MSIGOA: Enhanced Gazelle Optimization

• The paper introduces MSIGOA, which restructures GOA’s search dynamics using an Iteration-Based Updating Framework to balance global exploration and local exploitation.
• It employs Adaptive Parameter Tuning Strategies to scale Brownian and Lévy movements, enhancing the algorithm's adaptability throughout the optimization process.
• The Dominant Population-Based Restart Mechanism reinjects diversity and prevents premature convergence, leading to superior performance on benchmark and engineering design problems.

Searching arXiv for the specified paper and, if available, the base GOA paper for accurate citation context. Multi-Strategy Improved Gazelle Optimization Algorithm (MSIGOA) is a swarm-based meta-heuristic proposed as an enhancement of the Gazelle Optimization Algorithm (GOA) for numerical optimization and engineering design problems. It targets three stated shortcomings of GOA: the imbalance between exploration and exploitation, fixed Brownian-motion and Lévy-flight parameters with low adaptability, and insufficient population-wide information exchange that can induce stagnation and premature convergence. MSIGOA addresses these issues through three coordinated components: an Iteration-Based Updating Framework (IBUF), two Adaptive Parameter Tuning Strategies (APTS), and a Dominant Population-Based Restart Mechanism (DPRM) (Diao et al., 8 Sep 2025).

1. Algorithmic basis and design objective

MSIGOA builds directly on GOA, which is described as a swarm-based meta-heuristic inspired by gazelle foraging and escape behavior (Diao et al., 8 Sep 2025). The design objective is not to replace the behavioral basis of GOA, but to restructure its search dynamics so that exploration and exploitation are scheduled more deliberately across the run and so that information from strong individuals is propagated more effectively through the population.

The source formulation identifies three principal shortcomings in the basic GOA. First, GOA switches randomly between exploration and exploitation, which leads to poor balance between early global search and late local refinement. Second, its Lévy-flight and Brownian-motion parameters are fixed, which reduces adaptability across different stages of the optimization process. Third, the lack of population-wide information exchange can reduce diversity and increase susceptibility to local optima and premature convergence (Diao et al., 8 Sep 2025).

MSIGOA introduces three corresponding enhancements. IBUF divides the run into three stages: an early stage emphasizing exploration, a middle stage mixing exploration and exploitation, and a late stage focused on exploitation. APTS progressively scales the Brownian and Lévy step-size parameters over the course of the total iteration budget. DPRM periodically perturbs selected agents using both the current global best and a weighted summary of a dominant sub-population. This combination is presented as the mechanism by which the method enhances convergence speed while improving the balance between local exploitation and global exploration (Diao et al., 8 Sep 2025).

2. Core architecture and per-iteration workflow

The algorithm is organized into four phases per iteration (Diao et al., 8 Sep 2025):

1. Initialization (Eq. 1–3).
2. Exploitation/exploration step under IBUF + APTS (Eq. 4–12).
3. Gazelle “escape” step using local Brownian or Lévy motion (Eq. 5–8).
4. DPRM restart (Eq. 13–15).

The paper’s pseudocode specifies the following inputs and control flow: population size $N$, maximum iterations $T$, dimension $D$, and $\mathrm{PSR}=0.34$, with the output given as the best solution $\mathrm{Elite}$ (Diao et al., 8 Sep 2025). The population is initialized uniformly in $[Lb,Ub]^D$, objective values are evaluated, and the best individual is stored as $\mathrm{Elite}$. During each iteration, the algorithm first updates the Brownian and Lévy vectors through APTS, then performs the IBUF-directed update for each agent, then applies the basic GOA escape step based on $\mathrm{PSR}$, and finally executes the dominant population-based restart before boundary repair, reevaluation, and elite update.

In the pseudocode, the IBUF logic is stage dependent. For $t<T/3$, all agents undergo the update corresponding to Eq. 4. For $T/3 \le t < 2T/3$, agents with index $T$0 use the Lévy-based update of Eq. 5, while the remainder use the Brownian escape form of Eq. 7. For $T$1, all agents use the late-stage exploitation form of Eq. 10. After these moves, the basic GOA escape step is still applied, indicating that MSIGOA preserves a core behavioral layer from GOA while inserting higher-level scheduling and restart strategies around it (Diao et al., 8 Sep 2025).

This design suggests a hierarchical modification rather than a wholly new search rule: the base GOA dynamics remain present, but they are embedded in a more structured control framework. A plausible implication is that the method aims to preserve GOA’s behavioral diversity while reducing its stochastic inconsistency across search phases.

3. Iteration-Based Updating Framework and adaptive parameter tuning

The Iteration-Based Updating Framework is the central scheduling mechanism of MSIGOA. Let $T$2 denote the current iteration with $T$3, and let $T$4 denote the best solution found so far (Diao et al., 8 Sep 2025). The framework divides optimization into three stages.

In Stage 1, corresponding to global exploration for $T$5, the update uses Brownian-motion-based global search through Eq. 4:

$T$6

where $T$7 and $T$8 (Diao et al., 8 Sep 2025).

In Stage 2, corresponding to mixed search for $T$9, lower-indexed agents use the Lévy-flight form of Eq. 5,

$D$0

while higher-indexed agents use the Brownian-guided form of Eq. 7,

$D$1

where $D$2 alternates with $D$3, $D$4, and $D$5 is a decreasing scaling factor (Diao et al., 8 Sep 2025).

In Stage 3, corresponding to intensive local exploitation for $D$6, the update uses the elite-guided form of Eq. 10:

$D$7

(Diao et al., 8 Sep 2025).

The rationale given for IBUF is explicit. GOA originally selects exploration versus exploitation randomly at each iteration, whereas IBUF imposes a coarse schedule of explore $D$8 mix $D$9 exploit. According to the source description, this ensures thorough global search early, helps avoid premature convergence, and improves final precision through accurate local refinement late in the run (Diao et al., 8 Sep 2025).

The two Adaptive Parameter Tuning Strategies are defined by

$\mathrm{PSR}=0.34$0

$\mathrm{PSR}=0.34$1

These equations gradually reduce the effective Brownian and Lévy step sizes during the run (Diao et al., 8 Sep 2025). The stated purpose is to smooth the search process, reduce overshooting in later iterations, and preserve diversity in the middle stage. The paper characterizes fixed step sizes in basic GOA as too coarse for late Brownian search and too heavy-tailed for late Lévy-based refinement; APTS is introduced specifically to correct that mismatch (Diao et al., 8 Sep 2025).

4. Dominant population-based restart and information exchange

The Dominant Population-Based Restart Mechanism is the component that explicitly addresses insufficient information exchange and loss of diversity. The dominant sub-population $\mathrm{PSR}=0.34$2 is defined as the top 50% best agents, with size $\mathrm{PSR}=0.34$3 (Diao et al., 8 Sep 2025). Its weighted mean is

$\mathrm{PSR}=0.34$4

and the covariance matrix is

$\mathrm{PSR}=0.34$5

Selected agents are then restarted according to the pseudocode rule

$\mathrm{PSR}=0.34$6

with $\mathrm{PSR}=0.34$7 weighting the influence of the global best (Diao et al., 8 Sep 2025).

The mechanism is motivated by the tendency of GOA agents to follow a single global best, which can reduce diversity and trap the swarm in local minima. DPRM counters this by combining information from the current elite and the dominant half of the population, then injecting Gaussian perturbations governed by the covariance structure of that dominant set (Diao et al., 8 Sep 2025). In the source interpretation, this re-injects diversity and improves the ability to escape local optima.

The role of covariance is notable. Rather than perturbing agents with isotropic noise, DPRM uses a covariance matrix derived from the dominant population. This suggests that restart directions are informed by the current geometry of promising regions. A plausible implication is that the method attempts to preserve directional information from high-quality solutions while still disrupting stagnation.

5. Computational complexity and convergence characterization

The source summary defines $\mathrm{PSR}=0.34$8 as population size, $\mathrm{PSR}=0.34$9 as the number of iterations, $\mathrm{Elite}$0 as the problem dimension, and $\mathrm{Elite}$1 as the number of agents restarted in each generation (Diao et al., 8 Sep 2025). Under these definitions, the original GOA has complexity $\mathrm{Elite}$2 per run.

For MSIGOA, the added cost is decomposed by strategy. IBUF and APTS introduce only scalar or vector updates, yielding $\mathrm{Elite}$3 per iteration. DPRM requires computing $\mathrm{Elite}$4 and the covariance matrix $\mathrm{Elite}$5 from approximately $\mathrm{Elite}$6 dominant agents, which contributes $\mathrm{Elite}$7 per iteration. Applying restarts to $\mathrm{Elite}$8 agents contributes $\mathrm{Elite}$9 (Diao et al., 8 Sep 2025).

The resulting total complexity is stated as

$[Lb,Ub]^D$0

The source further notes that if $[Lb,Ub]^D$1, the additional $[Lb,Ub]^D$2 term is minor (Diao et al., 8 Sep 2025).

The convergence discussion is empirical rather than formal. MSIGOA is reported to converge more quickly and to higher accuracy than GOA on the benchmark suites used in the study, but no formal proof of asymptotic convergence is given (Diao et al., 8 Sep 2025). The paper instead attributes the observed gains to the staged framework and restart mechanism. This distinction is important: the method is supported by benchmark evidence and statistical testing, not by a theorem establishing global convergence.

6. Benchmark evaluation and engineering design performance

The experimental study evaluates parameter sensitivity, strategy effectiveness, convergence, and stability on two benchmark test sets: CEC2017 and CEC2022 (Diao et al., 8 Sep 2025). The setup comprises 29 CEC2017 functions in 50D and 100D, 12 CEC2022 functions in 10D and 20D, with 51 runs per function and the metrics best, mean, and Std (Diao et al., 8 Sep 2025).

The reported aggregate results are as follows:

Evaluation summary	CEC2017	CEC2022
Proportion of functions where MSIGOA is not worse than GOA	92.2%	83.3%
Proportion of functions where MSIGOA is not worse than other algorithms	88.57%	87.5%

The statistical analysis uses the Wilcoxon rank-sum test with $[Lb,Ub]^D$3 and Friedman average ranking (Diao et al., 8 Sep 2025). Against each competitor, MSIGOA has significantly more wins than losses in all four test settings. Its Friedman average ranking is reported as 1.0 in all four cases: CEC2017 50D, CEC2017 100D, CEC2022 10D, and CEC2022 20D. The next-best ranks are approximately 3.5, 4.4, 3.8, and 3.0, respectively (Diao et al., 8 Sep 2025).

The paper also evaluates extensibility on three constrained engineering design problems. For the tension/compression spring design, the variables are wire diameter $[Lb,Ub]^D$4, mean coil diameter $[Lb,Ub]^D$5, and number of coils $[Lb,Ub]^D$6, with objective

$[Lb,Ub]^D$7

as given in Eq. 16. The best MSIGOA result is reported as $[Lb,Ub]^D$8 (Diao et al., 8 Sep 2025). For pressure vessel design, with variables shell thickness $[Lb,Ub]^D$9, head thickness $\mathrm{Elite}$0, radius $\mathrm{Elite}$1, and length $\mathrm{Elite}$2, the best MSIGOA result is 5885.33277, compared in the source summary with GOA at 5906.02 (Diao et al., 8 Sep 2025). For welded-beam design, with variables weld thickness $\mathrm{Elite}$3, weld length $\mathrm{Elite}$4, beam width $\mathrm{Elite}$5, and beam thickness $\mathrm{Elite}$6, the best MSIGOA result is 1.6927682, compared in the source summary with GOA at 1.7353445 and other methods all above 1.70 (Diao et al., 8 Sep 2025).

These results are presented in the source as evidence of superior convergence and efficiency on both benchmark and engineering problems. A narrower interpretation, consistent with the data, is that the method performed best within the experimental comparisons reported in the paper.

7. Strengths, limitations, and prospective extensions

The paper identifies several strengths of MSIGOA. IBUF is described as providing a well-balanced exploration–exploitation schedule, APTS as enabling smooth adaptive scaling of search steps, and DPRM as improving the ability to escape local optima. Across the reported experiments, the method is characterized as consistently top-ranked on high-dimensional and real-world benchmark problems (Diao et al., 8 Sep 2025).

The stated limitations are comparatively modest but technically relevant. DPRM introduces extra overhead through the dominant population covariance calculation, leading to marginally higher runtime. In addition, on a small subset of highly multimodal functions, MSIGOA was not the best-performing algorithm, and the source notes that some hybridization may further help (Diao et al., 8 Sep 2025).

The future directions listed in the source are specific: extension to multi-objective and discrete optimization, including GKLS-style test suites; hybridization with classical local search or surrogate models for very expensive functions; and application to hyper-parameter tuning in deep learning, medical image segmentation, portfolio optimization, network design, resource allocation, and robotics path planning (Diao et al., 8 Sep 2025). These items are presented as prospective applications rather than validated deployments within the reported study.

A common misconception in reading such results is to equate empirical superiority on standard suites with a formal guarantee of universal dominance. The source does not support that conclusion. It supports instead a more limited claim: within the reported CEC2017, CEC2022, and engineering design experiments, the combination of staged search scheduling, adaptive step-size reduction, and dominant-population restart yielded stronger empirical performance than the compared variants and advanced algorithms (Diao et al., 8 Sep 2025).