Tardigrade Optimization Algorithm (TOA)

The Tardigrade Optimization Algorithm (TOA) is a contemporary bio-inspired metaheuristic optimization technique modeled after the adaptable survival behavior of tardigrades, microscopic organisms capable of enduring extreme environmental conditions through states like cryptobiosis and controlled energy conservation.

This algorithm integrates exploration–exploitation balancing, hunger-driven adaptation, and cooperative movement modeling to provide a versatile and robust optimization framework.

Algorithm Overview

TOA emulates five major biological behaviors:

1. Hunger-Driven Movement Mechanics

Tardigrades regulate mobility based on resource availability.
In TOA, each agent’s movement intensity depends on its hunger level ( H_i ), which adaptively changes according to its performance.

The hunger update follows:

$$ H_i(t+1) = \begin{cases} H_i(t) - \delta_h, & \text{if } f(\mathbf{x}_i) < \bar{f}(t) \\ H_i(t) + \delta_p, & \text{otherwise} \end{cases} $$

Lower hunger reduces activity, encouraging fine exploitation; higher hunger induces active searching.

2. Cryptobiosis Modeling

When hunger surpasses a threshold ( H_i > H_c ), the agent enters a reduced-mobility state.
Its position update becomes minimal or frozen:

$$ \mathbf{x}_i(t+1) = \mathbf{x}_i(t) + \epsilon \cdot \mathcal{U}(-\eta, \eta) $$

3. Elite-Based Learning and Cooperation

A subset of elite agents ( E \subset N ) are selected based on best fitness values.

Cooperation occurs either with the global best or a random elite member:

$$ \mathbf{x}_i(t+1) = \mathbf{x}_i(t) + \lambda \cdot H_i(t) \cdot (\mathbf{x}_E - \mathbf{x}_i(t)) $$

4. Lévy-Flight-Assisted Exploration

To ensure global diversity, some agents apply Lévy flight steps:

$$ \mathbf{x}_i(t+1) = \mathbf{x}_i(t) + \alpha \cdot \text{Levy}(\beta) $$

where:

• ( \alpha ): step size scaling factor,
• ( \text{Levy}(\beta) ): heavy-tailed Lévy distribution with parameter ( \beta \in (1,3] ),
  typically defined as:

$$ \text{Levy}(\beta) = \frac{\Gamma(1+\beta) \sin(\pi\beta / 2)} {\Gamma\left(\frac{1+\beta}{2}\right)\beta 2^{(\beta-1)/2}} \cdot \frac{u}{|v|^{1/\beta}} $$

This mechanism enables large exploratory jumps that help escape local minima.

5. Stagnation Detection and Escape

TOA monitors fitness improvement using a stagnation counter ( S_t ):

$$ S_t = \begin{cases} S_{t-1} + 1, & |F_{\text{best}}(t) - F_{\text{best}}(t-1)| < \varepsilon \\ 0, & \text{otherwise} \end{cases} $$

If ( S_t > S_{\max} ), the algorithm reinitializes some of the weakest agents:

$$ \mathbf{x}_i = \text{rand}(L_b, U_b) \quad \text{for } i \in W $$

where ( W ) represents the subset of low-performance agents.
This refresh restores diversity after stagnation.

6. Position Update Summary

By combining these dynamic mechanisms, the general agent update rule becomes:

$$\mathbf{x}_i(t+1) = \mathbf{x}_i(t) + \alpha_1 \cdot \mathcal{E}_1 + \alpha_2 \cdot \mathcal{E}_2 + \alpha_3 \cdot \mathcal{E}_3$$

Key Characteristics

• Exploration–Exploitation Balance: adaptive transition guided by hunger and iteration count.
• Elite Coordination: accelerates convergence through guided communication.
• Lévy-Based Jumps: preserve global diversity and exploration.
• Stagnation Escape: reintroduces randomness when stuck in local minima.
• Cryptobiosis Phase: precision refining near the global optimum.