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REVIEW 3 major objections 8 minor 40 references

Adaptive direct search algorithms with relaxable and quantifiable constraints

T0 review · 3 major / 8 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Mesh-free direct search handles constraints and converges to zero

desk verdict ADS-PB combines the mesh-free ADS framework with a Progressive Barrier for constrained DFO; convergence theory is sound but the algorithm spec is incomplete without the companion paper. read the letter →

arxiv 2607.05183 v1 pith:C4EZG2OK submitted 2026-07-06 math.OC

Charles Audet , Théo Denorme , Youssef Diouane , Sébastien Le Digabel , Christophe Tribes This is my paper · ORCID
classification math.OC
keywords adaptiveconstraintsdirectframeworkproposedsearchapproachbarrier
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

The paper introduces ADS-PB, an algorithm for optimizing blackbox functions subject to constraints that can be quantified and relaxed. The core idea is to combine a Progressive Barrier mechanism—which progressively tightens the allowed constraint violation, letting infeasible points contribute information—with a mesh-free direct search framework that replaces the traditional mesh discretization of MADS with a punctured-space exclusion approach. In this approach, trial points are rejected only if they fall within a shrinking radius of previously successful points, rather than being projected onto a mesh grid. The paper proves that both the exclusion radius and the frame size converge to zero (a stronger result than the liminf guarantee in MADS-PB) and that refined points satisfy Clarke stationarity. Computationally, ADS-PB consistently outperforms MADS-PB, with the advantage widening at stricter tolerances, because it can evaluate model-suggested trial points directly without mesh projection degrading their quality.

What carries the argument

Punctured space exclusion (Definition 1), progressive barrier threshold update (Equation 6), frame size and exclusion radius update rules (Equation 9), four iteration types (dominating, improving, reframing, unsuccessful)

What would settle it

Construct a problem where the search step repeatedly proposes dominating points along an unbounded sequence (e.g., f decreasing toward a limit at infinity); the compactness assumption fails and the convergence theorem does not apply.

Watch

Extended reading notes

Core claim

The central object is the punctured space: at each iteration, the algorithm excludes balls of radius delta_k around all previously successful points, and any trial point (from search or poll) falling inside these balls is discarded without evaluation. This replaces the mesh structure of MADS-PB. The convergence proof hinges on a Bolzano-Weierstrass argument: if the exclusion radius did not converge to zero, infinitely many successful points would be pairwise separated by at least epsilon inside a compact set, which is impossible. This yields the full limit (not just liminf) of both delta_k and Delta_k going to zero, from which Clarke stationarity of refined feasible and infeasible points fol

Load-bearing premise

The convergence proof requires that all evaluated points lie in a compact set, but the search step is described as completely free in the domain and could in principle propose points escaping to infinity, which would violate the assumption and break the proof.

Editorial extensions

If this is right

  • Removing mesh projection lets surrogate-model-suggested points be evaluated at their exact proposed location, which is why ADS-PB gains more from quadratic-model poll directions than MADS-PB does, especially at strict tolerances.
  • The reframing iteration type is a new algorithmic event unique to mesh-free direct search: a search point that dominates but lies too close to a prior success triggers a poll around it, and if that poll fails, the frame is repositioned rather than shrunk.
  • The full-limit convergence of delta_k to zero (versus liminf in MADS-PB) means the algorithm guarantees that poll neighborhoods eventually become arbitrarily fine around every accumulation point, not just along a subsequence.
  • The framework is designed to accommodate arbitrary search heuristics—surrogate models, Nelder-Mead, speculative search—without altering their proposed points, making it a natural substrate for hybrid model-based derivative-free optimization.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 8 minor

Summary. This paper introduces ADS-PB, an extension of the Adaptive Direct Search (ADS) framework to constrained blackbox optimization with relaxable and quantifiable constraints. The method combines the Progressive Barrier (PB) mechanism from MADS-PB with the mesh-free ADS structure, replacing mesh-based trial point generation with a punctured-space exclusion approach. The paper provides convergence analysis (Theorems 3.1–3.3) showing that both the exclusion radius and frame size converge to zero, and that refined points satisfy Clarke stationarity. Computational experiments on analytical and blackbox test problems compare ADS-PB with MADS-PB and other solvers within the NOMAD software.

Significance. The paper addresses a relevant problem in derivative-free optimization: removing mesh projections while maintaining convergence guarantees for constrained problems. The punctured-space mechanism is a natural idea, and the convergence analysis follows established DFO techniques. The implementation within NOMAD and comparison with MADS-PB under identical search components is a strength, as is the consistent computational advantage of ADS-PB at stricter tolerances. The stronger limit result in Theorem 3.1 (full limit to zero rather than liminf) is a nice theoretical contribution. The framework is falsifiable through the data profiles provided.

major comments (3)
  1. §3.3, Eq. (9): The increase/decrease rules for (Δ_k, δ_k) are defined as 'as in [10, Equations (2) and (3)]' but are not restated in the manuscript. These rules are load-bearing for the convergence proof of Theorem 3.1, particularly the claim that δ_k → 0 implies Δ_k → 0 and the argument that unsuccessful/reframing iterations decrease δ_k. Without restating the rules and their key properties (e.g., that the decrease factor is bounded away from 1), the algorithm is incomplete and the proof of Theorem 3.1 cannot be fully verified from the manuscript alone. The key properties needed should be stated explicitly.
  2. §3.4, Theorem 3.1, Case 1 (|S_ε| finite): The argument that 'there exists k_0 such that for all k ≥ k_0 the iteration is unsuccessful or reframing' leads to 'lim_{k∈K_ε} δ_k = 0, contradicting δ_k ≥ ε' requires justification that the decrease rule ensures δ_k actually falls below ε in finitely many steps. If the decrease rule allows δ_{k+1} arbitrarily close to δ_k, then δ_k ≥ ε could persist for infinitely many steps. The proof should either restate the decrease factor bound from [10] or explicitly argue finite-step decrease below ε. This is the concern raised in the stress-test note, and it lands: the proof as written does not close this gap without the decrease rule properties.
  3. §3.1 and §3.4, Assumption 1: The compactness assumption (all evaluated points belong to a compact set L) is load-bearing for the Bolzano-Weierstrass argument in Theorem 3.1. However, §3.1 describes the search step as 'completely free in X' and able to generate arbitrary trial points. The paper should discuss how Assumption 1 is enforced or motivated in practice, or at minimum note that the search strategy must be designed to respect this compactness for the convergence guarantee to apply.
minor comments (8)
  1. §3.3: The text contains a duplicated paragraph about V^k_succ replacing V^k from [10] and the update mechanism. The paragraph beginning 'This preserves the foundational principles...' and the following paragraph beginning 'The set of successfully visited points V^k_succ replaces...' overlap significantly. This redundancy should be removed.
  2. §3.3, Eq. (10): The intersection notation 't_update ∈ C_k ∪ S^k_eval ∪ P^k_eval ∩ ◦R^n_k' is ambiguous regarding operator precedence. Parentheses should clarify whether the intersection with the punctured space applies only to P^k_eval or to the entire union.
  3. §2.2: The barrier threshold update rule in Eq. (6) sets h^{k+1}_max = h^{k+1}_I after dominating or unsuccessful iterations, but h^{k+1}_I is not explicitly defined at that point in the text. A forward reference or brief definition would help the reader.
  4. §4.2: The data profiles in Figure 3 compare ADS-PB and MADS-PB variants, but the x-axis uses 'groups of (n+1) evaluations' for all variants including the 2n poll. Since 2n and n+1 variants use different numbers of evaluations per poll, the fairness of this normalization should be briefly discussed.
  5. §4.3: The paper states that LOGDS 'does not benefit from the full range of search and poll mechanisms available in NOMAD,' which explains part of its weaker performance. The comparison would be more informative if LOGDS were given access to equivalent search strategies, or if this limitation were noted more prominently as a caveat on the comparison.
  6. The reference [10] is a 2025 arXiv preprint by the same authors. Since the core ADS structure and parameter update rules are drawn from [10], the relationship between this paper and [10] should be clarified — specifically, what is genuinely new here versus what is established in [10].
  7. §3.2, Definition 2: The poll center definitions reference t_search, but the case where the search step produces no dominating point and the poll centers default to x^k_F and x^k_I could be stated more explicitly. A brief note that when no dominating search point exists, the poll centers are simply the current incumbents would help.
  8. The abstract states 'A convergence analysis of the proposed framework under mild assumptions is presented.' It would be helpful to name the key assumption (compactness of the evaluation set) explicitly, or at least hint at its nature (e.g., 'under a standard compactness assumption') in the abstract.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful reading and for identifying two genuine gaps in the convergence proof and one important clarification regarding Assumption 1. All three major comments are well-taken and will be addressed in the revised manuscript. Specifically, we will restate the increase/decrease rules from [10] with their key properties, close the finite-step decrease gap in the Case 1 argument of Theorem 3.1, and add a discussion of how Assumption 1 relates to the unrestricted search step. No standing objections remain.

read point-by-point responses

  1. Referee: §3.3, Eq. (9): The increase/decrease rules for (Δ_k, δ_k) are defined as 'as in [10, Equations (2) and (3)]' but are not restated in the manuscript. These rules are load-bearing for the convergence proof of Theorem 3.1, particularly the claim that δ_k → 0 implies Δ_k → 0 and the argument that unsuccessful/reframing iterations decrease δ_k. Without restating the rules and their key properties (e.g., that the decrease factor is bounded away from 1), the algorithm is incomplete and the proof of Theorem 3.1 cannot be fully verified from the manuscript alone. The key properties needed should be stated explicitly.

    Authors: The referee is correct. The update rules for (Δ_k, δ_k) are load-bearing for the convergence analysis, and the manuscript should be self-contained with respect to the key properties used in the proofs. In the revised manuscript, we will restate the increase and decrease rules from [10, Equations (2) and (3)] explicitly in Section 3.3. Specifically, the decrease rule ensures that δ_{k+1} ≤ γ_dec · δ_k for some constant γ_dec ∈ (0,1), and the increase rule ensures that δ_{k+1} ≤ γ_inc · δ_k for some constant γ_inc > 1, with the additional property that the ratio δ_k/Δ_k → 0 whenever δ_k → 0. These properties will be stated as a lemma or proposition immediately following the restated rules, so that the proofs of Theorems 3.1–3.3 can be verified without consulting [10]. revision: yes

  2. Referee: §3.4, Theorem 3.1, Case 1 (|S_ε| finite): The argument that 'there exists k_0 such that for all k ≥ k_0 the iteration is unsuccessful or reframing' leads to 'lim_{k∈K_ε} δ_k = 0, contradicting δ_k ≥ ε' requires justification that the decrease rule ensures δ_k actually falls below ε in finitely many steps. If the decrease rule allows δ_{k+1} arbitrarily close to δ_k, then δ_k ≥ ε could persist for infinitely many steps. The proof should either restate the decrease factor bound from [10] or explicitly argue finite-step decrease below ε.

    Authors: The referee has identified a genuine gap in the proof as written. The argument in Case 1 implicitly relies on the fact that the decrease factor is bounded away from 1, i.e., δ_{k+1} ≤ γ_dec · δ_k with γ_dec ∈ (0,1), but this property is not stated in the manuscript. Once this property is made explicit (as described in our response to the first major comment), the gap closes: if all iterations for k ≥ k_0 are unsuccessful or reframing, then δ_k decreases by at least a factor γ_dec < 1 at each such iteration, so δ_k ≤ γ_dec^{k−k_0} · δ_{k_0} → 0 as k → ∞, which means δ_k < ε for all sufficiently large k ∈ K_ε, contradicting the definition of K_ε. We will revise the proof of Theorem 3.1 to make this finite-step decrease argument explicit, referencing the bounded decrease factor. revision: yes

  3. Referee: §3.1 and §3.4, Assumption 1: The compactness assumption (all evaluated points belong to a compact set L) is load-bearing for the Bolzano-Weierstrass argument in Theorem 3.1. However, §3.1 describes the search step as 'completely free in X' and able to generate arbitrary trial points. The paper should discuss how Assumption 1 is enforced or motivated in practice, or at minimum note that the search strategy must be designed to respect this compactness for the convergence guarantee to apply.

    Authors: This is a fair point. The tension between the unrestricted search step and the compactness assumption should be acknowledged. We will add a remark after Assumption 1 clarifying two points. First, in practice, the blackbox domain X is typically bounded (e.g., bound-constrained), which naturally ensures that all evaluated points lie in a compact set. Second, when X is not bounded, the convergence guarantee applies only if the search strategy is designed so that all trial points remain within some compact set L ⊂ X. This is a standard assumption in the direct search literature (see, e.g., [8, 11]), and we will make this explicit. The freedom of the search step in ADS-PB refers to the absence of mesh projection, not to the absence of any restriction on where trial points may be placed. revision: yes

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity. The convergence proofs are self-contained given the update rules from [10], and the PB mechanism comes from independently published work [9].

full rationale

The paper's derivation chain is largely self-contained. Theorem 3.1's proof uses a packing/Bolzano-Weierstrass argument that is original to this paper: successful points in the punctured space are separated by at least δ_k ≥ ε, and compactness (Assumption 1) forces a contradiction. This argument does not reduce to any cited result by construction. The increase/decrease rules for (Δ_k, δ_k) are referenced to [10, Equations (2) and (3)] without restatement — this is a completeness gap (the algorithm is not fully specified without consulting [10]), but it is not circularity: the cited rules are standard direct search update rules (increase on success, decrease on failure), and the convergence proof uses them as inputs to derive a new conclusion (limit to zero, stronger than the liminf result in MADS-PB [8]). The PB mechanism itself comes from [9] (Audet & Dennis, 2009, SIAM J. Opt.), an independently published result with different co-authors. Theorems 3.2–3.3 (Clarke stationarity) follow standard direct search proof techniques applied to the punctured-space construction, with the key step (equation 12) being the decomposition y_k = p_k + Δ_k(v_k + (δ_k/Δ_k)w_k) that handles rejected poll points — this is a novel technical contribution, not a renaming. The computational experiments compare ADS-PB against MADS-PB and external solvers (CMA-ES, NSGA-II, LOGDS) on standard benchmark problems with no parameter fitting to the test data. The only self-citation concern is that [10] (same authors, 2025 arXiv preprint) provides the core ADS framework and update rules, but this citation provides independent algorithmic infrastructure rather than smuggling in the convergence conclusion. Score 1 reflects this minor dependence on an unpublished self-citation for algorithmic completeness, which is not load-bearing for the circularity of the proofs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 2 invented entities

The paper introduces two algorithmic constructions (punctured space, reframing iterations) that are well-motivated and empirically/tested. No new physical entities or unobservable postulates.

free parameters (3)
  • Δ_0 (initial frame size) = not specified
    Initial frame size parameter; set by user at algorithm initialization (Algorithm 2, Step 0).
  • δ_0 (initial exclusion radius) = not specified, constrained by δ_0 ≤ Δ_0
    Initial exclusion size parameter; set by user at algorithm initialization (Algorithm 2, Step 0).
  • increase/decrease rule parameters = not stated in this paper
    The rules for increasing and decreasing (Δ_k, δ_k) are defined 'as in [10, Equations (2) and (3)]' (Section 3.3, eq. 9) but not restated. These likely contain multiplicative factors that are free parameters.
assumptions (5)
  • domain assumption Assumption 1: All evaluated points belong to a compact set L.
    Stated in Section 3.4. Load-bearing for Theorem 3.1 (Bolzano-Weierstrass argument) and Theorems 3.2–3.3. Standard in DFO but not discussed in relation to the unrestricted search step.
  • domain assumption f is Lipschitz continuous near refined points.
    Invoked in Theorem 3.2 for the Clarke derivative to exist. Standard nonsmooth analysis assumption.
  • domain assumption h is Lipschitz continuous near refined infeasible points.
    Invoked in Theorem 3.3. Follows from constraint functions being Lipschitz, which is implicit.
  • standard math Refining directions are hypertangent to Ω at the refined point.
    Definition 4 (Rockafellar). Required for Theorem 3.2 to ensure p+tˆv ∈ Ω. Standard in constrained DFO convergence analysis.
  • domain assumption The increase/decrease rules from [10] ensure lim δ_k/Δ_k = 0 and lim Δ_k = 0 when lim δ_k = 0.
    Stated in Section 3.3 following eq. (9). The specific rules are not restated; the reader must consult [10].
invented entities (2)
  • Punctured space ◦R^n_k independent evidence
    purpose: Exclusion region around previously successful points to prevent redundant evaluations; replaces mesh structure.
    The punctured space is a construction, not a physical entity. Its effectiveness is empirically validated in Section 4 and its role in convergence is proven in Theorem 3.1. It is falsifiable: if it degraded performance, the data profiles would show it.
  • Reframing iteration type independent evidence
    purpose: Classifies iterations where a dominating search point is found outside the punctured space but the poll fails to find a dominating point inside it.
    Algorithmic classification, not a physical entity. Its behavior is observable in the algorithm's execution and affects incumbent updates.

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Cite this review

Pith. "Pith review of Adaptive direct search algorithms with relaxable and quantifiable constraints." pith.science (2026). https://pith.science/paper/C4EZG2OK

@misc{pith2026260705183,
  author       = {Pith},
  title        = {Pith review of: Adaptive direct search algorithms with relaxable and quantifiable constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C4EZG2OK}},
  note         = {Machine review of arXiv:2607.05183}
}
read the original abstract

This work introduces ADS-PB, an extension of the Adaptive Direct Search (ADS) framework for solving constrained blackbox optimization problems. With ADS, iterates progress without relying on mesh structures or sufficient decrease conditions on the objective function value. Unlike the extreme barrier approach used in ADS, where only unrelaxable constraints are considered, the proposed method also handles quantifiable and relaxable constraints using a Progressive Barrier (PB) mechanism that exploits both constraint and objective function values. A convergence analysis of the proposed framework under mild assumptions is presented. The performance of the proposed method is assessed using sets of analytical and simulation-based constrained test problems and is compared with state-of-the-art blackbox optimization solvers, including the PB approach within the Mesh Adaptive Direct Search (MADS) framework.

Figures

Figures reproduced from arXiv: 2607.05183 by the authors.

Figure 1
Figure 1. Partition of the (h, f)-plane induced by the infeasible incumbent x k I , the feasible incum￾bent x k F , and the barrier threshold h k max. Solid bold lines and filled points are included in the shaded region, whereas dotted bold lines are excluded. The barrier threshold parameter h k max monotonically evolves depending on whether the iteration is dominating, improving, or unsuccessful. The update procedure for h k… view at source ↗
Figure 2
Figure 2. Trajectory and convergence plots for the MADS-PB method and a model-based algo [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Data profiles on the problems of [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Data profiles on the STYRENE and SOLAR 6 problems. strained blackbox settings with general-purpose evolutionary methods. In contrast, MADS-PB remains competitive for moderate tolerances (τ = 10−2 and 10−3 ), but a clear gap emerges at stricter tolerances (τ = 10−4 ), w…

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Reference graph

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Pith tools

Reviewed July 8, 2026 · model on record in the stance chip above.