Genetic algorithms [9], differential evolution [10], particle swarms [11] and simulated annealing [12] provide different candidate-generation mechanisms. Their basic forms need admissible-variable and constraint-handling rules when applied to engineering models. GWO [13] is a named comparator in the supplied experiment; its presence does not make the comparison exhaustive of current optimization methods.

SHADE [14] adapts differential-evolution parameters from successful trials, while L-SHADE [15] adds population-size reduction. Subsequent mutation and hybrid variants [16] broaden this design space. Parameter adaptation and operator selection are related but distinct mechanisms; their performance depends on the specific executed variant. Covariance-adaptation methods [17] form a separate stateful comparison family. Hyper-heuristics [18] and adaptive operator selection [3], [4] already study online allocation among heuristics. AOPPS is positioned as a concrete engineering-oriented composition rather than the first framework to use structure-aware search. Automatic configuration through irace [19] can support such comparisons only when configurations and development costs are recorded.

The AOPPS credit is a normalized recent improvement from a selected operator attempt. A softmax distribution then allocates future attempts. This admits a bandit interpretation at the level of action selection [20–22], but the operator rewards depend on the evolving population and constraint state; they are not stationary independent arm rewards. The cited bandit literature [20], [21] proves guarantees for specified policies and assumptions. It does not supply an O(√(K log K/T)) regret rate for an arbitrary fixed-forgetting-factor, entropy-modulated softmax. No such rate is claimed for AOPPS. Logged credits can describe adaptation, but do not identify a phase-optimal policy without counterfactual operator comparisons.

Welded-beam, pressure-vessel, spring and speed-reducer models are useful engineering test cases [1], [2]. Their multiple published formulations can differ in bounds, admissible thicknesses, coil counts and constraint conventions. The source labels and reference values must therefore be tied to complete equations before feasibility and optimality claims are reproducible. These analytic test cases do not reproduce simulator cost, manufacturing tolerances or the full validation requirements of deployed engineering designs. They are used here as reported optimization cases. A small reference-relative gap does not certify an implementable or safe physical design.

AOPPS combines a local constraint-correction attempt with perturbation, discrete repair, refinement and crossover. Its evaluation question is whether this composition improves feasible outcomes under a fully counted objective-call budget. The supplied tables provide descriptive evidence, while the incomplete case specifications, tuning provenance and statistical inputs prevent a verified novelty or superiority certificate.

B1 is reported as a four-variable welded-beam cost model, f=1.10471h²L+0.04811tb(14+L), with five constraints and reference 1.724852. The abbreviated description supplies nominal bounds 0.1–10.0 but not the full stress, deflection, buckling and geometric equations. These missing details must be supplied before the reference can be matched to the executed variant.

B2 is reported as pressure-vessel cost minimization in shell thickness, head thickness, radius and cylinder length. Thicknesses are multiples of 0.0625 inch, so a continuous move must be mapped back to admissible thickness indices. The source reference is 5885.3327. Complete bounds, volume and thickness constraints, tolerance conventions and returned vectors are needed to validate feasible comparisons.

B3 is a three-variable spring-volume case with wire diameter, coil diameter and a continuous active-coil count, four reported constraints and reference 0.012665. Treating coil count as continuous defines a relaxation of a manufactured integer-coil design. The result therefore applies only to the executed relaxation unless an integer design and its feasibility are separately established.

B4 is a seven-variable speed-reducer case with eleven reported constraints and reference 2994.4207. The description includes a tooth-count variable alongside continuous shaft and gear dimensions. The full objective, bounds and constraint equations remain necessary to reproduce the source outcomes.

B5 is unresolved at the model-definition level. The prose names four tooth-count variables despite saying three, while Table 1 reports dimension six. It gives target ratio 4.611, integer bounds [5] and reference 1.49×10⁻⁷, but not the ratio formula or a six-variable mapping. We retain the reported numerical results without inventing the two missing variables or asserting that they correspond to a standard planetary-gear benchmark.

AOPPS maintains N evaluated candidates and a global K=5 credit table. Candidate generation, admissible-variable repair and nonlinear feasibility checking are separate steps. Entropy modulates allocation temperature; the reported feasibility rate is a diagnostic unless an explicit control dependence is specified. The global credit table uses recent outcomes without fitting a neural policy. Its overhead is small in K, but its total cost depends on the selected operators. SQP iterations, derivative estimates and constraint-correction trials may consume multiple objective calls for one population member and must be counted.

Its theorem inset is not a verified convergence result; the conditional statement and required safeguards are given in Section 4.5. The schematic does not establish the exact executed operator dictionary or state updates.

The source portfolio contains a heavy-tailed perturbation, integer-increment repair, local feasibility correction, up to five SQP steps, and arithmetic crossover. Integer repair and bound clipping are mandatory admissibility operations after every move, even when also available as a selectable operator. The local correction uses constraint normals but cannot certify nonlinear feasibility.

SQP applies to differentiable continuous subproblems with discrete coordinates fixed; numerical derivative calls consume budget. Crossover needs an explicit discrete inheritance rule. The five labels describe intended roles. Heavy tails do not prove escape from every basin, rounding does not enforce nonlinear constraints, and an arithmetic blend of feasible points need not remain feasible in a nonconvex region. Claims about extra operators or preliminary diminishing returns require their omitted experimental outputs.

Let the population entropy at iteration t be defined as

H_t = − Σ_b p_b(t) · log p_b(t), (3)

Here p_b(t) is the fraction of candidates in bin b of a specified projected histogram. Binning, scaling and the projection basis must be documented; a moving principal-component basis can change entropy without a change in global reachability. Lower entropy lowers the source temperature and concentrates allocation toward higher-credit operators, rather than selectively increasing exploration.

Pr(op = k | t, i) = exp(Q_k(t)/τ_t) / Σ_{k'} exp(Q_{k'}(t)/τ_t), (4)

The source schedule τ_t=τ₀H_t/H₀ is undefined if H₀=0 and singular if H_t=0. A well-defined implementation requires positive denominator and temperature safeguards, for example τ_t=max{τ_min,τ₀H_t/max(H₀,h_min)} with τ_min,h_min>0. This is a consistency specification whose execution must be checked against source code. No fixed lower operator-probability bound follows from the unsafeguarded schedule.

Q_k(t+1) = (1 − η) · Q_k(t) + η · ΔF_k(t), (5)

For an operator attempt at a fixed t, compare parent and offspring using the same F_t. A well-defined credit is max{0,F_t(parent)−F_t(offspring)}/max(m_t,m_min), where m_t is the recorded median absolute improvement over the last 50 iterations and m_min>0. Each selected operator must receive an update for the attempt, including zero improvement, if the credit is intended to estimate average attempt utility. Multiple attempts require a stated update order; the other credit components remain unchanged.

Eq. (5) is recency-weighted reward estimation followed by Boltzmann allocation [21]. It is not an exact equivalence to a policy with a transferable generic regret bound. A decreasing temperature increases exploitation; bounded rewards and positive temperature alone do not establish useful candidate coverage or finite-time optimization rates.

Algorithm 1. AOPPS.

Input: N candidates, generation cap T, hard new-objective-call budget B, operators and constraints.

Output: best evaluated feasible design if one exists; otherwise report no feasible incumbent.

1. Initialize exactly N admissible candidates; evaluate objectives and constraints, and count every call.

2. Set Q_k=0; initialize a feasible incumbent and the stated histogram/credit history.

3. For t=1,…,T while budget remains:

4. Compute the specified entropy and report the feasibility rate.

5. Update a positive safeguarded temperature; confirm the executed safeguards.

6. Set λ(t) and compare all candidates using scores at this same t.

7. For each budget-permitted candidate attempt:

8. Sample operator k using Eq. (4).

9. Generate an offspring, holding integer coordinates fixed during continuous SQP.

10. Repair bounds and integer increments; apply the specified local correction policy.

11. Evaluate the original objective and constraints; count derivative and repair trials.

12. Calculate pre-replacement reward and update selected credit, including unsuccessful attempts.

13. Accept under the stated same-time score rule; retain any improved feasible incumbent.

14. End candidate loop.

15. Apply the recorded stagnation rule, reporting actual evaluations if stopped early.

16. End generation loop at T or the hard budget B.

17. Return the feasible incumbent and all residual violations, or a feasibility-failure status.

For an objective-call cost C_f, vector operations for a population cycle can be O(Nd), and allocation uses O(NK). Local projections and SQP may require Jacobians, linear or quadratic subproblem solves and several calls, so they are not generally O(d). With N=50 and T=1,000, a single call per particle per cycle would already require 50,000 calls, exceeding the reported B=20,000. The hard call counter must therefore control execution independently of the generation cap. A coverage result needs assumptions about feasible candidate proposals, not merely compactness, bounded rewards, operator-label selection or ergodicity of discrete coordinates. Assume exact evaluations of one fixed original objective and a finite value f★=inf of the original objective on the feasible mixed domain, and let Aε be its feasible ε-sublevel set. Suppose an elitist feasible incumbent is retained and each eligible proposal trial has conditional probability at least qε>0 of producing a point in Aε. Proposition 1 (conditional hitting). After m such trials, the probability that the best feasible incumbent has not reached Aε satisfies

Pr(best_m>f★+ε, or no feasible incumbent) ≤ (1−qε)^m. (6)

Repeated conditional expectation bounds consecutive miss probabilities by their product; after a hit, the feasible incumbent is retained. If eligible trials continue indefinitely for every ε>0, objective values approach f★ almost surely. A positive-temperature softmax over operators and discrete rounding do not establish the required probability on continuous feasible sublevel sets, especially with equality constraints or zero-volume feasible sets. The finite stopping rule, bounded penalty and source outputs do not verify these assumptions, and no Hoeffding objective-error rate is claimed.

The original B1 plot is retained but cannot presently be interpreted as a fixed-objective best-so-far feasible trace. Its median rises near the annotated SQP transition, which an elitist fixed-objective incumbent cannot do. Its endpoint also differs from the principal B1 summary. A changing penalty score, current candidate or another campaign would be a different quantity; the raw trace must identify which was plotted. The source’s phase annotations describe an intended exploration-to-refinement interpretation. They do not establish that a component caused the depicted changes or that the same phases occurred on other cases. Iteration counts also do not provide equal objective-call locations when selected operators have different costs.

The labelled best-so-far median is nonmonotone near the SQP transition and its endpoint is not reconciled with Table 2. The plotted quantity, campaign and interval inputs require confirmation; no convergence certificate is inferred.

The original probability plot describes reported shifts among operator labels. A high selected share indicates allocation by the controller, not that the operator is optimal at that stage. Demonstrating tracking of the best operator would require contemporaneous counterfactual evaluations or a defined reference policy. Aggregated selection profiles can hide variation across runs. Credit initialization, histogram definition and zero-improvement updates must be supplied to attribute the plotted profile to the stated recurrence. The plot does not independently verify causal roles of perturbation, projection, crossover or SQP.

Selection logs and the implemented credit/temperature update are required to verify its relationship to Eqs. (4) and (5).

AOPPS has the lowest displayed mean relative gap on all five cases in Table 2. The supplied mean ranks give AOPPS 1.02, but their aggregation differs from simply ranking the displayed case means. Conditional on the reported CD=3.37, the differences from GA (4.14) and SA (4.77) exceed the threshold; differences from DE, GWO and PSO do not. This corrects the broader claim attached to the source abstract. The source Friedman statistic and p-value require the rank matrix. Table 3’s very small aggregate p-values do not establish all 25 per-case contrasts. Its displayed Holm entries also fail the step-down monotonicity requirement for the stated family when applied to the displayed raw p-values. Rebuilding the analysis requires original differences with their case/run mapping; a pooled sample count of 150 does not create 150 independent benchmark problems.

Feasibility, case equations and aggregation require verification.

Using its source CD=3.37, AOPPS exceeds the threshold only against GA and SA. This conditional reading is not a reconstruction of the test inputs.

Table 4 retains source effect sizes and intervals. Cliff’s δ [6] measures ordering of the defined outcomes, and Hedges’ g [7] standardizes a difference with a small-sample correction. Their strongly negative values favor AOPPS under the stated difference direction, but require the actual feasible-outcome arrays. Threshold labels and nonzero intervals are conditional descriptions, not validation of missing case definitions or the pooling design. Averaging standardized effects across only five cases is a descriptive summary whose weighting and uncertainty must be justified. It does not establish a general advantage over an engineering-problem population or demonstrate that the controller caused the observed gaps.

Mean differences and interval limits are in relative-gap ratio units, whereas Table 2 uses percentages. Negative differences favor AOPPS; raw arrays and resampling provenance are required.

The heatmap supplies reported case-level ranks. It must be tied to either per-run rankings or ranks of case means; these are different summaries. Favorable displayed ordering across five cases motivates further evaluation without establishing generalization to unseen engineering models.

Morris [23] screening uses elementary effects over specified parameter ranges and a defined response. The source reports 30 trajectories and four levels and ranks λ₀ and N highest by mean absolute effect. Those rankings are conditional on the ranges, normalization and response. A nonzero dispersion can arise from nonlinearity or interaction and does not separate them.

Low elementary effects within sampled ranges do not establish transferable safe defaults or low sensitivity outside those ranges. The design matrix, response definition, repeated-run aggregation and uncertainty are needed to reproduce the reported screening and compare integer parameters with continuous ones.

Parameter ranges, normalized steps and response construction are unresolved; the plot does not certify robust deployment settings.

Table 5’s full B1 mean is 0.04%, while Table 2 reports 4.54% for B1. This unresolved campaign or metric discrepancy prevents treating the ablation as a decomposition of the principal result. Arithmetic relative to the displayed 0.04 baseline gives increases of 875% for uniform allocation, 2,550% without projection, 750% without SQP, 350% without Lévy perturbation and 200% without crossover. These correct the prose’s smaller percentages but do not validate the ablation data. Projection removal has the largest displayed ablation gap. Its magnitude exceeds the increase under uniform allocation, contradicting the claim that every individual removal hurts less than disabling adaptation. Equal call budgets, actual residual violations and unchanged remaining settings are required for causal interpretation. The supplied interval bounds are retained, but their scale and estimand do not reconcile with the displayed percentage changes.

The principal-result baseline and interval definition require reconciliation; no new ablation was executed.

The supplied timing increases from 12.3 s at D=30 to 489.7 s at D=200, a factor of approximately 39.8; dimension increases by approximately 6.7. The gap changes from 0.04% to 0.21%, a factor of 5.25. These are endpoint arithmetic, not a proved asymptotic scaling law. The claimed replication by D/4 is not an integer number of four-variable units at D=30, so the generator and reference scaling must be supplied.

The replicated-case generator, evaluation accounting and timing protocol require confirmation.