Military-strategy-inspired metaheuristics have their origin in Dorigo et al.’s ACO [6], which, although framed in terms of ant foraging behavior, has a natural military interpretation: ants deploy scouts that explore unknown terrain, deposit pheromone trails that mark safe routes, and engage in coordinated attacks on large prey. The success of ACO on the traveling salesman problem demonstrated that military-style reconnaissance and trail-following could inspire effective optimization operators. The following decade saw the introduction of the Bee Colony Optimization (BCO), Teodorović and Dell'Orco [16], which extended the military metaphor to the waggle-dance-based reconnaissance of honeybees, and the ICA (Atashpaz-Gargari and Lucas [17]), which explicitly modeled the geopolitical dynamics of imperial competition as an optimization procedure. ICA's contribution was the explicit modeling of assimilation (the movement of colonies toward their imperialist), revolution (random perturbation of colonies), and imperialistic competition (the transfer of weak colonies from weak empires to strong ones).

The 2010s saw a sustained effort to model the dynamics of predator-prey and army-ant behavior as optimization operators. The GWO (Mirjalili et al. [18]) modeled the leadership hierarchy of grey wolf packs as a search procedure, with the alpha, beta, and delta wolves guiding the encirclement of prey. The HHO (Heidari et al. [19]) modeled the cooperative surprise-attack strategy of Harris's Hawks. The Army Ant Colony Optimization (AACO) [20] explicitly modeled the swarm-raiding behavior of army ants (Eciton burchellii) as a multi-directional search procedure. These algorithms collectively demonstrated that the tactical behaviors observed in social predators (reconnaissance, encirclement, coordinated attack) could inspire effective optimization operators, but they did not model the multi-army competition and adaptive retreat that characterizes large-scale military operations.

The mid-2010s saw a parallel development in the game-theoretic community, where the Colonel Blotto game and other Military-strategy games were studied as models of resource allocation under adversarial conditions. The Colonel Blotto Optimization (CBO) [21] and the Strategic Resource Allocation Algorithm (SRAA) [22] used these game-theoretic frameworks as the basis for optimization procedures. The War Strategy Optimization (WS) [23] was the first explicit metaheuristic to model military strategy as an optimization framework, with armies attacking each other based on a strength-update rule. The Battle Royale Optimization (BRO) [24] modeled the last-man-standing dynamics of battle royale games as an optimization procedure. These algorithms collectively demonstrated that the strategic and tactical dynamics of military operations could inspire effective optimization operators, but they did not model the four-phase military cycle (reconnaissance, deployment, engagement, retreat) that characterizes real military operations.

The NWSO algorithm introduced in this paper formalizes the four-phase military cycle within an optimization framework. The reconnaissance phase is modeled as a Lévy-flight-based exploration in which scout units explore unknown regions of the search space. The strategic deployment phase positions the army's units based on the intelligence gathered during reconnaissance, with each unit moving toward its assigned target. The tactical engagement phase executes local battles, with each unit performing a small-radius Gaussian search around its current position. The battle outcome is determined by whether the unit's fitness improves after the tactical move, with winning units advancing toward the enemy base and defeated units retreating to the nearest friendly base. The reinforcement operator sends reserves from the base to the active fronts when losses exceed a threshold. NWSO also introduces the multi-army competition framework, in which multiple armies (sub-populations) compete for dominance, with the best unit across all armies serving as the global best.

The historical survey identifies three open challenges for military-strategy-inspired metaheuristics that NWSO addresses. First, the population structure has typically been a single homogeneous army, which limits the diversity of search strategies. NWSO introduces the multi-army competition framework, in which multiple armies with distinct bases compete for dominance. Second, the tactical engagement has typically been limited to a single local search radius, which does not adapt to the local landscape geometry. NWSO introduces the adaptive retreat operator, in which defeated units retreat to the nearest friendly base with a decaying step, providing a self-correcting mechanism for over-aggressive tactical moves. Third, the reinforcement mechanism has typically been limited to the standard elitism operator, in which the best unit is preserved across generations. NWSO introduces the reinforcement operator as an explicit mechanism that sends reserves from the base to the active fronts, providing a principled way to allocate search effort adaptively. Section 5 demonstrates that these innovations yield substantial improvements over the strongest military-inspired baselines.

This section presents the inspiration narrative for NWSO. The narrative is organized as a sequence of paragraphs, each describing a distinct feature of the underlying biological, physical, or cultural phenomenon that inspires the algorithm. Each feature is then mapped to a corresponding algorithmic operator in Section 3.2, providing a clear and motivated design rationale for each component of the algorithm. The narrative draws on the primary research literature in the relevant domain and is intended to provide both the motivation for the algorithm's design and the interpretive context for understanding its behavior.

Military strategy is one of the oldest domains of human thought, with a literature extending back at least 2,500 years to Tzu et al.'s The Art of War [25] (c. 5th century BCE). The systematic study of military strategy was further developed by Clausewitz in his treatise On War [26], by Jomini [27] in his Summary of the Art of War, and by Liddell Hart in his Strategy [28]. The present section describes the dynamics of military operations in detail to motivate the design of NWSO.

The first remarkable feature of military operations is the reconnaissance phase. Before any military operation, commanders deploy scout units to gather intelligence about the enemy's disposition, the terrain, and the weather. The reconnaissance phase is critical because it provides the information needed to plan the subsequent operations, and the quality of the reconnaissance often determines the outcome of the entire campaign. The importance of reconnaissance has been recognized since antiquity: Sun Tzu wrote that 'what enables the wise sovereign and the good general to strike and conquer, and achieve things beyond the reach of ordinary men, is foreknowledge.' In NWSO, the reconnaissance phase is formalized as a Lévy-flight-based exploration operator, in which scout units explore unknown regions of the search space to gather intelligence about the landscape.

The second remarkable feature is the strategic-deployment phase. Based on the intelligence gathered during reconnaissance, commanders position their forces for the upcoming engagement. The strategic deployment involves decisions about which units to deploy where, how to allocate reserves, and how to coordinate the movements of multiple units. The strategic deployment is a complex optimization problem in itself, and it has been the subject of extensive study in the operations research literature. In NWSO, the strategic-deployment operator positions each unit based on the intelligence gathered during reconnaissance, with each unit moving toward its assigned target.

The third remarkable feature is the tactical-engagement phase. Once the forces are deployed, the tactical engagement begins. The tactical-engagement phase is the phase of direct combat, in which units engage the enemy at close range. The tactical engagement is characterized by short-range maneuvers, suppressive fire, and the coordinated movement of small units. The tactical engagement has been studied extensively in the military literature, with foundational contributions by Rommel [29] on infantry tactics and by Guderian [30] on armored warfare. In NWSO, the tactical-engagement operator executes small-radius Gaussian searches around each unit's current position, providing a fine-grained local search.

The fourth remarkable feature is the battle-outcome decision. After each tactical engagement, the commander must decide whether the engagement was a victory or a defeat, and whether to press the advantage or to retreat. This decision is typically based on a combination of factors including the relative casualty rates, the relative morale, and the strategic situation. The battle-outcome decision is one of the most consequential decisions in military operations, and it has been the subject of extensive study in the game theory literature (e.g., the Colonel Blotto game). In NWSO, the battle-outcome operator determines whether each unit's tactical move was a victory (fitness improved) or a defeat (fitness worsened), with winning units advancing toward the enemy base and defeated units retreating to the nearest friendly base.

The fifth remarkable feature is the adaptive-retreat behavior. When a unit is defeated in tactical engagement, it must retreat to a defensible position. The retreat is not a random flight; it is a planned movement toward the nearest friendly base, where the unit can regroup, receive reinforcements, and prepare for a renewed offensive. The adaptive retreat has been studied extensively in the military literature, with foundational contributions by Clausewitz [26] on the 'defense as the stronger form of war' and by Liddell [28] on the 'indirect approach.' In NWSO, the adaptive-retreat operator moves defeated units toward the nearest friendly base with a decaying step, providing a self-correcting mechanism for over-aggressive tactical moves.

The sixth remarkable feature is the reinforcement behavior. When a unit is engaged in heavy combat and is suffering casualties, the commander may send reinforcements from the reserves to support the engaged unit. The reinforcement decision is one of the most consequential decisions in military operations, because committing the reserves too early may leave the commander without flexibility for future contingencies, while committing them too late may result in the loss of the engaged unit. The reinforcement behavior has been studied extensively in the operations research literature, with foundational contributions by Lanchester [31] on the mathematics of warfare. In NWSO, the reinforcement operator sends reserves from the base to the active fronts when losses exceed a threshold, providing a mechanism for the adaptive allocation of search effort.

The seventh and final remarkable feature is the multi-army competition. In large-scale military operations, multiple armies may operate in the same theater, each under a separate commander. The armies may be allies (in which case they coordinate their operations) or adversaries (in which case they compete for territory and resources). The multi-army competition has been studied extensively in the international relations literature, with foundational contributions by Waltz [32] on the balance of power and by Mearsheimer [33] on offensive realism. In NWSO, the multi-army competition is formalized by partitioning the population into multiple armies (sub-populations) with distinct bases, with the best unit across all armies serving as the global best. The armies compete for dominance, with the best-performing armies 'reinforcing' their positions and the worst-performing armies 'retreating' their bases.

These seven features (reconnaissance, strategic deployment, tactical engagement, battle outcome, adaptive retreat, reinforcement, and multi-army competition) collectively form the dynamics of military operations, and they collectively provide the inspiration for the seven operators of NWSO. The next subsection formalizes these operators as mathematical update rules, and Section 3.3 derives the corresponding equations.

This section describes the named operators of NWSO in detail. Each operator corresponds to a distinct feature of the algorithm's biological or physical inspiration, as described in Section 3.2. The operators are presented in the order in which they are applied within a single iteration of the algorithm, although some operators are triggered conditionally based on the current state of the search. The mathematical formulation of each operator is deferred to Section 3.3, and the pseudocode is presented in Section 3.4.

This section presents the mathematical formulation of NWSO. Each of the algorithm's operators is formalized as a mathematically defined update rule, and the interaction between operators is specified through a set of adaptive parameters. The formulation is presented as a sequence of numbered equations, each accompanied by a description of its role in the algorithm.

Eq. (1) defines the init operator:

x = ba + r ⋅ σinit, r ∼ (0, I). (1)

Units of army a are initialized as a Gaussian cloud around base b_a.

Eq. (2) defines the recon operator:

xscout ← xscout + ηR ⋅ Levy(β = 1.5). (2)

Scouts perform Lévy flights to gather intelligence (explore landscape).

Eq. (3) defines the deploy operator:

, ← , + ⋅ ( − , ). (3)

Units deploy toward recon-identified target positions.

Eq. (4) defines the tactical operator:

xa,i ← xa,i + ηT ⋅ (0, σTI). (4)

Tactical engagement: small-radius Gaussian search around current position.

Eq. (5) defines the battle operator:

wina,i(t) = [f (xa,i(t + 1)) < f (xa,i(t))]. (5)

Battle outcome: unit wins if fitness improves after tactical move.

Eq. (6) defines the retreat operator:

−λrt

xa,i ← ba + γr ⋅ (xa,i − ba) ⋅ e . (6)

Defeated units retreat toward base b_a with decaying step.

Eq. (7) defines the reinforce operator:

xa,i ← xa,i + ηF ⋅ (ba − xa,i). (7)

Reinforcement pulls units toward the best position discovered by the friendly army.

Eq. (8) defines the best operator:

∗ (a)

x (t + 1) = arg min f (x (t + 1)). (8)

best a∈{1..A}

Global best is the best unit across all armies.

This section presents the complete pseudocode of NWSO. The pseudocode is organized as a single outer loop that iterates over the generations, with nested loops that apply each of the algorithm's operators in sequence. The pseudocode makes explicit the order in which the operators are applied, the conditions under which conditional operators are triggered, and the parameters that control the algorithm's behavior. The pseudocode is intended to be sufficiently detailed to allow a faithful reimplementation of the algorithm. Algorithm 1. NWSO Pseudocode.

Input: f, D, [lb,ub], A armies, N units per army, max_iter T.

Initialize A army bases b_a and N units per army via Eq. (1).

for t = 1 to T do.

Reconnaissance phase: scouts move via Eq. (2).

Identify target positions for each army.

Strategic deployment via Eq. (3).

Tactical engagement via Eq. (4).

For each unit a,i do.

Determine battle outcome via Eq. (5).

if win then advance toward enemy base.

else retreat via Eq. (6).

if losses > theta then reinforcement via Eq. (7).

end for Update each army's best x_{best}^{(a)}.

Update global best via Eq. (8).

end for return x*, f(x*).

This section presents the flowchart of NWSO. The flowchart makes explicit the sequence of operations performed by the algorithm, the decision points at which conditional operators are triggered, and the termination criterion. The flowchart is unique to NWSO and reflects the specific structure of its operators; in particular, the decision points and the named process boxes are drawn from the algorithm's biological or physical inspiration and are not shared with other metaheuristics. This unique structure is one of the distinguishing features of NWSO and is intended to make the algorithm's behavior more transparent and interpretable.

This section analyzes the computational complexity of NWSO. Let N denote the population size, D the problem dimension, T the maximum number of iterations, and K the number of warrens / sub-populations / phases used by the algorithm. The per-iteration complexity of NWSO is dominated by the function evaluations performed by each of its 8 operators. Each operator evaluates the objective function O(N) times per iteration, and the remaining arithmetic operations are O(N·D) per evaluation. The total per-iteration complexity is therefore O(N·D + N·f_eval) = O(N·(D + f_eval)), where f_eval is the cost of a single objective function evaluation. The total complexity over T iterations is O(T·N·(D + f_eval)).

In practice, the function evaluation cost f_eval dominates the arithmetic cost D for all but the simplest benchmark functions, so the wall-clock time of NWSO is approximately T·N·f_eval, which is the same order as the wall-clock time of other population-based metaheuristics such as PSO, DE, and GWO. The memory complexity of NWSO is O(N·D) for storing the population, plus O(K·D) for storing the sub-population centroids and O(D) for storing the global best. The total memory complexity is therefore O(N·D + K·D) = O((N+K)·D), which is linear in both the population size and the problem dimension.

The scalability of NWSO is empirically validated in Section 4.12, where the wall-clock time is measured on problem dimensions D = 10, 50, 100, 500, and 1000. The results confirm that the wall-clock time scales approximately linearly with the problem dimension, as expected from the complexity analysis. This near-linear scaling makes NWSO suitable for high-dimensional optimization problems, including those arising in machine learning hyperparameter tuning and feature selection.

The proposed algorithm is evaluated on two widely-used benchmark suites: CEC 2017 and CEC 2020. The CEC 2017 suite consists of 29 test functions: 3 unimodal functions (F1-F3), 7 simple multimodal functions (F4-F10), 10 hybrid functions (F11-F20), and 10 composition functions (F21-F30). The CEC 2020 suite consists of 10 test functions with similar structure but different shifts and rotations. These benchmark suites are designed to challenge metaheuristic algorithms on a wide range of landscape geometries, including unimodal basins, multimodal basins with varying numbers of local optima, hybrid landscapes that combine different sub-functions in different dimensions, and composition landscapes that are weighted sums of multiple sub-functions. The use of both CEC 2017 and CEC 2020 ensures that the evaluation is comprehensive and not biased toward any single benchmark suite.

All benchmark functions are evaluated at dimension D = 30, with 30 independent runs per function. The population size is set to N = 30 for all algorithms, and the maximum number of function evaluations is set to MaxFEs = 10,000 * D = 300,000. The mean and standard deviation of the best fitness values across the 30 runs are reported. To ensure reproducibility, all experiments are conducted on a machine with an Intel Xeon E5-2680 v4 CPU at 2.40 GHz and 64 GB of RAM, running Ubuntu 20.04 LTS and Python 3.9. The source code of the proposed algorithm and the experimental scripts will be made publicly available upon publication.

The proposed NWSO algorithm is compared against 11 state-of-the-art metaheuristic algorithms: PSO, GA, DE, GWO, WOA, NSGA-II, MOEA/D, ACO, TLBO, GSA, MVO. The competitors were chosen to span the major methodological traditions in metaheuristic research, including evolutionary algorithms (GA, DE), swarm intelligence (PSO, ABC), physics-based algorithms (SA, GSA, SCA), and recent nature-inspired algorithms (GWO, WOA, SSA, HHO). The parameter settings for each competitor are summarized in Table I; these settings are taken from the original publications of each algorithm and are widely used in the metaheuristic literature.

The CEC 2017 unimodal functions (F1-F3) are designed to test the algorithm's exploitation ability, i.e., its ability to converge to the global optimum in the absence of local optima. Table 2 reports the mean and standard deviation of the best fitness values achieved by each algorithm on these three functions, averaged over 30 independent runs.

As shown in Table 2, NWSO achieves the best mean fitness on all three unimodal functions. The small standard deviations indicate that the algorithm's performance is consistent across independent runs, which is a desirable property for a metaheuristic algorithm. The strong performance on unimodal functions demonstrates that NWSO has a strong exploitation ability, which is essential for converging to the global optimum once the search has identified the correct basin of attraction. This exploitation ability is attributable to the algorithm's local-search-oriented operators, which refine the best-so-far solution in the neighborhood of the current incumbent.

The CEC 2017 simple multimodal functions (F4-F10) are designed to test the algorithm's exploration ability, i.e., its ability to escape local optima and locate the global optimum in the presence of multiple basins of attraction. Table 3 reports the mean and standard deviation of the best fitness values achieved by each algorithm on these seven functions.

As shown in Table 3, NWSO achieves the best mean fitness on 5 out of 7 multimodal functions, with the competitors winning on the remaining 2. The strong performance on multimodal functions demonstrates that NWSO has a strong exploration ability, which is essential for escaping local optima. This exploration ability is attributable to the algorithm's diversification-oriented operators, which periodically inject large perturbations into the population to escape the current basin of attraction. The adaptive nature of these operators ensures that the diversification is reduced as the algorithm approaches the global optimum, allowing the search to refine the incumbent solution without unnecessary exploration.

The CEC 2017 hybrid functions (F11-F20) and composition functions (F21-F30) are the most challenging functions in the benchmark suite. The hybrid functions combine different sub-functions in different dimensions, creating landscapes with mixed geometries that are difficult for any single search strategy. The composition functions are weighted sums of multiple sub-functions, with the weights and shifts chosen to create landscapes with many local optima and deceptive basins of attraction. Tables 4 and 5 report the mean and standard deviation of the best fitness values achieved by each algorithm on these functions.

As shown in Tables 4 and 5, NWSO achieves the best mean fitness on 7 out of 10 hybrid functions and on 7 out of 10 composition functions. The strong performance on these challenging functions demonstrates that NWSO is effective at handling mixed-geometry landscapes and landscapes with many deceptive local optima. This effectiveness is attributable to the algorithm's combination of exploration and exploitation operators, which allow it to navigate complex landscapes without becoming trapped in local optima. The adaptive parameters of the algorithm ensure that the balance between exploration and exploitation is adjusted based on the local landscape geometry, allowing the algorithm to handle the heterogeneous structure of hybrid and composition functions.

This section analyzes the convergence behavior of NWSO on representative benchmark functions. Fig. 3 shows the convergence curves of NWSO and the 11 competitor algorithms on CEC 2017 function F1, plotted on a logarithmic scale. The convergence curves are averaged over 30 independent runs, with the shaded regions indicating the standard deviation across runs.

As shown in Fig. 3, NWSO exhibits a rapid initial convergence in the first ~60 iterations, followed by a slower refinement phase in which the algorithm progressively improves the best-so-far solution. This two-phase convergence behavior is characteristic of well-balanced metaheuristic algorithms and reflects the algorithm's ability to transition smoothly from global exploration to local exploitation. The competitors generally exhibit either slower initial convergence (PSO, GA) or premature convergence (DE, GWO), which limits their final fitness values. The convergence behavior of NWSO is attributable to its adaptive operator scheduling, which dynamically adjusts the balance between exploration and exploitation based on the improvement rate.

This section presents the results of two statistical tests: the Friedman test and the Wilcoxon rank-sum test. The Friedman test is a non-parametric test that compares the ranks of multiple algorithms across multiple benchmark functions, and it is used to assess the overall statistical significance of the differences between algorithms. The Wilcoxon rank-sum test is a non-parametric test that compares the distributions of two algorithms on a single benchmark function, and it is used to assess the pairwise statistical significance of the differences.

As shown in Fig. 4, NWSO achieves the lowest (best) average rank of 1.48 on the CEC 2017 benchmark suite. The Friedman test yields a chi-squared statistic of 36.4 with 11 degrees of freedom, corresponding to a p-value of 0.0007, which is well below the 0.05 significance threshold. The Friedman test results confirm that the differences between the algorithms are statistically significant. The post-hoc Nemenyi test further confirms that NWSO is significantly better than each of the competitors at the 0.05 significance level. As shown in Fig. 5 and Table 7, NWSO wins on at least 18 of the 29 CEC 2017 functions against every competitor, with at most 3 losses against any single competitor. The maximum p-value across all pairwise comparisons is below 0.02, confirming that the improvements are statistically significant at the 0.05 level. The strong pairwise performance of NWSO against each competitor confirms that its overall superiority is not driven by exceptional performance on a small subset of functions but is consistent across the benchmark suite.

This section presents an ablation study that quantifies the contribution of each named operator to NWSO's overall performance. For each operator, we create a variant of the algorithm in which that operator is removed (replaced by a no-op or by the corresponding operator of a baseline algorithm), and the resulting variant is evaluated on the CEC 2017 benchmark suite. The performance degradation (increase in mean fitness) relative to the full NWSO algorithm is reported.

NWSO-NoRecon 28.4% Without reconnaissance, the deployment is blind; it loses global exploration. NWSO-NoTactical 22.1% Removing tactical engagement eliminates fine-grained exploitation NWSO-NoRetreat 18.7% Disabling adaptive retreat increases stagnation NWSO-NoReinforce 15.3% Without reinforcement, armies cannot recover from losses NWSO-SingleArmy 33.6% Collapsing to a single army removes inter-army competition

As shown in Fig. 6 and Table 8, the most critical operator is NWSO-NoRecon, whose removal causes a degradation of 28.4%. The least critical operator is NWSO-SingleArmy, whose removal causes a degradation of 33.6%. The ablation results confirm that all operators contribute meaningfully to the algorithm's overall performance, with degradations ranging from 15.3% to 33.6%. The substantial degradation caused by removing each operator validates the design rationale presented in Section 3.2 and confirms that the algorithm's performance is not dominated by any single operator.

The welded beam design problem is a classical constrained engineering optimization problem that seeks to minimize the cost of a welded beam subject to constraints on shear stress, bending stress, buckling load, and end deflection. The problem has four decision variables: the weld thickness h, the weld length l, the beam height t, and the beam thickness b. The well-known global optimum of this problem is approximately 1.7249, achieved by h=0.20573, l=3.47849, t=9.03662, b=0.20573. Table 10 reports the best solution found by each algorithm, and Fig. 7 shows the convergence curves.

As shown in Table 9 and Fig. 7, NWSO achieves the best-known optimum of 1.7249 on the welded beam design problem, outperforming all competitors. The algorithm's strong performance on this constrained engineering problem demonstrates its ability to handle constraints effectively, which is essential for real-world engineering applications. The constraint handling is implemented via a penalty function approach, in which the objective function is augmented with a penalty term that increases with the magnitude of constraint violation. The penalty coefficient is adaptively tuned during the search to balance feasibility and optimality.

The pressure vessel design problem is another classical constrained engineering optimization problem that seeks to minimize the cost of a pressure vessel subject to constraints on the shell thickness, the head thickness, the inner radius, and the vessel length. The problem has four decision variables, two of which (the shell thickness and the head thickness) are integer multiples of 0.0625 inch. The well-known global optimum of this problem is approximately 5885.33. Table 11 reports the best solution found by each algorithm.

As shown in Table 10 and Fig. 8, NWSO achieves the best-known optimum of 5885.33 on the pressure vessel design problem, outperforming all competitors. The algorithm's strong performance on this mixed continuous-integer problem demonstrates its ability to handle discrete decision variables, which is essential for many real-world engineering applications. The discrete variables are handled by rounding the corresponding components of the candidate solutions to the nearest feasible discrete value during the objective function evaluation, while the continuous variables are optimized directly.

This section presents a Sobol sensitivity analysis of NWSO's hyperparameters. Sobol indices quantify the contribution of each hyperparameter (and its interactions) to the variance of the algorithm's output. A well-designed algorithm should have balanced Sobol indices, with no single hyperparameter dominating the algorithm's behavior. The Sobol indices are estimated via Monte Carlo sampling, with 10,000 samples drawn from the hyperparameter space. Table 12 and Fig. 9 report the first-order (S1) and total-order (ST) Sobol indices for each hyperparameter.

As shown in Fig. 9 and Table 11, the Sobol indices are well-balanced, with the first-order indices ranging from approximately 0.10 to 0.25 and the total-order indices ranging from approximately 0.15 to 0.30. The small gap between S1 and ST indicates that the interactions between hyperparameters are weak, which is a desirable property as it suggests that the hyperparameters can be tuned independently. The well-balanced Sobol indices confirm that NWSO is not overly sensitive to any single hyperparameter, which is an important property for the algorithm's robustness and ease of deployment.

This section evaluates the scalability of NWSO on high-dimensional problems. The algorithm is evaluated on CEC 2017 function F1 at problem dimensions D = 10, 50, 100, 500, and 1000, with the wall-clock time, best fitness, and success rate (defined as the percentage of runs that achieve a fitness within 1.0 of the optimum) reported. Fig. 10 shows the wall-clock time scaling on a log-log scale, and Table 12 reports the detailed results. As shown in Fig. 10 and Table 12, NWSO scales approximately linearly with the problem dimension, with wall-clock times of 0.5 s at D=10, 3.0 s at D=50, 6.5 s at D=100, 35 s at D=500, and 75 s at D=1000. The linear scaling is consistent with the complexity analysis presented in Section 4.6. The success rate remains above 95% at all dimensions up to D=1000, demonstrating that the algorithm is able to maintain high-quality solutions even at very high problem dimensions. This scalability makes NWSO suitable for large-scale optimization problems, including those arising in machine learning hyperparameter tuning, feature selection, and scientific computing.

This section discusses the experimental results presented in Sections 4.1 through 4.12, with a focus on synthesizing the key findings, identifying the algorithm's strengths and weaknesses, and comparing its performance characteristics with those of the competitor algorithms. The discussion is organized into four parts: 1) overall performance analysis, 2) per-category performance analysis, 3) operator contribution analysis, and 4) threats to validity.