{"id":"ae099027-11ae-45a5-b635-e50374d1876d","arxiv_id":"2607.05183","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":3,"one_line_summary":"ADS-PB extends mesh-free adaptive direct search to constrained blackbox optimization by incorporating a progressive barrier mechanism, with convergence guarantees and improved practical performance over mesh-based methods.","lead":"The paper introduces ADS-PB, a new algorithm for constrained blackbox optimization that combines mesh-free adaptive direct search with a progressive barrier for handling relaxable constraints. It matters for engineers and scientists optimizing expensive simulations where constraints can be violated during search but carry useful information.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The convergence proof of Theorem 3.1 has a gap: the case |S_ε| finite does not handle infinitely many improving iterations, which may prevent δ_k from decreasing.","rationale":"The reader correctly identified that the deferred increase/decrease rules from [10] are a problem, but framed it as an incompleteness/reproducibility issue. My analysis suggests a more specific concern: the proof of Theorem 3.1 Case 1 may actually depend on a property of those unstated rules — specifically that the decrease factor is bounded away from 1. Without this, the argument that finitely many successful iterations in K_ε leads to a contradiction may not hold. The reader's other points (compactness assumption, NOMAD reproducibility) are valid but less load-bearing. The compactness assumption (Assumption 1) is standard in DFO and the unrestricted search step is a feature, not a bug — the assumption simply scopes the theoretical guarantee. The NOMAD version issue is a practical reproducibility concern, not a theoretical one. The convergence theory is the paper's core contribution, and the gap I identified (or its resolution once [10]'s rules are checked) determines whether the central claim holds. If the decrease rules from [10] do guarantee a bounded decrease factor, the proof is fine and the verdict should remain CONDITIONAL only for the reproducibility issues. If not, the convergence guarantee needs revision. Either way, the paper should restate the update rules for self-containedness.","tokens_in":18776,"tokens_out":928,"duration_ms":429982,"concrete_test":"Retrieve and restate the increase/decrease rules from [10, Equations (2) and (3)]. Then verify: if δ_k ≥ ε and the iteration is unsuccessful or reframing, does the decrease rule guarantee δ_{k+1} ≤ γ·δ_k for some fixed γ ∈ (0,1)? If yes, the proof of Theorem 3.1 Case 1 holds. If the decrease factor can approach 1 (e.g., δ_{k+1} = δ_k / (1 + 1/k)), then δ_k ≥ ε could persist indefinitely and the proof breaks.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 3.1 partitions K_ε = {k : δ_k ≥ ε} into two cases based on S_ε, the subset of dominating and improving iterations. Case 1: |S_ε| finite. The paper argues that for k ≥ k_0, all iterations are unsuccessful or reframing, so δ_k decreases, contradicting δ_k ≥ ε. But this is incomplete. The 'otherwise' branch of (9) (which triggers decrease) covers only unsuccessful AND reframing iterations. If there are infinitely many improving iterations with δ_k ≥ ε but only finitely many dominating ones, then S_ε is infinite (improving iterations are in S_ε), so this falls into Case 2, not Case 1. However, Case 2's argument only addresses the packing of successful points t_k via Bolzano-Weierstrass. The real gap is subtler: in Case 1, the claim that 'there exists k_0 such that for all k ≥ k_0 the iteration is unsuccessful or reframing' does not follow from |S_ε| finite alone. S_ε is defined as the subset of K_ε iterations that are dominating or improving. If |S_ε| is finite, then for k ∈ K_ε with k ≥ k_0, the iteration is indeed unsuccessful or reframing. But the decrease rule (9) for 'otherwise' (which includes both reframing and unsuccessful) must ensure δ_k actually decreases below ε in finitely many steps. The increase/decrease rules from [10] are not restated, so we cannot verify that the decrease factor is bounded away from 1 (e.g., δ_{k+1} ≤ γ·δ_k with γ < 1). If the decrease rule allows δ_{k+1} arbitrarily close to δ_k, then δ_k ≥ ε could persist for infinitely many unsuccessful/reframing iterations without contradiction. This is the load-bearing issue: the proof's Case 1 implicitly assumes that consecutive decreases drive δ_k below any fixed ε in finitely many steps, but this property is not established without seeing the actual decrease rule from [10, Equations (2) and (3)].","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","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.","tokens_in":19797,"tokens_out":1404,"duration_ms":177718,"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":[{"comment":"§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.","section":null},{"comment":"§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.","section":null},{"comment":"§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.","section":null}],"minor_comments":[{"comment":"§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.","section":null},{"comment":"§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.","section":null},{"comment":"§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.","section":null},{"comment":"§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.","section":null},{"comment":"§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.","section":null},{"comment":"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].","section":null},{"comment":"§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.","section":null},{"comment":"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.","section":null}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution from an experienced group, and the core ideas are sound. The main issue is the incomplete restatement of the update rules from [10], which makes the algorithm and proof not fully self-contained. This is fixable by restating the key properties. The stress-test concern about Theorem 3.1 Case 1 is valid but likely resolvable once the decrease rule properties are stated. I would encourage the authors to make the paper as self-contained as possible given that [10] is a recent preprint."},"author_rebuttal":{"model":"glm-5.2","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"§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."},{"response":"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_made":"yes","referee_comment":"§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."},{"response":"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_made":"yes","referee_comment":"§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."}],"tokens_in":18553,"tokens_out":1160,"duration_ms":97885,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"Bottom line: this paper extends the Adaptive Direct Search (ADS) framework to handle relaxable, quantifiable constraints via a Progressive Barrier mechanism. The core idea — replacing mesh projection with a punctured-space exclusion approach so model-based trial points are evaluated directly — is well-motivated and the convergence analysis is correct. The computational results show a consistent advantage over MADS-PB, especially at strict tolerances. It deserves a serious referee, but there are two real gaps to address before acceptance on the algorithm spec and the proof's dependence on an external reference.","headline":"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.","tokens_in":19896,"tokens_out":185,"would_cite":true,"duration_ms":57318,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Mesh-free direct search handles constraints and converges to zero","keywords":[],"falsifier":"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.","tokens_in":19004,"feed_emoji":"🎯","tokens_out":757,"duration_ms":73470,"temperature":0.7,"pith_summary":"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.","feed_headline":"Mesh-free search handles constraints and converges to zero","feed_subtitle":"ADS-PB replaces mesh projection with punctured-space exclusion, letting model-suggested points be evaluated directly—proving both radius and","key_machinery":"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)","core_discovery":"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","pith_inferences":[],"forward_implications":["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."],"fun_headline_variants":["Mesh-free direct search discards near-duplicate trial points","Punctured space method enforces convergence in constrained search","Excluding prior successes drives direct search convergence","ADS-PB ditches mesh projection for punctured-space exclusion","Progressive barrier direct search converges without meshes"],"cache_read_input_tokens":0,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Mesh-free direct search discards near-duplicate trial points","Punctured space method enforces convergence in constrained search","Excluding prior successes drives direct search convergence","ADS-PB ditches mesh projection for punctured-space exclusion","Progressive barrier direct search converges without meshes","Discarding nearby points proves convergence for adaptive search"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":860,"prompt_tokens":452,"completion_tokens":408,"prompt_tokens_details":null},"tokens_in":452,"tokens_out":408,"duration_ms":8088,"temperature":1.0,"reasoning_tokens":384,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T00:30:07.509433+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"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.","supporting_citations":[],"review_version":1}