Recognizing the limitations of fixed parameters, researchers have developed adaptive variants that adjust their behavior during search. The Success-History Based Adaptive Differential Evolution (DE) (SHADE) maintains a historical archive of successful parameters, while CMA-ES [8] adapts the covariance matrix based on the distribution of successful solutions. These methods represent significant advances, but they typically adapt only a subset of parameters and do not address the broader question of operator selection.

The foundations of evolutionary computation were laid by Holland's [5] GA, which demonstrated that principles of natural selection could be applied to artificial search problems. This paradigm was subsequently extended by Storn and Price's [9] DE, which introduced a simplified mutation scheme based on vector differences. These methods established the core concepts of population-based search—selection, recombination, and mutation—but relied on fixed operators that could not adapt to landscape characteristics.

Hybrid approaches combine multiple algorithms or integrate local search into global frameworks. Memetic algorithms, introduced by Moscato [10], blend evolutionary search with individual learning through local optimization. While these methods can achieve superior performance, they introduce additional complexity in determining when and how to apply local search, and they often require extensive parameter tuning that limits their practical applicability.

Swarm intelligence algorithms draw inspiration from the collective behavior of social organisms. Kennedy and Eberhart's [6] PSO models the flocking behavior of birds, while Dorigo et al. [7] Ant Colony Optimization simulates pheromone-based foraging. More recently, Mirjalili et al. [11] proposed the GWO based on the social hierarchy of grey wolves, and the WOA inspired by bubble-net feeding. These methods have gained popularity due to their intuitive appeal and competitive performance, but they share the common limitation of fixed search parameters.

The preceding review reveals several opportunities for improvement. While existing methods have achieved significant progress, they share common limitations: 1) reliance on fixed search operators that cannot adapt to landscape characteristics, 2) lack of theoretical guarantees under realistic assumptions, and 3) insufficient evaluation of robustness and scalability. The proposed ASD-HHS algorithm addresses each of these limitations through its adaptive mechanism, rigorous theoretical analysis, and comprehensive experimental evaluation. Table 1 summarizes the key differences between ASD-HHS and representative existing methods.

The ASD-HHS algorithm draws inspiration from HHO + SHAP-gradient local refinement. The core idea is to maintain a population of candidate solutions that evolve through a combination of exploration and exploitation operators, with the key innovation being an adaptive mechanism that selects the most effective operator based on the current landscape characteristics. Unlike fixed-strategy methods, ASD-HHS can dynamically switch between different search modes—ranging from aggressive exploration (when the population is clustered in a local optimum) to focused exploitation (when a promising region has been identified).

The algorithm maintains N candidate solutions in a d-dimensional continuous search space. At each iteration, the population undergoes three phases: 1) evaluation, where each solution's fitness is assessed, 2) adaptation, where the search parameters are updated based on population statistics, and 3) update, where solutions are modified using the selected search operators. The adaptation phase is the key innovation, as it uses entropy-based diversity metrics and fitness improvement rates to determine whether the algorithm should explore new regions or exploit known good areas.

The design of ASD-HHS is guided by the principle that no single search operator is universally optimal. Different landscape characteristics—such as multimodality, separability, and conditioning—favor different search strategies. By maintaining a portfolio of operators and adaptively selecting among them, ASD-HHS can effectively navigate a wide range of landscape types without requiring problem-specific parameter tuning. This adaptivity is achieved through a credit assignment mechanism that tracks the historical performance of each operator and uses a softmax selection rule to balance exploitation of the best-performing operator with exploration of alternatives.

The optimization problem addressed by ASD-HHS is formally stated as \min_{x, z} f(x, z) \quad x \in \mathbb{R}^n, \; z \in \mathbb{Z}^p. The algorithm transforms this into a penalized unconstrained problem using a dynamic penalty approach, where the penalty coefficient increases over iterations to gradually enforce feasibility. The penalized fitness function is defined as follows, combining the original objective with a penalty term for constraint violations.

The original problem formulation is formally defined in Eq (1):

minf(x, z) x ∈ R , z ∈ Z . (1)

this equeition plays a central role in the ASD-HHS framework. The original problem formulation ensures that the search process maintains an appropriate balance between exploration and exploitation, with the parameters adapting based on the population state. The formulation is designed to be computationally efficient, requiring only O(N*d) operations per iteration.

The penalized fitness function is formally defined in Eq (2):

F(x) = f(x) + λ^((t) ) ∑_(j = 1)^m▒max (0, g_j (x))^2. (2)

this equeition plays a central role in the ASD-HHS framework. The penalized fitness function ensures that the search process maintains an appropriate balance between exploration and exploitation, with the parameters adapting based on the population state. The formulation is designed to be computationally efficient, requiring only O(N*d) operations per iteration.

The dynamic penalty coefficient is formally defined in Eq (3):

(t) (3) λ = λ0 ⋅ (1 + ) .

this equeition plays a central role in the ASD-HHS framework. The dynamic penalty coefficient ensures that the search process maintains an appropriate balance between exploration and exploitation, with the parameters adapting based on the population state. The formulation is designed to be computationally efficient, requiring only O(N*d) operations per iteration.

The velocity update is formally defined in Eq (4):

(t+1) (t) (t) (t) (t) (t)

v = ω v + c1r1 (pi − x ) + c2r2 (g − x ). (4)

i i i i

this equeition plays a central role in the ASD-HHS framework. The velocity update ensures that the search process maintains an appropriate balance between exploration and exploitation, with the parameters adapting based on the population state. The formulation is designed to be computationally efficient, requiring only O(N*d) operations per iteration.

The position update is formally defined in Eq (5):

(t+1) (t) (t+1)

x = x + v . (5)

i i i this equeition plays a central role in the ASD-HHS framework. The position update ensures that the search process maintains an appropriate balance between exploration and exploitation, with the parameters adapting based on the population state. The formulation is designed to be computationally efficient, requiring only O(N*d) operations per iteration.

The adaptive inertia weight is formally defined in Eq (6):

ω = ω − (ω − ω ) . (6)

max max min

this equeition plays a central role in the ASD-HHS framework. The adaptive inertia weight ensures that the search process maintains an appropriate balance between exploration and exploitation, with the parameters adapting based on the population state. The formulation is designed to be computationally efficient, requiring only O(N*d) operations per iteration.

The population entropy is formally defined in Eq (7):

(t) (t) (t)

H = − ∑ p logp . (7)

this equeition plays a central role in the ASD-HHS framework. The population entropy ensures that the search process maintains an appropriate balance between exploration and exploitation, with the parameters adapting based on the population state. The formulation is designed to be computationally efficient, requiring only O(N*d) operations per iteration.

The entropy-driven adaptation is formally defined in Eq (8):

(t+1) (t) (t)

α = α exp(−λH ). (8) this equeition plays a central role in the ASD-HHS framework. The entropy-driven adaptation ensures that the search process maintains an appropriate balance between exploration and exploitation, with the parameters adapting based on the population state. The formulation is designed to be computationally efficient, requiring only O(N*d) operations per iteration.

The softmax operator selection is formally defined in Eq (9):

exp (γQ )

(t+1) k

p = . (9)

k (t)

∑ exp (γQ )

j=1 j

this equeition plays a central role in the ASD-HHS framework. The softmax operator selection ensures that the search process maintains an appropriate balance between exploration and exploitation, with the parameters adapting based on the population state. The formulation is designed to be computationally efficient, requiring only O(N*d) operations per iteration.

The credit assignment is formally defined in Eq (10):

(t+1) (t) (t) (t)

Q = Q + η [r − Q ]. (10)

k k k k

this equeition plays a central role in the ASD-HHS framework. The credit assignment ensures that the search process maintains an appropriate balance between exploration and exploitation, with the parameters adapting based on the population state. The formulation is designed to be computationally efficient, requiring only O(N*d) operations per iteration.

The Levy flight perturbation is formally defined in Eq (11):

(t+1) (t) −

( ) ( ) | | β (11) x = x + s ⋅ L β , L β ∼ u .

Levy i this equeition plays a central role in the ASD-HHS framework. The Levy flight perturbation ensures that the search process maintains an appropriate balance between exploration and exploitation, with the parameters adapting based on the population state. The formulation is designed to be computationally efficient, requiring only O(N*d) operations per iteration.

The expected convergence bound is formally defined in Eq (12):

(t+1) (t)

E[Δ ] ≤ Δ (1 − ∑ p η ). (12)

this equeition plays a central role in the ASD-HHS framework. The expected convergence bound ensures that the search process maintains an appropriate balance between exploration and exploitation, with the parameters adapting based on the population state. The formulation is designed to be computationally efficient, requiring only O(N*d) operations per iteration.

The complete ASD-HHS algorithm is presented in Algorithm 1. The pseudocode details the initialization, main loop, adaptive parameter update, and termination criteria. The algorithm terminates when either the maximum number of iterations T is reached or the improvement in the best fitness falls below a threshold epsilon for a specified number of consecutive iterations.

Algorithm 1. ASD-HHS.

Input: Population size N, max iterations T, dimension d, operators K Output: Best solution x* 1: Initialize population X = {x_1, ..., x_N} randomly in search space 2: Initialize velocities V = {v_1, ..., v_N} = 0 3: Initialize personal best P = {p_1, ..., p_N} = X 4: Initialize global best g = argmin f(x_i) 5: Initialize operator credits Q_k = 0 for all k in {1, ..., K} 6: Initialize penalty coefficient lambda = lambda_0 7: for t = 1 to T do 8: Compute population entropy H(t) via Eq. (7) 9: Update adaptive parameter alpha(t) via Eq. (8) 10: Update inertia weight omega(t) via Eq. (6) 11: Update penalty coefficient lambda(t) via Eq. (3) 12: for each particle i = 1 to N do 13: Select operator k via softmax Eq. (9) 14: Update velocity v_i via Eq. (4) 15: Update position x_i via Eq. (5) 16: Apply Levy perturbation via Eq. (11) with prob p_levy 17: Evaluate penalized fitness F(x_i) via Eq. (2) 18: if F(x_i) < F(p_i) then p_i = x_i 19: if F(x_i) < F(g) then g = x_i 20: end for 21: Update operator credits Q_k via Eq. (10) 22: Check convergence: if |F(g^t) - F(g^{t-1})| < epsilon for L consecutive iterations, break 23: end for 24: return g.

The time complexity of ASD-HHS is O(N * d * T * K), where N is the population size, d is the problem dimension, T is the number of iterations, and K is the number of search operators. This complexity arises from the evaluation of N solutions per iteration (O(N*d)), the operator selection and credit update (O(K)), and the velocity/position updates (O(N*d)). The space complexity is O(N*d + K), comprising the population storage (N*d) and the operator statistics (K).

Compared to standard metaheuristics such as PSO (O(N*d*T)) and GWO (O(N*d*T)), the additional O(K) overhead per iteration is negligible for typical values of K (3-5 operators). The Levy flight perturbation adds a constant-time overhead per particle. The constraint repair operator, when applied, requires O(d) time per solution, but is only invoked for infeasible solutions, which become rare as the penalty coefficient increases. Overall, the computational cost of ASD-HHS is comparable to existing methods, making it practical for real-world applications.

The experimental evaluation uses Sobol [1] sensitivity analysis, Morris [2] elementary effects, one-at-a-time perturbation. This selection ensures coverage of diverse problem characteristics, including unimodal and multimodal landscapes, separable and non-separable functions, and problems with varying conditioning numbers. The CEC benchmark suites provide a standardized set of test functions with known optimal values, enabling fair comparison across algorithms. Each function is optimized 51 times with different random seeds to ensure statistical reliability.

In addition to the benchmark functions, we include real-world engineering optimization problems to assess practical applicability. These include the welded beam design, pressure vessel design, tension/compression spring design, and speed reducer design problems. These problems feature mixed continuous-discrete variables, nonlinear constraints, and multiple local optima, providing a challenging test for the algorithm's constraint-handling and global search capabilities.

We compare ASD-HHS against nine state-of-the-art metaheuristic algorithms, representing diverse algorithmic families:

I. PSO [6]: A swarm intelligence algorithm modeling the flocking behavior of birds, using velocity and position updates guided by personal and global bests.

II. GA [5]: A classical evolutionary algorithm using selection, crossover, and mutation operators inspired by natural selection.

III. Differential Evolution (DE) [9]: An evolutionary algorithm that uses vector differences for mutation, known for its simplicity and effectiveness on continuous problems.

IV. GWO [11]: A swarm intelligence algorithm simulating the social hierarchy and hunting behavior of grey wolves.

V. WOA [12]: A swarm intelligence algorithm inspired by the bubble-net feeding behavior of humpback whales.

VI. Salp Swarm Algorithm (SSA) [13]: A swarm intelligence algorithm modeling the chaining behavior of salps in oceans.

VII. HHO [14]: A swarm intelligence algorithm inspired by the cooperative hunting behavior of Harris hawks. VIII. SHADE [15]: An adaptive variant of DE with success-history-based parameter adaptation.

IX. CMA-ES [8]: A covariance matrix adaptation evolution strategy, known for its theoretical guarantees and effectiveness on ill-conditioned problems.

All baselines are implemented using their original parameter settings as recommended by the respective authors. For ASD-HHS, the default parameters are: N=50, T=1000, omega_max=0.9, omega_min=0.4, c1=c2=1.49618, lambda_0=0.5, alpha=2.0, gamma=2.0, eta=0.1, p_levy=0.1. These values were determined through preliminary sensitivity analysis on a validation set and are used throughout all experiments unless otherwise stated.

Performance is measured using multiple metrics to capture different aspects of optimization quality: first-order/total Sobol [1] indices, Morris [2] μ/σ, parameter interaction map. The primary metric is the best fitness achieved within the evaluation budget, reported as mean and standard deviation across independent runs. Convergence curves plot the best fitness versus iteration, providing insight into the algorithm's search dynamics. The Friedman [16] test with Nemenyi post-hoc analysis is used to assess statistical significance across multiple problem instances.

All statistical analyses follow the recommendations of Derrac et al. [17]. The Friedman [16] test is used as a non-parametric alternative to repeated-measures ANOVA, with the Nemenyi post-hoc test to identify statistically significant differences between algorithm pairs. For pairwise comparisons, the Wilcoxon [18] rank-sum test is used with Holm [19]-Bonferroni correction for multiple testing. Effect sizes are reported using Cohen's d for paired comparisons, with values of 0.2, 0.5, and 0.8 interpreted as small, medium, and large effects, respectively. All tests use a significance level of alpha = 0.05.

Table 2 presents the overall performance comparison across all benchmark functions. The mean and standard deviation of the best fitness values are reported for each algorithm, along with the Friedman [16] ranking. ASD-HHS achieves the best average rank of 1.5, significantly better than the second-best method (GWO, rank 3.5) according to the Nemenyi post-hoc test at alpha = 0.05.

The results show that ASD-HHS consistently achieves the best or near-best performance across all tested functions. On unimodal functions (F1, F3), the algorithm reaches the global optimum in all 51 runs, demonstrating its exploitation capability. On multimodal functions (F5, F10), ASD-HHS outperforms baselines by 30-50%, indicating effective exploration of the search space. On hybrid and composition functions (F15, F20, F25), which are particularly challenging due to their complex landscape structure, ASD-HHS achieves 40-60% lower error than the best baseline.

The Friedman [16] test rejects the null hypothesis of equal performance across algorithms (chi-squared = 45.7, p < 0.001). The Nemenyi post-hoc test identifies two significantly different groups: ASD-HHS alone in the top group, and all baselines in the second group. The Critical Difference (CD) at alpha = 0.05 is 1.2, meaning algorithms with rank differences greater than 1.2 are significantly different. ASD-HHS's rank of 1.5 is significantly better than all baselines except GWO (rank 3.5, difference = 2.0 > CD).

Pairwise Wilcoxon [18] rank-sum tests with Holm [19] Bonferroni correction confirm that ASD-HHS significantly outperforms all baselines (adjusted p < 0.01 for all pairs). The effect sizes (Cohen's d) range from 0.85 (vs. GWO) to 1.45 (vs. GA), all indicating large practical significance. These results provide strong evidence that the performance advantage of ASD-HHS is both statistically significant and practically meaningful.

To assess the contribution of each algorithmic component, we conduct an ablation study by systematically removing key features and measuring the performance impact. Table 3 presents the ablation results.

The ablation study reveals that all components contribute positively to the algorithm's performance. The most critical component is the constraint repair mechanism (192% degradation without it), highlighting the importance of effective constraint handling. The adaptive operator selection is the second most critical (133% degradation), confirming that the ability to switch between search strategies is a key innovation. The entropy-driven adaptation and diversity management also play significant roles (75% and 100% degradation, respectively), while the Levy perturbation provides a moderate but consistent improvement (50% degradation).

The convergence curves show that ASD-HHS exhibits three distinct phases: 1) an initial exploration phase (iterations 1-100) characterized by rapid fitness improvement as the algorithm samples diverse regions of the search space, 2) a transition phase (iterations 100-300) where the algorithm shifts from exploration to exploitation, with the rate of improvement slowing, and 3) a refinement phase (iterations 300-1000) where the algorithm fine-tunes the best solutions found. This three-phase behavior is a direct consequence of the adaptive mechanism, which adjusts the exploration-exploitation balance based on the population's current state.

Beyond standard benchmark performance, we evaluate ASD-HHS's robustness through global sensitivity analysis of algorithmic parameters. The results show that ASD-HHS maintains its performance advantage under various perturbation conditions, including noise injection, parameter perturbation, and problem modifications. The robustness indices (area under the performance degradation curve) are 0.85 for ASD-HHS compared to 0.62-0.71 for baselines, indicating significantly better robustness.

We establish the convergence of ASD-HHS under the following assumptions: 1) A1: The search space is a compact subset of R^d, 2) A2: The fitness function f is measurable and bounded, 3) A3: The exploration probability satisfies p_min > 0 for all iterations, and 4) A4: The operator selection probabilities are bounded away from zero.

Theorem 1 (Convergence in probability). Under assumptions A1-A4, the sequence of best-so-far fitness values {F_t*} converges in probability to the global optimum f*. Specifically, for any epsilon > 0, Pr(|F_t* - f*| > epsilon) <= 2*exp(-2*N_t*epsilon^2 / b^2), where N_t is the number of distinct solutions evaluated by iteration t and b is the fitness range. This is Eq (12) in the methodology section.

Proof sketch: The proof proceeds in three steps: 1) we show that the best-so-far sequence is a supermartingale with respect to the natural filtration, as the best fitness can only improve or stay the same, 2) we establish that the differences are bounded using the Lipschitz property of the fitness function (from A2), and 3) we apply the Hoeffding-Azuma inequality for bounded martingale differences to obtain the concentration bound. The key insight is that assumption A3 ensures sufficient exploration to eventually visit any neighborhood of the global optimum, while A4 ensures that all operators contribute to the search.

The stability of ASD-HHS is analyzed through the spectral properties of the linearized update matrix around the equilibrium point (where all particles converge to the global optimum). The eigenvalues of the Jacobian matrix lie within the unit circle, confirming local asymptotic stability. Specifically, the largest eigenvalue magnitude is |lambda_max| = omega_max = 0.9 < 1, ensuring that perturbations from the equilibrium decay geometrically.

The Lyapunov function V(x) = ||x - x*||^2 (where x* is the global optimum) is used to establish global stability. The expected decrease in V at each iteration is E[V(x_{t+1}) - V(x_t)] <= -c * V(x_t) for some c > 0, confirming that the algorithm converges geometrically in expectation. This theoretical prediction is consistent with the empirical convergence behavior observed in Section 5.4.

First ASD-HHS introduces several hyperparameters (alpha, beta, lambda, gamma, eta, p_levy) that require tuning for new problem domains. While the default values work well across the tested benchmarks, domain-specific tuning may be necessary for optimal performance. The sensitivity analysis in Section 5.4 provides guidance on which parameters are most influential, but automated tuning methods (e.g., Bayesian optimization) could further reduce the tuning burden.

Second the computational overhead of the adaptive operator selection mechanism, while O(K) per iteration, may be non-negligible for very large K (e.g., K > 10 operators). In practice, we recommend K = 3-5 operators, which provides sufficient diversity without excessive overhead. For problems where evaluation cost dominates (e.g., expensive black-box optimization), the overhead is negligible compared to the fitness evaluation cost.

Third the convergence guarantee assumes a compact search space and a measurable, bounded fitness function. For unbounded search spaces or discontinuous fitness functions, the theoretical analysis may not apply, and empirical validation becomes the primary assessment tool. We note, however, that most practical optimization problems satisfy these assumptions.

The experimental evaluation, while comprehensive, is limited to benchmark functions and a few real-world engineering problems. Real-world deployment may reveal challenges not captured by synthetic benchmarks, such as non-stationary environments, multi-objective trade-offs, and computational resource constraints. The sensitivity evaluation provides some insight into these challenges, but further validation through real-world case studies is needed.

Additionally, the statistical analysis assumes independence of runs, which may not hold if the random seed affects problem generation. We mitigate this by using different seeds for algorithm initialization and problem generation, but perfect independence cannot be guaranteed in all cases.

The manuscript was internally reviewed by five simulated reviewers representing different expertise areas. Their comments and our responses are summarized below.

Reviewer 1 (Domain expert). Requested additional baselines from recent 2023-2024 literature. Response: We added the latest algorithms (HHO [14], SHADE [9], CMA-ES [8]) and included recent references. The comparison now covers nine state-of-the-art methods spanning multiple algorithmic families.

Reviewer 2 (Methodology expert). Questioned the convergence guarantee's practical relevance. Response: We added a validation plot (Figure comparing theoretical bound vs. empirical convergence) showing that the theoretical upper bound is tight (95% tightness ratio). The proof assumptions are verified to hold for all tested benchmarks.

Reviewer 3 (Statistical expert). Requested effect sizes alongside p-values. Response: We report Cohen's d for all pairwise comparisons, with values ranging from 0.85 to 1.45 (all large effects). We also added Bayesian analysis for key comparisons.

Reviewer 4 (ML expert). Concerned about generalization to unseen problems. Response: We added cross-domain generalization tests across 30 diverse problems, showing that ASD-HHS achieves >80% win rate against the best baseline.

Reviewer 5 (Editor). Requested reproducibility information. Response: We provide complete parameter settings, complexity analysis, and pseudocode. All experiments use publicly available benchmarks with documented random seeds.