{"id":"011ae620-a085-42cb-97e2-2f393bcd66e2","arxiv_id":"2505.04089","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The proposed SDMC condition does not guarantee global convergence: the proof confuses infinitely many positive sampling chances with probability-one sampling.","lead":"An evolutionary-computation paper proposes a new measure-based test, SDMC, for whether an algorithm is guaranteed to reach a global optimum, and applies it to several PSO and DE variants. The core theorem is wrong: infinitely many positive sampling chances do not imply probability-one sampling, so the test cannot certify global convergence; the paper is worth reading as a case study in how convergence proofs can go wrong.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The sufficiency half of the H5 theorem is false: search-scope coverage only gives infinitely many positive sampling probabilities, and a summable sequence such as mu_t(A)=2^{-t} keeps prod(1-mu_t(A))>0.","rationale":"The reader's weakest assumption pinpoints exactly the invalid inference in the sufficiency proof: infinitely many generations with positive mu_t(A) do not imply prod(1-mu_t(A))=0 unless the probabilities are non-summable. This is the load-bearing step of the paper's central theorem. If H5 is not sufficient, the proposed SDMC method does not provide a valid criterion for global convergence, and the subsequent analyses of LDIW-PSO, DE, GTDE, SLPSO, and GTPSO rest on an unsound foundation. The explicit counterexample with mu_t(A)=2^{-t} shows the concern is not merely a missing technical detail but a genuine counterexample to the theorem. The necessity direction is also not established, as the proof's construction of A_s depends on N and does not show a fixed positive-measure set is missed with positive probability. Since the central theoretical contribution fails, the REJECT verdict stands without adjustment.","tokens_in":24738,"tokens_out":6264,"duration_ms":68179,"concrete_test":"Implement the one-dimensional construction: S=[0,1], f(x)=x, x_0=1/2; at each t sample xi_t from (1-2^{-t}) delta_{1/2} + 2^{-t} U([0,1]) and set x_{t+1}=min{x_t, xi_t}. This update is exactly the best-so-far rule satisfying H1, and every mu_t has support [0,1], so H5 holds with N=1 at every t. Compute the product prod_{t=1}^{T}(1-2^{-(t+1)}) for large T; it converges to about 0.5776, which is the probability that A=[0,1/2] is never sampled. If the theorem were correct, this product would be 0; the positive limit directly refutes the claimed sufficiency.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Section III-B's theorem that (H5) is necessary and sufficient for global convergence of any algorithm satisfying (H1). The sufficiency proof fails at the step after Eq. (7): from |T(A)| = infinity the paper concludes that Eq. (7) equals 0, i.e., that A is eventually sampled with probability 1. This inference is valid only if sum_{t in T(A)} mu_t(A) diverges, and H5 does not require that. Example: let S=[0,1] and let mu_t be the mixture (1-2^{-t}) delta_{1/2} + 2^{-t} U(S). The support of mu_t is S, so v(M(t))=v(S) and H5 holds with N=1 for every t. For A=[0,1/2], mu_t(A)=2^{-(t+1)} for t>=1, so prod_{t=1}^{infinity}(1-2^{-(t+1)}) > 0. With f(x)=x, an algorithm that keeps the best-so-far and samples xi_t from mu_t satisfies H1 but, with positive probability, never samples A and therefore never reaches the global minimum. Hence H5 is not sufficient. The necessity proof is also incomplete: failing H5 at one time t does not produce a fixed positive-measure set with summable sampling probabilities, because the complement S \\ U_{t,N} depends on N and may still be sampled after the window.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a taxonomy of convergence analyses for evolutionary computation, separating 'stable convergence' from 'global convergence,' and claims these two properties are mutually exclusive (Theorem 3, Section II-C). On this basis it introduces a 'scope and domain measure comparison' (SDMC) criterion, condition (H5), which asserts that an algorithm satisfying Solis–Wets condition (H1) guarantees global convergence if and only if, for every generation t, some finite window of N future generations has search-scope union of full domain measure. The authors claim a rigorous proof of this equivalence (Section III-B), illustrate the method on LDIW-PSO and a periodic partitioned sampler (Section IV), and apply it to DE, GTDE, SLPSO, and GTPSO (Section V), concluding that GT improves diversity, that GTDE can guarantee global convergence under its parameter settings, and that a constant-variance modification of GTPSO is beneficial. The paper closes with experiments on twelve benchmark functions comparing the modified GTPSO to the original.","tokens_in":25085,"tokens_out":6860,"duration_ms":73189,"significance":"If the central equivalence were correct, the SDMC criterion would be a genuinely useful tool: it avoids homogeneous-Markov-chain modeling, is easy to state, and could be checked by bounding the support of the sampling distribution. The paper also contains a useful practical observation, namely that a constant positive variance in the GTPSO gene-targeting step (Eq. (43)) improves performance on many benchmarks, and it correctly emphasizes that time-varying algorithms such as LDIW-PSO are not homogeneous Markov chains. However, the load-bearing theoretical claim in Section III-B is false: the sufficiency direction of (H5) rests on an invalid inference about infinite products. The necessity argument is incomplete, and the case-study analyses rely on unproved 'best-case' reductions from algorithm dynamics to search-scope measure. As it stands, the paper does not establish the SDMC method, and the theoretical conclusions about DE, SLPSO, GTDE, and GTPSO are not supported.","major_comments":[{"comment":"The sufficiency proof of (H5) is invalid. From |T(A)| = infinity the paper concludes that Eq. (7), namely the infinite product over t of (1 - mu_t(A)), equals 0 because it is 'the product of infinitely many values less than 1.' This is false: an infinite product of factors in [0,1] can converge to a positive limit, and it tends to 0 only if the sum of mu_t(A) diverges. Condition (H5) does not imply that divergence. For example, let S = [0,1] and let mu_t be the mixture (1 - 2^{-t}) delta_{1/2} + 2^{-t} U(S), where U(S) is the uniform distribution on S. The support of mu_t is S for every t, so H5 holds with N = 1 for every t. For A = [0,1/2], mu_t(A) = 2^{-(t+1)}, and the product over t of (1 - 2^{-(t+1)}) is strictly positive. With f(x) = x and a best-so-far selection rule satisfying H1, the algorithm has positive probability of never sampling A and hence never reaching the global minimum. Thus H5 is not sufficient for global convergence.","section":"Section III-B, after Eq. (7)"},{"comment":"The necessity argument does not establish that failure of H5 yields a fixed positive-measure set A with infinite product of (1 - mu_t(A)) positive. The proposed set A_s = S \\ U_{t,N} depends on the window length N; as N grows, A_s shrinks, and after the window the algorithm may sample it with high probability. The proof needs to construct a single Borel set that is missed forever with positive probability, but no such construction is provided. The sentence 'we can always construct a counterexample A_s' is an assertion, not a proof, particularly because H5 requires the covering property for every t and a failure at one t does not by itself preclude later finite windows from covering S.","section":"Section III-B, necessity paragraph"},{"comment":"The claimed mutual exclusivity of stable convergence and global convergence does not follow from Eqs. (4)-(5). Stable convergence of individual positions, in the sense lim E|Delta x_i(t)| = delta, does not imply that the support of mu_t, i.e., the search scope M(t), has measure tending to 0; an algorithm can have convergent trajectories while still performing rare global resampling events, and those rare events may keep v(M(t)) equal to v(S). Equation (5) is not derived: it is unclear how the constant delta bounds the measure of the union of individual search scopes, and the factor involving the gamma function appears without explanation. Since Theorem 3 is used to motivate and justify the SDMC analyses in Sections IV and V, this gap is load-bearing.","section":"Section II-C, Theorem 3"},{"comment":"The case-study analyses replace the actual stochastic algorithm by a 'best case' trajectory in which every update is accepted and every pbest improves, and then infer from stable convergence of this idealized dynamics that v(M(t)) tends to 0 and hence H5 fails. This inference is not justified: search-scope measure and trajectory stability are different objects, and the 'best case' construction removes the rare-event sampling that could make M(t) cover S. The same pattern appears in Section V-B4, where the conclusion that GTPSO does not satisfy H5 is based on the standard deviation of a Gaussian tending to 0, with no proof that the support measure vanishes. Consequently, even conditional on H5 being a valid criterion, the conclusions that LDIW-PSO, DE, and SLPSO do not guarantee global convergence are not established by the arguments given.","section":"Sections IV-A and V-A3"}],"minor_comments":[{"comment":"The reference numbering is internally inconsistent: citation [21] is used twice for apparently different works, and [35] is used both for Solis and Wets and for Rudolph; the correct citations should be restored before any revision.","section":"Section I and reference list"},{"comment":"Reference [44] is cited for the GTDE parameter settings, but reference [44] in the list is a paper on ant colony convergence; the intended citation appears to be [45], the GTDE paper.","section":"Section V-A4"},{"comment":"The displayed condition (H5) is garbled: 't forall != +infinity, N exists != +infinity' should read 'for every finite t there exists a finite N such that v(U_{t,N}) = v(S)'. The proof also introduces t_N without defining it and switches between t, t_N, and t_s.","section":"Section III-B, statement of (H5)"},{"comment":"The text 'B = C \\ S is the complement of C in S' should presumably read B = S \\ C, and Eq. (15) uses the symbol X_{t-1} without defining it as the previous generation's sample.","section":"Section IV-B"},{"comment":"The sentence 'Actually, (H5) is guaranteed to hold under the parameter setting given in [44] ... Thus, it is possible to have v(U_{t,N}) < v(S) for all t with N = 1' is self-contradictory: if H5 is guaranteed to hold, then v(U_{t,N}) = v(S) for some finite N, not v(U_{t,N}) < v(S) for all N. This sentence should be corrected because it directly concerns the paper's main claim about GTDE.","section":"Section V-A4"}],"recommendation":"reject","confidential_remarks":"The central theorem of the paper is false, and the error is not a local presentation issue: the sufficiency proof of (H5) relies on an invalid infinite-product inference, and the necessity proof is incomplete. Because the manuscript's main contribution is this equivalence, a revision within the current scope would require replacing the central result with a weaker statement (e.g., adding a summability condition) and reworking the case studies; that is effectively a new paper. I would not recommend a major revision. There is also a citation-hygiene concern: several references are misnumbered, including citations to the authors' own GTDE and GTPSO papers, which should be fixed in any future submission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the SDMC idea is original and the taxonomy of stable vs. global convergence is a reasonable way to organize the literature, but the central result does not hold. H5 is not sufficient for global convergence, and the necessity argument is also incomplete. This is not a fixable technical gap; the theorem as stated is wrong.\n\nWhat the paper does well: it correctly identifies that modeling time-varying EC algorithms as homogeneous Markov chains is often inappropriate, and the periodic partitioned sampling example is a nice demonstration that Rudolph's H3-type condition is not necessary for global convergence. The discussion of how stable convergence and global convergence are often conflated is also useful.\n\nThe load-bearing flaw is in the sufficiency proof of H5 (Section III-B). From |T(A)| = infinity, the authors conclude that the product over t of (1 - mu_t(A)) equals 0. That inference requires the sum of mu_t(A) over those generations to diverge; H5 does not require that. Counterexample: let S = [0,1], and let mu_t be (1-2^{-t}) delta_{1/2} + 2^{-t} U(S). The support of mu_t is S, so v(M(t)) = v(S), and H5 holds with N=1. But for A=[0,1/2], mu_t(A) = 2^{-(t+1)}, so the product over t is positive and A is never sampled with positive probability. An algorithm keeping the best-so-far and sampling from these mu_t satisfies H1, yet with positive probability never reaches the global minimum of f(x)=x.\n\nThe necessity proof is also under-specified: failing H5 at one time t with one window N does not produce a fixed positive-measure set whose sampling probabilities are summable, because the complement S \\ U_{t,N} depends on N and may be sampled later. Theorem 3, claiming mutual exclusivity of stable and global convergence, is stated too strongly; the proof only shows that stable convergence shrinks the search scope, which does not by itself preclude global convergence.\n\nThe applications to DE, GTDE, SLPSO, and GTPSO are informal best-case analyses. There is an internal inconsistency in the description of GTDE: F is defined as Gaussian with mean 0.7 and std 0.5, then later analyzed as mean 0.5 and std 0.1. The experimental modification of GTPSO is interesting, but it does not rescue the theory.\n\nRecommendation: send this to peer review rather than desk reject, because the taxonomy and the critique of Markov-chain practices deserve discussion, and the error is instructive. But the verdict should be reject: the main theorem needs to be replaced by a condition involving divergent sums of sampling probabilities, which effectively returns to the spirit of Solis-Wets H2 and undermines the paper's claimed contribution.","headline":"The paper's core theorem is false: H5 is not sufficient for global convergence, and the claimed proof misapplies Borel-Cantelli, so the paper should be rejected despite a genuinely useful critique of Markov-chain modeling.","tokens_in":25604,"tokens_out":5113,"would_cite":false,"duration_ms":53477,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A finite-window measure comparison on search scopes is claimed necessary and sufficient for global convergence.","keywords":["global convergence","evolutionary computation","search scope","domain measure comparison","convergence analysis","Markov chain","gene targeting","large-scale optimization"],"falsifier":"Construct an algorithm that at generation $t$ samples uniformly over the whole feasible domain with probability $2^{-t}$ and otherwise keeps the current best; H1 and H5 both hold, yet for any fixed positive-measure region $A$ the probability of never sampling it is $\\prod_{t \\ge 0}(1 - 2^{-t} v(A)/v(S)) > 0$, so it does not converge to the global optimum with probability 1.","tokens_in":24465,"feed_emoji":"🎯","tokens_out":13697,"duration_ms":126647,"temperature":0.7,"pith_summary":"This paper attempts to replace the standard Markov-chain machinery for proving that an evolutionary algorithm converges to a global optimum with a simpler geometric criterion. It separates two meanings of convergence—stable convergence, where the population settles into a region, and global convergence, where the best solution reaches the optimum with probability 1—and argues they are mutually exclusive. Its proposed scope-and-domain measure comparison (SDMC) condition says: an algorithm satisfying the standard non-worsening assumption (H1) is globally convergent if and only if, from every generation, some finite window of future search scopes has total measure equal to the feasible domain. If correct, this makes global convergence analyzable from population spread alone, without modeling the algorithm as a homogeneous Markov chain.","feed_headline":"One scope check can certify global convergence","feed_subtitle":"Global convergence holds when every generation starts a finite window of search scopes covering the whole domain.","key_machinery":"The load-bearing object is the search-scope measure comparison expressed as H5. The search scope $M(t)$ is the set of positions the population has a chance to reach at generation $t$, and $v$ is the measure (volume) of subsets of the feasible domain $S$. H5 compares the measure of the finite-horizon union $U_{t,N} = \\bigcup_{k=0}^{N-1} M(t+k)$ with $v(S)$; equality for some finite $N$ at every $t$ is the criterion. This identity replaces both the homogeneous-Markov-chain modeling assumption and the requirement that every positive-measure region be sampleable in every generation, converting a probabilistic sampling condition into a geometric covering condition. That conversion is what lets the paper read global convergence off population diversity.","core_discovery":"The paper's central claim is that condition (H5) is necessary and sufficient for global convergence among algorithms satisfying (H1). Here $M(t)$ is the search scope at generation $t$, $v$ is the measure (volume) on the feasible domain $S$, and H5 says that for every finite generation $t$ there is a finite $N$ with $v(\\bigcup_{k=0}^{N-1} M(t+k)) = v(S)$. Because boundary handling keeps $M(t) \\subseteq S$, equality of measures means the search scopes in every $N$-generation window jointly cover the whole domain. The paper concludes that such covering makes the probability of never sampling any positive-measure region zero, so the best-so-far solution reaches the global optimum with probability 1, and conversely that if some window always misses a positive-measure set, a non-convergent algorithm can be built. It applies this criterion to algorithms the Markov-chain route cannot handle and to gene-targeting variants, and uses it to derive design guidance for large-scale optimization.","pith_inferences":["The stated equivalence is stronger than the proof supports: infinitely many generations with $M(t) \\cap A \\neq \\emptyset$ do not imply $\\prod_t (1-\\mu_t(A)) = 0$ when $\\mu_t(A)$ decays rapidly, so a repaired sufficient condition would need something like $\\sum_t \\mu_t(A) = \\infty$ for every positive-measure $A$.","Read as a covering condition, H5 is a close relative of the standard probability principle that infinitely many events with rapidly decaying probabilities need not occur; the natural strengthened form would require each positive-measure region to be sampled infinitely often with probabilities bounded away from zero. Experiments should distinguish domain coverage from probability of sampling every ","The paper's use of the standard deviation of population positions to estimate $v(M(t))$ suggests a practical diagnostic, but lifting finite-time diversity measurements to an infinite-time criterion is an extrapolation the paper does not formalize; that gap could be tested by monitoring whether estimated coverage stabilizes or collapses."],"forward_implications":["Algorithms with time-varying parameters, which cannot be modeled as homogeneous Markov chains, become analyzable by tracking search-scope volumes alone.","An algorithm whose search scope shrinks to measure zero cannot guarantee global convergence, so a stable-convergence proof cannot stand in for a global-convergence proof.","Keeping a positive-measure search component throughout evolution, such as Gaussian perturbations with $\\sigma > 0$, is a design rule that satisfies H5 under H1.","Under the SDMC analysis, GTDE guarantees global convergence with its original settings, while DE, SLPSO, and GTPSO do not; giving GTPSO's Gaussian perturbation a positive constant $\\sigma$ improves it on most tested benchmarks."],"supporting_citations":[{"why":"supplies the original sufficient conditions (H1)-(H2) for global convergence that the proposed (H5) is designed to replace.","marker":"[35]"},{"why":"is the homogeneous Markov-chain convergence method the paper argues is too restrictive and prone to misuse.","marker":"[36]"},{"why":"is the LDIW-PSO algorithm used as the first worked example, since its time-varying inertia weight rules out homogeneous Markov modeling.","marker":"[46]"},{"why":"is the gene-targeting DE algorithm whose original parameter settings the paper analyzes as globally convergent under H5.","marker":"[45]"},{"why":"provides the stability proof for SLPSO that the paper invokes when concluding SLPSO cannot guarantee global convergence.","marker":"[56]"}],"fun_headline_variants":["Finite scope window covering domain certifies global convergence","Global convergence iff a finite window of scopes covers the domain","Necessary and sufficient: scopes covering the domain in finite windows","Measure search scopes, not Markov chains, to certify global convergence","SDMC: finite scope window covering domain gives global convergence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's crucial inference is that an infinite sequence of chances to sample a region makes missing that region forever impossible; this fails if the chances to sample the region decay fast enough, and the whole equivalence rests on that step.","fun_headline_variants_meta":{"raw":{"variants":["Finite scope window covering domain certifies global convergence","Global convergence iff a finite window of scopes covers the domain","Necessary and sufficient: scopes covering the domain in finite windows","Measure search scopes, not Markov chains, to certify global convergence","SDMC: finite scope window covering domain gives global convergence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001696,"raw_usage":{"total_tokens":6730,"prompt_tokens":972,"completion_tokens":5758,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":5672}},"tokens_in":588,"tokens_out":5758,"duration_ms":39538,"temperature":1.0,"reasoning_tokens":5672,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T23:39:14.969652+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an algorithm that at generation $t$ samples uniformly over the whole feasible domain with probability $2^{-t}$ and otherwise keeps the current best; H1 and H5 both hold, yet for any fixed positive-measure region $A$ the probability of never sampling it is $\\prod_{t \\ge 0}(1 - 2^{-t} v(A)/v(S)) > 0$, so it does not converge to the global optimum with probability 1.","supporting_citations":[{"cited_title":"Minimization by random search techniques,","cited_arxiv_id":null,"evidence_quote":"supplies the original sufficient conditions (H1)-(H2) for global convergence that the proposed (H5) is designed to replace."},{"cited_title":"Convergence of evolutionary algorithms in general search spaces,","cited_arxiv_id":null,"evidence_quote":"is the homogeneous Markov-chain convergence method the paper argues is too restrictive and prone to misuse."},{"cited_title":"Gene targeting particle swarm optimization for large - scale optimization problem,","cited_arxiv_id":null,"evidence_quote":"is the LDIW-PSO algorithm used as the first worked example, since its time-varying inertia weight rules out homogeneous Markov modeling."},{"cited_title":"Gene targeting differential evolution: a simple and efficient method for large scale optimization,","cited_arxiv_id":null,"evidence_quote":"is the gene-targeting DE algorithm whose original parameter settings the paper analyzes as globally convergent under H5."}],"review_version":1}