete, if modest, demonstration that the combination can help.\n\nThe soft spots: Eq. (6) says J(x1,x2)=-(x1-x2)^4+0.5 and the text claims the global minimum is at (1,-1) with J=0. Evaluating the printed function at (1,-1) gives -15.5, and the true minimum on [-1,1]^2 is 0.5 along x1=x2. The reported convergence values in §3.1 (e.g., J≈9×10^-5 near (0.9999,-0.9998)) are impossible for that function. So the experiments in Figs. 4-5 and Table 5 appear to have used a different objective. That is a load-bearing error, not a typo: it is the evidence for the headline claim that rDSM finds optima where DSM fails.\n\nThe noise concern is less serious than it first looks. The uniform noise U[0,0.02] has positive mean, so the averaged value is biased, but the bias is constant for all vertices, so it doesn't change their relative ranking. The reevaluation still helps. More important is the absence of a baseline: since degeneracy correction is taken from Luersen and Le Riche, the paper should compare rDSM to that method. Without that, the added value of the combination is unclear.\n\nAlso, \"high-dimensional\" is a stretch—the only higher-dimensional test is n=5. A 20- or 50-dimensional problem would be needed to support the abstract's claim.\n\nWho this is for: practitioners doing derivative-free optimization with noisy experimental evaluations, who might use the code as a building block. The paper deserves a serious referee because the software is usable and the ideas are sensible, but the central example must be corrected. I'd send it to peer review, asking for a corrected 2D test, a comparison with [13], and at least one n>10 benchmark.","headline":"Central 2D test is irreproducible as written; the package itself is plausible and deserves a corrected round.","tokens_in":11617,"tokens_out":4724,"would_cite":false,"duration_ms":43085,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C56","65K05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["downhill simplex method","derivative-free optimization","simplex degeneracy","volume maximization","vertex reevaluation","noisy optimization","valley-shaped test function","high-dimensional optimization"],"falsifier":"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","tokens_in":10697,"feed_emoji":"📐","tokens_out":11936,"duration_ms":123220,"temperature":0.7,"pith_summary":"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.","feed_headline":"rDSM reaches global minima where downhill simplex stalls","feed_subtitle":"Degeneracy correction and vertex re-evaluation extend derivative-free search to noisy, higher-dimensional problems.","key_machinery":"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,","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Defines the classic downhill simplex method that rDSM builds on and uses as the baseline in all comparisons.","marker":"[1]"},{"why":"Provides the edge-degenerated and volume-degenerated simplex characterizations and the small-threshold guidance that rDSM's detection criteria implement.","marker":"[13]"},{"why":"Shows that DSM coefficients should be dimension-dependent, informing the parameter settings for the high-dimensional runs.","marker":"[12]"},{"why":"Supplies the determinant-based simplex volume formula used in the constrained volume-maximization correction.","marker":"[20]"},{"why":"Gives the two-dimensional linear-gradient-with-obstacle objective used to demonstrate that rDSM escapes simplex collapse.","marker":"[21]"},{"why":"Sources for the bounded uniform perturbation used in the noisy 2D tests that evaluate the reevaluation mechanism.","marker":"[22, 23, 24]"},{"why":"Provide the valley-shaped test function used in the 5D comparison of rDSM against DSM.","marker":"[25, 26, 27]"}],"fun_headline_variants":["Robust simplex solver conquers noise and high dimensions","Downhill simplex gets a fix for degeneracy and noise","rDSM: derivative-free optimization that won't stall early","Fixing simplex degeneracy and noise for higher-dimensional search","rDSM revives downhill simplex for noisy high-dim problems"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Robust simplex solver conquers noise and high dimensions","Downhill simplex gets a fix for degeneracy and noise","rDSM: derivative-free optimization that won't stall early","Fixing simplex degeneracy and noise for higher-dimensional search","rDSM revives downhill simplex for noisy high-dim problems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000681,"raw_usage":{"total_tokens":2901,"prompt_tokens":687,"completion_tokens":2214,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":431,"completion_tokens_details":{"reasoning_tokens":2132}},"tokens_in":431,"tokens_out":2214,"duration_ms":16974,"temperature":1.0,"reasoning_tokens":2132,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T04:48:19.673589+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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","supporting_citations":[{"cited_title":"Globalized nelder–mead method for engineering optimization.Comput Struct, 82(23-26):2251–2260, 2004","cited_arxiv_id":null,"evidence_quote":"Provides the edge-degenerated and volume-degenerated simplex characterizations and the small-threshold guidance that rDSM's detection criteria implement."},{"cited_title":"Springer Berlin, Heidelberg, 2009","cited_arxiv_id":null,"evidence_quote":"Gives the two-dimensional linear-gradient-with-obstacle objective used to demonstrate that rDSM escapes simplex collapse."}],"review_version":1}