A Hybrid Five-Algorithm Metaheuristic Framework for Unconstrained Non-Linear Optimization: Design, Implementation and Benchmark Evaluation

Metaheuristic optimization algorithms have become indispensable for solving highly nonlinear optimization problems in which gradient information is either inaccessible or unreliable [1-3]. These nature-inspired algorithms—including evolutionary algorithms, swarm intelligence algorithms, and physics-based algorithms—provide approximate solutions within a reasonable computational time. The No Free Lunch theorem, however, establishes that no single algorithm can outperform all others across every problem type, thereby motivating the development of hybrid methods [4].

1.1 Research gap and motivation

Despite numerous efforts to develop hybrid metaheuristic approaches, current methods exhibit three major deficiencies: (1) most hybrids integrate only two or three algorithmic components, which restricts the diversity of search behaviors; (2) existing hybrid frameworks lack explicit integration or fusion strategies, limiting the effective utilization of the distinct search and exploitation capabilities of each sub-algorithm; (3) most studies do not provide theoretical justification for algorithm selection and weight allocation [5]. The present study addresses these shortcomings by proposing a five-algorithm hybrid system that explicitly incorporates complementary algorithmic roles.

1.2 Justification of algorithm choice (theoretical justification)

The five optimization algorithms were selected based on their complementary search mechanisms. (1) The Enhanced Jaya Algorithm (EJA) [6] provides a parameter-free framework that enables strong intensification through best-worst solution guidance; (2) The Improved Whale Optimization Algorithm (IWOA) [7] incorporates a spiral-based local search mechanism and employs adaptive exploration within the bubble-net attack strategy; (3) Adaptive Differential Evolution (ADE) [8] excels at exploration in continuous domains through mutation and crossover operators with time-varying parameters; (4) Advanced Particle Swarm Optimization (APSO) [9] enhances velocity-based social learning through a nonlinear inertia weight to achieve a balanced search; and (5) the Modified Grey Wolf Optimizer (MGWO). The resulting combination ensures comprehensive coverage of diverse search strategies, namely: gradient-free intensification, spiral exploitation, differential mutation, social cooperation, and hierarchical hunting [10].

3.1 Algorithmic contributions of novelty

The main achievements of this work are:

(1) The development of a systematic framework for integrating five metaheuristic algorithms, that addresses the complementarity between the search mechanisms of these algorithms;

(2) A theoretically inspired weight-allocation strategy based on the algorithmic characteristics;

(3) A sequential execution strategy to optimize the arrangement order of algorithm application;

(4) A comprehensive benchmark evaluation with quantitative convergence assessment.

Although the constituent algorithms have been individually reported in the literature, the specific integration methodology and the structured approach to leveraging complementary strengths constitute the novel contribution of this work.

3.2 Algorithm framework

The hybrid algorithm was implemented sequentially, applying five metaheuristic techniques in each iteration. Figure 1 presents a flowchart illustrating the sequential coordination of the five constituent algorithms. Each algorithm contributes to population evolution according to a predetermined weight: EJA (25%), IWOA (20%), ADE (20%), APSO (20%), and MGWO (15%). The corresponding pseudocode is presented in Algorithm 1.

5.1 Statistical comparison of solution quality

Table 1 presents a comprehensive statistical summary, including the best, mean, and worst fitness values, as well as the standard deviation, computed over 30 independent runs. The hybrid algorithm achieved superior mean fitness values across all ten benchmark functions. For the Sphere function, the hybrid algorithm attained a mean fitness value of $3.67 \times 10^{-41}$, which represents an improvement of nine orders of magnitude over the EJA $\left(7.84 \times 10^{-31}\right)$ and 14 orders of magnitude over MGWO $\left(2.14 \times 10^{-24}\right)$. These results confirm the superior optimization capability of the proposed hybrid framework.

Table 1. Statistical results of the final solutions (30 independent runs, dimension = 30)

5.2 Quantitative analysis of the rate of convergence

To provide a rigorous convergence analysis beyond qualitative visual assessment, the convergence rate was defined as the ratio of the number of iterations required to achieve a specified threshold value. Convergence rate as the ratio between the number of iterations (say) achieve a certain predefined value. Table 2 gives the values of the average number of iterations required to reach the specified thresholds. The hybrid algorithm required 37 iterations to reach a threshold of $10^{-20}$ for the Sphere function compared to 72 iterations for EJA, 81 iterations for IWOA, 58 iterations for ADE, representing convergence speed improvements of 48.6%, 54.3%, and 36.2%, respectively. The Relative Convergence Efficiency (RCE = iterations_individual / iterations_hybrid) averaged 1.89 across all functions, indicating that the hybrid algorithm converges approximately twice as fast as the individual component algorithms.

Table 2. Average iterations to reach threshold values (30 runs)

5.3 Results of the iteration for Rastrigin function

By the 25^th iteration would be the global optimum (0,0), and the average fitness would still be improving as more members of the population would tend to move towards optimum areas. Table 3 presents the iteration-by-iteration results for the Rastrigin function, illustrating the progressive improvement in both best and mean fitness values throughout the optimization process.

Table 3. Iteration results obtained on the Rastrigin function: An extensive discussion of the optimal run

As shown in Figure 1, the convergence curves shown for the Rastrigin function are used to show the convergence dynamics of the investigated algorithms. The hybrid approach shows a significantly faster rate of convergence and reaches better solutions than the individual constituent algorithms.

5.4 Numerical analysis of optimization trajectories

The trajectories visualized in three dimensions are quantified via two main metrics: Trajectory Diversity (TD) which is defined to be the average Euclidean distance between the consecutive search points, i.e., $T D=(1 / T) \sum\|\mathrm{x}(t+1)-\mathrm{x}(t)\|$ Exploitation Intensity (EI): which is the fraction of iterations where the fitness value is improved, i.e., EI = (number of improving iterations)/T. In the case of the Sphere function, the hybrid algorithm achieves TD of 12.4, which is higher than TD of 8.7 achieved by EJA and TD of 15.2 achieved by APSO and EI of 0.89, which is also higher than 0.72 achieved by EJA and 0.65 achieved by APSO, and this demonstrated that the hybrid algorithm possesses a balanced exploration process and an excellent exploitation process.

Figure 2 shows the contribution analysis where the Enhanced Jaya component is responsible for 27.3% of the overall improvement, the Improved Whale algorithm makes a contribution of 21.8%, ADE has a contribution of 23.5%, the APSO contributes with 16.9%, and the MGWO contributes with 10.5%.

Figure 3 presents the 3D optimization trajectories for all ten benchmark functions, illustrating how the hybrid algorithm navigates the fitness landscape. Each subplot illustrates the path taken by the search path from the random initial positions to reaching the global optimum. Key observations are as follows: (a) For unimodal functions (Sphere, Rosenbrock, Dixon-Price, Zakharov), the trajectories follow direct convergence paths with minimal exploratory deviation; (b) For multimodal functions (Rastrigin, Ackley, Griewank, Schwefel, Levy, Michalewicz), the trajectories exhibit broad initial exploration followed by focused search around promising regions; and (c) The algorithm successfully avoids local optima, as evidenced by its convergence to the known global optimum in all cases.

Figure 4 illustrates the population distribution at four stages of optimization (iterations 1, 10, 50, and 100) for the Rastrigin function. This visualization reveals the progressive transition from exploration to exploitation: (1) At iteration 1, the population is uniformly distributed across the search space, reflecting strong exploratory coverage; (2) By iteration 10, clusters begin to form around promising regions, including areas near local optima; (3) By iteration 50, the population is predominantly concentrated near the global optimum region, with reduced diversity; (4) By iteration 100, all individuals have converged to the global optimum at (0, 0). This progressive concentration confirms the effective exploration–exploitation balance achieved by the hybrid framework [10, 14].

(a) Sphere function

(b) Rastrigin function

(c) Rosenbrock function

(d) Ackley function

(e) Griewank function

(f) Schwefel function

(g) Levy function

(h) Dixon-price function

(i) Zakharov function

(j) Michalewicz function

Figure 3. 3D optimization trajectories for benchmark functions showing the search behavior of the hybrid algorithm

Figure 4. Population distribution during various optimization stages for the Rastrigin function, showing the regulation of passing from the uniform exploration stage (iteration 1) to focused exploitation stage (iteration 100) [29]

5.5 Comparison with state-of-the-art hybrid

To substantiate the claimed competitive performance, direct comparative experiments were conducted against three recent state-of-the-art hybrid methods, under identical conditions (d = 30, N = 50, T = 100, 30 runs): DE-PSO [22], GWO-WOA [23], and Jaya-MFO [24]. For the Sphere function, the proposed hybrid obtained the mean fitness of 3.67e-41 in comparison to DE-PSO (4.23e-35), GWO-WOA (8.91e-28) and Jaya-MFO (2.15e-33). Wilcoxon rank-sum tests confirmed statistically significant differences (p < 0.05) in 8 out of 10 functions between the proposed hybrid and each competing method. The proposed hybrid achieved the best average rank of 1.2, compared with DE-PSO (2.8), GWO-WOA (3.4), and Jaya-MFO (2.6), across all benchmark functions.

This study presented a novel hybrid optimization framework integrating five metaheuristic algorithms: EJA, IWOA, ADE, APSO, and MGWO are integrated. Empirical evaluation on ten benchmark functions demonstrated an average improvement of 94.7% in solution quality and a convergence speed increase of up to 47.1% relative to the individual constituent algorithms.

Based on the findings of this study, the following design principles are proposed for the development of future hybrid metaheuristic frameworks: (1) Complementarity principle: constituent algorithms should be selected to exhibit distinct search mechanisms (e.g., gradient-free, spiral, differential, social, and hierarchical), rather than similar ones; (2) Weight allocation principle: algorithms with stronger intensification characteristics should be assigned higher weights within the integration scheme; (3) Sequencing principle: algorithms should be ordered from exploration-dominant to exploitation-dominant within each optimization iteration; (4) Diversity preservation principle: at least one algorithm that maintains population diversity throughout the optimization process should be included.

Future research directions include adaptive weight adjustment based on real-time performance feedback, extension to constrained optimization problems, application to real-world engineering design challenges, parallel implementation to reduce computational overhead, and theoretical analysis of convergence guarantees.