algorithm_details.md

Immortal Jellyfish Algorithm — Mathematical Specification

1. Biological Inspiration

Turritopsis dohrnii — the "immortal jellyfish" — is the only known animal capable of reverting from its fully-grown adult medusa form back to its juvenile polyp stage, bypassing death entirely. This remarkable lifecycle provides a natural template for balancing exploration and exploitation:

Life stage	Biological behaviour	Optimisation role
Polyp	Sessile, Lévy-walk feeding tentacles	Broad global exploration
Strobilation	Asexual budding of ephyra clones	Diversification + crossover
Medusa	Free-swimming, attracted to light	Exploitation toward elites
Senescence	Degradation and cell-level reset	Fine-grained local search

Additionally, the lighthouse metaphor supplies the attraction force: a point source of light whose intensity obeys the inverse-square law, decaying both with distance and with time (as the jellyfish ages).

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2. Problem Statement

$$\min_{\mathbf{x} \in \Omega}; f(\mathbf{x}), \qquad \Omega = \prod_{j=1}^d [l_j, u_j]$$

A population of $n$ agents $\mathbf{P} = {\mathbf{x}_1, \dots, \mathbf{x}n} \subset \Omega$ is evolved over $T{\max}$ iterations.

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3. Phase Scheduler

Normalised progress: $\tau = t / T_{\max} \in [0,1]$.

Phase	Condition	Operator
0 — Polyp	$\tau < 0.15$	Lévy-flight exploration
1 — Strobilation	$0.15 \le \tau < 0.50$	Ephyra budding
2 — Medusa	$0.50 \le \tau < 0.85$	Lighthouse attraction + pulse
3 — Senescence	$0.85 \le \tau \le 1$	Gaussian refinement

Sigmoidal blend weight (controls step-size decay):

$$\sigma(\tau) = \frac{1}{1 + e^{-12(\tau - 0.5)}}$$

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4. Mathematical Operators

4.1 Lévy-Flight Exploration (Polyp)

Steps generated by Mantegna's algorithm ($\beta = 1.5$):

$$\sigma_u = \left(\frac{\Gamma(1+\beta)\sin(\pi\beta/2)}{\Gamma!\left(\frac{1+\beta}{2}\right)\beta,2^{(\beta-1)/2}}\right)^{1/\beta}$$

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

Position update:

$$\mathbf{x}_i^{t+1} = \mathbf{x}_i^t + 0.01(u_j - l_j) \cdot L_j \cdot (\mathbf{x}_i^t - \mathbf{x}^*)$$

4.2 Strobilation with Adaptive Ephyra Count

The number of ephyra offspring decreases as the strobilation phase progresses:

$$n_{\text{eph}}(\tau) = \max!\left(2,; \text{round}!\left(5 - 3 \cdot \frac{\tau - 0.15}{0.35}\right)\right)$$

Each ephyra is:

$$\mathbf{c}_k = \text{clip}!\left(\mathbf{x}_i + \mathcal{N}(0,1) \cdot L(\beta) \cdot \frac{\mathbf{u} - \mathbf{l}}{6},; \mathbf{l},; \mathbf{u}\right)$$

Best among ${\mathbf{x}i, \mathbf{c}1, \dots, \mathbf{c}{n\text{eph}}}$ is retained.

4.3 Lighthouse Attraction (Medusa)

Intensity at agent $i$ from lighthouse $\mathbf{g}_s$:

$$I_{i,s} = \frac{I_0}{1 + \kappa|\mathbf{x}_i - \mathbf{g}_s|^2} \cdot e^{-\lambda\tau}$$

Adaptive step-size:

$$\alpha_t = \alpha_{\max}(1-\sigma(\tau)) + \alpha_{\min}\sigma(\tau)$$

Update:

$$\mathbf{x}_i^{t+1} = \mathbf{x}_i^t + \alpha_t I_{i,s}(\mathbf{g}_s - \mathbf{x}_i^t) + 0.01\mathcal{N}(0,1)|\mathbf{g}_s - \mathbf{x}_i^t|$$

4.4 Pulsed Swimming (Medusa)

Sinusoidal bell contraction with quadratic amplitude decay:

$$\delta_j^t = A_0(1-\tau)^2 \sin(2\pi f_p \tau + \phi_i) \cdot (u_j - l_j)$$

where $\phi_i \sim \mathcal{U}(0, 2\pi)$ is a fixed personal phase.

4.5 Transdifferentiation (Medusa, stagnation)

When $\text{age}i \ge T\text{age}$:

$$\mathbf{x}_i^{\text{opp}} = \mathbf{l} + \mathbf{u} - \mathbf{x}_i$$

Accept $\mathbf{x}_i^{\text{opp}}$ if better; otherwise resample uniformly.

4.6 Archive-Based Budding (Medusa / Senescence)

$$\mathbf{x}_i' = \text{clip}(\mathbf{x}_i + L(\beta) \cdot (\mathbf{x}_i - \mathbf{a}),; \mathbf{l},; \mathbf{u})$$

where $\mathbf{a}$ is a random sample from the elite archive.

4.7 Senescence Refinement

$$\mathbf{c} = \text{clip}!\left(\mathbf{x}^* + 0.05(u_j - l_j)(1-\tau)^2 \cdot \mathcal{N}(0,1),; \mathbf{l},; \mathbf{u}\right)$$

Accept $\mathbf{c}$ if $f(\mathbf{c}) < f(\mathbf{x}_i)$.

4.8 Diversity Guard

Normalised mean absolute deviation:

$$D = \frac{1}{nd}\sum_{i=1}^n\sum_{j=1}^d \frac{|x_{ij} - \bar{x}_j|}{u_j - l_j}$$

If $D < \delta_{\min} = 0.005$, the worst 20 % of agents are uniformly resampled.

4.9 Crowding-Distance Archive Update

The elite archive (capacity $|\mathcal{A}|$) prunes excess entries by a combined criterion: primary sort by fitness, tie-breaking by crowding distance (larger distance = more diverse = preferred to retain).

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5. Parameters

Symbol	Parameter	Default
$n$	Population size	50
$T_{\max}$	Max iterations	500
$k$	Lighthouse count	3
$\beta$	Lévy exponent	1.5
$|\mathcal{A}|$	Archive capacity	15
$T_\text{age}$	Stagnation threshold	25
$\lambda$	Intensity decay	0.5
$\alpha_{\max}$	Max step-size	2.0
$\alpha_{\min}$	Min step-size	0.05
$\delta_{\min}$	Diversity threshold	0.005

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6. Time Complexity

$$\mathcal{O}(T_{\max} \cdot n \cdot d)$$

Identical to PSO, GWO, and WOA — IJA is a drop-in replacement.