Marine Predators Algorithm

• Marine Predators Algorithm is a nature-inspired optimization method that mimics predator-prey dynamics using Brownian motion and Lévy flights.
• It employs a structured, multi-phase search strategy with an elite matrix to balance global exploration and local exploitation effectively.
• MPA has demonstrated strong performance in complex scenarios like interplanetary trajectory optimization by outperforming several benchmark algorithms.

The Marine Predators Algorithm (MPA) is a nature-inspired metaheuristic optimization method rooted in the foraging tactics of marine predators and their prey, notably modeling the interplay of intensive local searches and long-distance exploration via stochastic motion patterns. Its algorithmic framework explicitly leverages Brownian motion and Lévy flights to dynamically balance exploration and exploitation throughout the optimization process. A multi-phase search architecture and an elite matrix memory mechanism underpin its robustness in navigating complex, nonlinear, and high-dimensional search spaces, making it particularly effective for challenging scenarios such as interplanetary trajectory optimization (Dehkordi et al., 18 May 2025).

1. Biological and Conceptual Foundations

MPA’s design is closely modeled on the predation–evasion dynamics observed in marine ecosystems, featuring key analogies:

• Predator-Prey Dynamics: Top predators, capable of faster movement, perform rapid local foraging (exploitation), while slower or comparable prey escape via erratic, Brownian-like movements (exploration).
• Lévy Motifs: When predator and prey have comparable speeds, foraging shows both local and nonlocal search patterns, statistically described by Lévy flights.
• Algorithmic Mapping: MPA interprets search agents as “predators” and optimal solutions as “prey locations.” The population is guided to exploit known good solutions while being able to undertake global exploration, directly reflecting observed ecological strategies.

This bio-inspired analogy provides a rigorous conceptual schema for search mechanisms adaptable to the exploration–exploitation trade-off fundamental to black-box optimization (Dehkordi et al., 18 May 2025).

2. Multi-Phase Search Structure and Position Update Rules

MPA divides the optimization trajectory into three sequential phases. Each phase models a distinct predator-prey speed regime, operationalized by specialized update equations:

• Phase 1 (Exploration: Predator Faster than Prey, $t < T/3$):

$$\text{stepsize}_i = B_i \odot (X_\alpha - B_i \odot X_i),\quad X_i^{t+1} = X_i^t + P \odot \text{stepsize}_i$$

Agents undertake Brownian motion-centered exploration around $X_\alpha$ (the elite solution).

• Phase 2 (Transition: Equal Speeds, $T/3 \le t < 2T/3$):
  • First half: Lévy flights for global jumps:

  $$\text{stepsize}_i = L_i \odot (X_\alpha - L_i \odot X_i),\quad X_i^{t+1} = X_i^t + P \odot \text{stepsize}_i$$ - Second half: Brownian steps for local refinement:

  $$\text{stepsize}_j = B_j \odot (B_j \odot X_\alpha - X_j),\quad X_j^{t+1} = X_\alpha + P \odot \text{stepsize}_j$$

• Phase 3 (Exploitation: Predator Slower than Prey, $t \ge 2T/3$):

$$\text{stepsize}_i = L_i \odot (L_i \odot X_\alpha - X_i),\quad X_i^{t+1} = X_\alpha + P \odot \text{stepsize}_i$$

The parameter $P$ is a vector for scaling or stochastic perturbation, and $X_\alpha$ denotes the leader or global best in the elite matrix.

3. Elite Matrix and Memory Storage

A defining feature of MPA is its elite matrix, an $m \times d$ structure containing the $m$ highest-quality solutions at each iteration in $d$-dimensional search space. The update rule is: $$\text{if } f(x_i) < \max_{k} f(E_k),\quad \text{replace the worst elite E}_k\text{ with } x_i$$ where $f(\cdot)$ denotes the objective function. The elite matrix both preserves solution diversity and focuses exploitation around high-fitness regions. During position updates, samples from $E$ (including $E_1 = X_\alpha$) serve as guides for the entire swarm, thereby codifying both memory and adaptive pressure (Dehkordi et al., 18 May 2025).

4. Stochastic Motion Modeling: Brownian and Lévy Dynamics

MPA’s exploration and exploitation mechanisms are powered by two distinct stochastic processes:

• Brownian Motion:

$B = \sigma\, \mathcal{N}(0, I_d)$

A Gaussian random vector with scaling $\sigma$, supporting local, isotropic search.

• Lévy Flights:

$$L = \mu\, \frac{u}{|v|^{1/\beta}},\quad u \sim \mathcal{N}(0, \sigma_u^2), \quad v \sim \mathcal{N}(0,1)$$

Heavy-tailed step sizes, parameterized by $\beta \in (1,2]$, typically $\beta=1.5$, generating infrequent long jumps to explore remote regions and escape deep local optima.

Elementwise application of these motion models in position updates provides a statistically principled means for multimodal exploration and refined search intensification.

5. Exploration–Exploitation Dynamics

The three-phase design governs the transition from global search to local refinement:

• Brownian-dominated updates in Phase 1 maximize broad sampling and reduce premature convergence risk.
• Hybrid regime in Phase 2 stratifies the population, allowing simultaneous local and global probing.
• Lévy-driven exploitation in Phase 3 achieves deep refinement with punctuated, long-range escape routes from local minima.
• The elite matrix and “FADs” (Fish Aggregating Devices) mechanisms inject additional random jumps when improvement stagnates, further safeguarding against entrapment.

This structured alternation results in a robust balance, widely considered a key determinant of MPA’s efficacy (Dehkordi et al., 18 May 2025).

6. Parameterization, Strengths, and Limitations

The MPA framework is characterized by a small set of main parameters:

• Population size $N$: 30–50.
• Elite matrix size $m$: 3–5.
• Lévy exponent $\beta$: typically 1.5, adaptable.
• Brownian scaling $\sigma$: $0.1$ – $1.0$ of search range.
• Max iterations: 500–2000, based on problem dimensionality.

Strengths:

• Multi-phase control aligns exploration/exploitation with search dynamics.
• Elite matrix prevents information loss, with minimal overhead.
• Stochastic motion foundations (Brownian/Lévy) are theoretically and empirically robust.

Limitations:

• Performance is sensitive to the elite matrix size $m$ and $P$-scaling.
• Lévy parameter $\beta$ often requires empirical tuning.
• MPA's convergence may lag highly problem-tailored local optimizers.

7. Hybridization and Applications: GMPA and GTOPX Performance

The Grey Wolf Optimizer–Marine Predators Algorithm (GMPA) hybrid replaces GWO’s canonical update rules with MPA's three-phase mechanism. Key hybridization points:

• Three-phase updates replace/augment GWO's encircling and hunting rules, using Brownian motion and Lévy flights for exploration and exploitation.
• Elite memory: GWO’s best $\alpha$, $\beta$, $\delta$ wolves are iteratively tracked with the MPA-style elite matrix.
• FADs: Integrated to disrupt symmetry and prevent search stagnation.
• Adaptive step sizes: Modified from MPA’s constants to iteration-dependent forms, coupled with GWO's convergence parameters.

Empirical Findings (Dehkordi et al., 18 May 2025):

• On GTOPX interplanetary trajectory benchmarks (e.g., Cassini-1, Rosetta), GMPA outperformed vanilla GWO, pure MPA, PSO, DE, and LSHADE, achieving lower fuel-costs ($\Delta V$) and superior convergence rates.
• Robustness to high multimodality and dimensionality was evidenced by consistent performance gains (5–15% improvement in average $\Delta V$) and reduced stagnation across independent trials.

This suggests that the incorporation of MPA's exploration–exploitation protocol and memory structure into hybrid metaheuristics is particularly beneficial in large-scale, complex aerospace optimization contexts.