At the beginning of the algorithm, a population array P∈RN×(1+d) is initialized, where N denotes the number of individuals in the population and d is the number of decision variables of the optimization problem. Each row of P corresponds to a candidate solution: the first column stores the fitness value fi, while the remaining d columns contain the decision variables xi,1,xi,2,…,xi,d. The initial values of these variables are generated using a uniform random distribution within the predefined lower and upper bounds of the search space. More formally, each individual can be expressed as: (1)Pi=(fi,xi,1,xi,2,…,xi,d)where fi is the fitness associated with the i-th individual and xi,j represents the j-th variable of that solution. The array is sorted in ascending order based on fitness, so that the first row P1 corresponds to the current best solution.

All individuals except the best one are updated through a movement mechanism that drives them closer to the current best solutions. These directionally-guided moves are primarily responsible for the exploitation phase, providing a structured path toward optimality compared to pure random exploration. Furthermore, the framework allows new directional operators to be added to the move pool seamlessly due to its modular architecture. Four distinct strategies have been implemented and tested. In the following formulations, the movement is described with respect to xbest, but it can equivalently be directed toward x¯elite, the centroid of the top-performing individuals.

Linear Movement Each variable xi,j is updated using a weighted vector pointing toward the best-known solution xbest,j (see Eq. (2), where t denotes the current iteration or analysis step). (2)xi,j(t+1)=xi,j(t)+r⋅(xbest,j(t)−xi,j(t))where r∈[−1,1] is a random number in the range [−1,1].

Spiral Movement Inspired by natural spiral navigation patterns [9], this strategy updates each variable by combining both radial and angular components, enabling the solution to follow a helical trajectory around a target point (see Eq. (3)). (3)xi,j(t+1)=xi,j(t)+ρ⋅cos⁡(θ)⋅(xbest,j(t)−xi,j(t))+ρ⋅sin⁡(θ)⋅ξwhere θ∈[0,2π] is a random angle (which is applied component-wise for highly multidimensional navigation) determining the direction of rotation, ρ is a decaying amplitude that decreases linearly with the iteration count to ensure convergence, and ξ∈[−0.5,0.5] is a uniformly distributed random scalar that introduces controlled perturbation along the spiral path. The maximum absolute value is 0.5, so ξ acts as a moderate scaling factor; it is sufficient to introduce diversification, while preserving convergence stability.

Adaptive Data-Driven Movement. This is a newly introduced strategy designed to enhance the responsiveness of the algorithm to the evolving state of the population. Instead of applying a fixed movement rule, each variable is updated through a directionally biased mechanism driven by real-time population statistics. The update formulation integrates two complementary indicators:

Ct: the fitness gap between the worst and the best individuals in the current population;

The procedure involves the computation of two dynamic coefficients, α and β, based on Ct and σ¯t, respectively. (4)Ct=fworst(t)−fbest(t),α=1−tT,β=σ¯txmax,j(t)−xmin,j(t)where t is the current iteration and T is the maximum number of iterations. The coefficient α controls the exploitation pressure, decreasing linearly over time, while β scales the exploratory component based on the diversity of the population. The average standard deviation σ¯t is computed by first evaluating the standard deviation σj(t) of each decision variable xj across the current population, and then averaging over all d variables: (5)σ¯t=1d∑j=1dσj(t)

This metric reflects the overall dispersion of the population in the decision space at iteration t, and enables the adaptive modulation of the search intensity. Each variable xi,j is updated according to the following equation: (6)xi,j(t+1)=xi,j(t)+α⋅r⋅(xbest,j(t)−xi,j(t))+β⋅rwhere r∈[−1,1] is a uniformly distributed random scalar. The first term directs the solution toward the current best individual with strength modulated by α, while the second introduces a diversity-preserving perturbation, scaled by β. Although this movement rule is structurally focused on exploiting promising regions, the inclusion of the uniformly random scalar r classifies it as a directionally-guided stochastic update rather than a purely deterministic one. This balances the algorithm’s capacity to intensify search near optima and maintain sufficient exploration to escape local traps.

Although Ct does not appear directly in the update rule of Eq. (6), it governs the dynamic adjustment of the parameters α and β as described below. In fact, to further enhance adaptability, a conditional adjustment mechanism is introduced in the algorithm based on the convergence indicator Ct. This mechanism alters the values of α and β according to three distinct scenarios:

If Ct≤0.01, diversification mechanisms are activated by increasing the exploratory component β, which is doubled (β→2⋅β). The convergence factor α remains unchanged and is computed as α=1−t/T. This configuration enhances the algorithm’s ability to escape premature convergence.

To transition from empirical justification to a formally robust configuration, a targeted sensitivity analysis was conducted on the adjustment factors (e.g., the α decay and β scaling) and the Ct threshold values. By evaluating diverse search landscapes (unimodal, multimodal, and highly constrained), the analysis verified that the Ct thresholds of 0.01 and 0.1 provide the most reliable trigger points to shift between exploitation and exploration, while the scaling bounds prevent divergent oscillatory behavior. This rigorous robustness analysis ensures that the adaptive rules are not merely empirical estimates, but structurally sound control mechanisms for maintaining population diversity. These values ensured a stable balance between exploration and exploitation across heterogeneous test cases. More aggressive adjustments resulted in unstable oscillatory behavior, whereas milder variations had a negligible impact on convergence dynamics. To further prevent instability near the threshold conditions, the control rules are evaluated at each iteration, while the effects of parameter updates are naturally smoothed by the progressive decay of α and the statistical averaging embedded in β.

Directional Ensemble Movement. In the work, a fourth strategy aims to combine the strengths of the previous mechanisms by integrating them into a unified ensemble update. Rather than relying on a single movement model, this approach blends the linear, spiral, and adaptive components into a single formulation: (7)xi,j(t+1)=ω1⋅xi,j(linear)+ω2⋅xi,j(spiral)+ω3⋅xi,j(adaptive)

By combining multiple directional components, the directional ensemble strategy enables a richer and more balanced navigation of the search space. This formulation mitigates the weaknesses of any single method and increases robustness across different search space topologies. The weights can either be fixed or adaptively adjusted according to real-time feedback from the search dynamics. To ensure a robust baseline performance without problem-specific tuning, a limited sensitivity analysis was conducted on a representative subset of structurally distinct functions (F1 for unimodality, F10 for multimodality, and F22 for asymmetrical local traps). The analysis evaluated variations in Ct thresholds, ensemble weights (ω1,ω2,ω3), and population size distribution. Results confirmed that a balanced deterministic-to-stochastic ratio and an ensemble weighting skewed toward the adaptive data-driven movement (ω1=0.2, ω2=0.3, ω3=0.5) yield the most stable convergence across diverse topologies. To explicitly substantiate this, Table 1 reports a summary of the sensitivity analysis, demonstrating how variations in these hyper-parameters impact the mean convergence.

While this static configuration demonstrated strong generalization capabilities and was adopted for all subsequent experiments, we explicitly acknowledge that an exhaustive, problem-specific tuning of the Ct thresholds and ensemble weights could potentially unlock further performance improvements for specific optimization tasks.

To maintain diversity and exploration, a second set of solutions is generated using random movements based on predefined or adaptive strategies. These new individuals are appended to the population P and evaluated using the objective function. The algorithm supports nine random movement strategies, selected via a control parameter. Moreover, if required and in automatic mode, the algorithm permits a fully adaptive mode (strategy 9), where the choice of strategy depends on the convergence and diversity of the population: when the population is stagnant, global exploration is favored; when it’s highly dispersed, localized refinement is applied. In the work, the implemented strategies are: 1.

Global Random Sampling (GRS): Uniform random variables in the search space;

Each strategy, whose underlying mechanism is discussed in detail in Section 3, aims to address a specific trade-off between diversification and intensification, enabling a coherent balance between global exploration and local refinement.

After each distinct phase (firstly the directional movement, and subsequently the stochastic exploration), the algorithm merges the newly generated candidate solutions with the current population. The extended array is immediately re-sorted by fitness, and only the best N individuals are retained. This two-stage intra-generational elitism accelerates convergence by ensuring that the stochastic phase strictly operates on the refined outcomes of the deterministic phase. This strict retention of the top-performing individuals ensures that the sequence of best solutions is monotonically non-increasing in terms of fitness, i.e., f(xbest(t+1))≤f(xbest(t)). This elitist property is fundamental for theoretical stability.

Furthermore, the algorithmic architecture provides a formal probabilistic guarantee of global convergence. According to stochastic search theory, a metaheuristic converges to the global optimum in probability if two conditions are met: (1) the process is elitist, and (2) the probability of generating a solution in any subset of the search space with a positive measure (including the global optimum region) is strictly greater than zero at every iteration. In SRCA, condition (1) is satisfied by the aforementioned sorting and truncation mechanism. Condition (2) is mathematically guaranteed by the stochastic exploration phase: the continuous availability of the Global Random Sampling (Strategy 1) and the heavy-tailed Lévy flight (Strategy 8) ensures that every point in the domain S has a probability P(x∈S)>0 of being sampled. Consequently, as the number of evaluations approaches infinity, the probability of finding the global optimum approaches one, providing a rigorous probabilistic stability guarantee for the SRCA framework.

The entire process, movement, exploration, fitness evaluation, and sorting, is repeated for a fixed number of iterations or until a convergence criterion is met (e.g., negligible improvement in fitness). Throughout the process, the best solution found so far is tracked and returned at the end of the optimization.

A critical aspect of any metaheuristic is its computational complexity. The time complexity of SRCA is primarily determined by the objective function evaluation, the sorting mechanism, and the update rules. Let N be the population size, D the problem dimension, and MaxIt the maximum number of iterations.

The complexity of the initialization is O(N⋅D). In the main loop, SRCA performs two major phases: 1.

Deterministic Update: Requires O(N⋅D) operations.

Consequently, SRCA evaluates 2N candidate solutions per iteration. To ensure a strictly fair and unbiased comparison across all tested algorithms, the evaluation budget is rigorously governed by a predefined maximum number of function evaluations (MaxFEs) rather than a fixed iteration count. Therefore, while SRCA processes more candidates per cycle compared to algorithms like PSO or GSO, the total computational budget remains identical across all comparators. This rigorous normalization ensures that any performance superiority is exclusively attributable to the algorithmic efficiency and the structural integration of directionally-guided and stochastic behaviors.

To synthesize the overall execution flow and facilitate full reproducibility, the step-by-step procedure of the proposed framework is detailed in Algorithm 1.

The algorithm SRCA was fully developed using Python 3.12 environment. While platforms such as MATLAB are frequently used for numerical optimization due to their extensive libraries and integrated visualization tools, the choice of Python was driven by its open-source nature, rich ecosystem, and flexibility. To ensure rigorous comparative conditions across all tested algorithms, the termination criterion was defined strictly by the maximum number of function evaluations (MaxFEs) rather than a fixed number of iterations. The MaxFEs budget was dynamically determined based on the problem’s dimensionality (D): 100,000 for D≤10; 200,000 for D≤30; 400,000 for D≤50; 800,000 for D≤150; and 1,000,000 otherwise. For the SRCA algorithm, the population size was maintained at 100 individuals, with an additional 100 randomly generated candidates evaluated at each iteration, operating strictly within the assigned MaxFEs limit. All comparator algorithms were rigorously evaluated under these exact same MaxFEs constraints over 25 independent runs to guarantee a completely normalized, statistically robust, and equitable performance assessment.

To evaluate the performance and robustness of the SRCA algorithm, a comprehensive suite of benchmark functions was employed [9,27,28]. These functions (named F1-F23), widely adopted in the literature, are used to assess the global exploration capabilities, convergence speed, and resilience to local optima across various problem types, including unimodal, multimodal, separable, and non-separable search spaces. Table 2 reports the benchmark functions, including their respective domains and known global minima. For each function, a 3D plot in two variables is also included.

Tables 3 and 4 report the comparative performance of SRCA against well-established metaheuristic algorithms, highlighting its relative effectiveness on benchmark functions. In the tables, the symbol [*] refers to results obtained from Python implementations specifically developed as part of this study. Conversely, the symbol [**] denotes results sourced directly from established literature [9]. Relying on peer-reviewed baseline data for canonical algorithms (e.g., PSO, WOA, GSA) strictly prevents potential “implementation bias”, ensuring these comparators are represented at their objectively verified best under the specified computational constraints. In contrast, the GSO and FA implementations required local execution because they incorporate specific recent extensions [22] for which comprehensive baseline data across all 23 functions was unavailable. In particular, The GSO algorithm was extended beyond its standard formulation through several methodological enhancements [22]. Four distinct exploration strategies were introduced for the rangers (global, local, cycle-by-cycle hybrid, and periodic switching) enabling a dynamic balance between exploration and exploitation. Additionally, producers execute directional searches via adaptive perturbations in spherical coordinates, with a decaying amplitude mechanism that encourages convergence while mitigating premature stagnation. A directional reset mechanism embedded within the producer simulates scouting behavior, further improving adaptability during extended optimization processes. The GSO algorithm was configured with 50 producers, each associated with 20 scroungers and 50 rangers, resulting in a structured and hierarchical search process that balances directed exploitation with distributed exploration. A customized Firefly Algorithm (FA) was also implemented in Python to reflect the exploration logic adopted in SRCA and the revised GSO. In addition to the canonical attractiveness-and-randomness scheme, the proposed FA integrates four interchangeable random exploration strategies controlled by the same parameters used in GSO. At each iteration, every firefly is also attracted toward the centroid of the xelite brightest individuals (xelite=5 by default), yielding a multi-leader attraction mechanism similar to the ensemble option in SRCA. The base attractiveness β0, light absorption coefficient γ, random step amplitude α, and local search radius all decay linearly over time, promoting broad exploration in early stages and fine-grained exploitation in later iterations. An elitist replacement step ensures that the current global best (queen firefly) is never lost, while on-the-fly random scouting injects diversity and mitigates stagnation. The FA was configured with an initial population of 100 fireflies, complemented by 100 additional random solutions evaluated at each iteration under various random movement strategies.

Analyzing the results presented in Tables 3 and 4, it can be observed that SRCA consistently approaches the global optimum across a wide variety of search spaces. It is important to objectively note that SRCA does not universally dominate the benchmarks. On certain unimodal and continuous functions (such as F1, F2, F3, and F9), highly specialized algorithms like DE and WOA exhibit superior precision, reaching extreme minima or exact zeroes. Similarly, GSO and PSO outperform SRCA on specific multimodal topologies (e.g., F7, F11, F20). However, SRCA achieves the best results among the compared methods on 14 out of the 23 functions, showcasing a highly balanced exploitation capacity and robust convergence behavior. The deterministic movement module, when combined with the stochastic cycle-based exploration and the adaptive strategy selection, creates a powerful synergy that prevents stagnation and fosters reliable general-purpose optimization.

However, it is important to rigorously analyze the specific behavioral dynamics observed on function F12, a complex benchmark characterized by sharp boundary effects and severe penalty terms. The structural difficulty SRCA occasionally encounters here stems directly from its architectural mechanics. Specifically, the heavy-tailed stochastic operators (such as the Lévy flight) frequently generate large exploratory steps. In a landscape bounded by steep penalty walls, these massive jumps cause frequent boundary violations. While algorithms like PSO naturally mitigate this issue through velocity clamping (which smoothly decelerates particles near the boundaries), SRCA’s current boundary-handling mechanism simply truncates variables at the bounds (clipping). This approach can temporarily cause an artificial concentration of solutions on the boundary, resulting in localized losses of population diversity. However, it must be noted that when SRCA is evaluated under a strictly normalized and equitable computational budget (MaxFEs), its adaptive strategies are granted sufficient evaluations to recover from these boundary collisions, yielding significantly improved final results. As seen in Table 3, while SRCA does not surpass the extreme precision of GSO or PSO on this specific function, adopting the proper MaxFEs budget substantially narrowed the performance gap, validating the algorithm’s resilience on penalized spaces. Nevertheless, this analysis highlights that the integration of advanced boundary-handling or gradient-aware repair mechanisms remains a valuable future improvement for navigating sharp penalty boundaries.

While this isolated issue is observed, SRCA demonstrates a competitive trade-off between convergence speed and solution quality, with low standard deviation values across most runs, indicating high robustness and reliability.

To rigorously assess the practical applicability and constraint-handling capabilities of the proposed SRCA framework, a suite of six classic structural engineering design problems was selected [29–31]. These problems are widely recognized in the literature for their highly non-convex search spaces, non-linear constraints, and mixed continuous-discrete decision variables, making them ideal stress tests for modern metaheuristics. The evaluated problems are: Tension/Compression Spring Design (F1), Pressure Vessel Design (F2), Three-Bar Truss Design (F3), Welded Beam Design (F4), Multiple Disk Clutch Brake Design (F5), and Speed Reducer Design (F6). The detailed mathematical formulations, including the objective functions, constraint inequalities, and decision variable bounds for all six optimization problems, are thoroughly documented in existing literature [32,33].

In this experimental phase, SRCA was comprehensively compared against an extensive set of advanced algorithms from the state-of-the-art, including SASS, COLSHADE, IUDE, BiPopEpsMAgES, εMAgES, iLSHADE, SRA, GA3, GA4, HPSO, CPSO, and MBA. The performance metrics evaluated across multiple independent runs include the Best, Mean, Worst, and Standard Deviation (Std_Dev) of the objective function evaluations. The consolidated numerical results are presented in Table 5.

An analysis of the results presented in Table 5 reveals that SRCA is a reliable and competitive framework for real-world engineering design. While highly specialized algorithms occasionally report marginally lower extreme minima, SRCA distinguishes itself through operational stability and robustness across independent evaluations, which are critical features for industrial applications where reproducibility is paramount.

For the Tension/Compression Spring Design (F1), SRCA achieves a best cost of 0.012734, grouping tightly with top-tier methods like COLSHADE and IUDE (0.01270). Notably, SRCA maintains a low standard deviation (4.50⋅10−5), confirming that its adaptive stochastic operators effectively explore the narrow feasible region of this specific problem without undergoing divergent behaviors.

In the Pressure Vessel Design (F2), the algorithm locates a highly competitive minimum of 6070.58, approaching the prevalent optimum cluster established around 6060. While top-tier variants like COLSHADE and IUDE achieve a superior mean cost (6060), SRCA’s operational consistency remains noteworthy: its mean cost of 6279.28 outperforms several complex competitors, such as SASS (6410), εMAgES (7380), and iLSHADE (8480). These latter algorithms suffer from severe standard deviations (up to 3140 for iLSHADE), indicating occasional convergence failures. Conversely, SRCA systematically secures high-quality feasible solutions without such extreme variance.

The Three-Bar Truss Design (F3) highlights SRCA’s precision. The algorithm perfectly matches the optimal global value of 263.896 discovered by top-performing approaches like COLSHADE and BiPopEpsMAgES. The near-zero standard deviation (0.000931) underlines its capacity to manage non-linear inequality constraints seamlessly.

Regarding the Welded Beam Design (F4), SRCA records a best cost of 1.72852. While slightly higher than the 1.67 optimum found by differential evolution variants, the tight gap between its Best (1.728) and Worst (1.798) runs proves the absence of stagnation in sub-optimal local traps, a common issue for this highly multimodal benchmark.

For the Multiple Disk Clutch Brake (F5), SRCA’s performance is virtually indistinguishable from the known global optimum. It identifies a minimum of 0.235243 (compared to 0.235242), achieving an exceptional standard deviation of 1.31⋅10−7. This demonstrates that the algorithm’s continuous boundary handling and dynamic parameter scaling are calibrated for discrete/continuous mixed spaces.

Finally, in the Speed Reducer Design (F6), SRCA achieves a best result of 2996.61. While methods like SRA (2995) and MBA (2994.48) locate marginally lower extreme minima, a notable aspect of SRCA’s performance in this benchmark is the consistency across independent runs, evidenced by a standard deviation of 0.4526. In comparison, several competing methods (such as COLSHADE, IUDE, and BiPopEpsMAgES) record significantly higher variance levels (Std_Dev ≈1013) for this specific problem, which may indicate sensitivity to initialization or difficulties in managing the problem’s constraints. The numerical results suggest that SRCA’s dynamic scaling of stochastic steps helps maintain search stability, preventing divergent behaviors and facilitating reliable convergence within the feasible domain.

To exhaustively address the verification of structural constraints and provide complete reproducibility, Table 6 explicitly reports the optimal decision vectors (design variables) corresponding to the best solutions achieved by SRCA for each evaluated problem. Furthermore, the table formally confirms the feasibility status, verifying that all reported optimal solutions strictly satisfy the respective non-linear mechanical and geometric inequality constraints without any violations.

In conclusion, the structural benchmark analysis confirms that the proposed integration of directional deterministic exploitation and dynamically scheduled stochastic exploration prevents premature convergence and ensures stable constraint satisfaction across varied engineering topologies.

To justify the architectural complexity of SRCA and scientifically validate the inclusion of its distinct movement and exploration strategies, an ablation study was conducted. Following standard experimental practices, three representative benchmark functions were selected: F1 (unimodal, testing pure convergence speed), F10 (multimodal, testing local optima avoidance), and F22 (highly asymmetrical local traps).

The fully equipped SRCA (“Full”) was compared against three structurally disabled variants: SRCA “w/o Spiral” (disabling the spiral directional movement, ω2=0), SRCA “w/o Adapt” (disabling the adaptive data-driven movement, ω3=0), and SRCA “w/o Lévy” (preventing the adaptive controller from utilizing heavy-tailed stochastic jumps). All variants were executed for 25 independent runs under the strict MaxFEs budget normalization detailed in Section 4. The results are consolidated in Table 7.

The data demonstrates that every architectural component contributes significantly to the framework’s overall robustness. Disabling the spiral movement (w/o Spiral) causes severe performance degradation and extreme variance across all landscape types, proving its necessity for localized spatial navigation. Disabling the heavy-tailed stochastic jumps (w/o Lévy) crucially impairs the algorithm’s ability to escape local minima, as evidenced by the dramatic loss of precision and stability on the multimodal F10 and F22 benchmarks. Finally, disabling the adaptive movement (w/o Adapt) leads to a consistent loss of precision and sharply increased standard deviations, particularly on asymmetrical landscapes. Ultimately, this ablation study empirically demonstrates that the optimal performance of SRCA relies on the synergistic integration of all its structural components.