Melvin Alvarez
190 lines
5.0 KiB
Markdown
190 lines
5.0 KiB
Markdown
# Differential Evolution
|
||
|
||
**Differential Evolution (DE)** is a powerful evolutionary algorithm for global optimization of continuous, non-linear, non-convex functions. It's particularly effective for multimodal optimization landscapes.
|
||
|
||
## Algorithm Overview
|
||
|
||
DE works by maintaining a **population** of candidate solutions and iteratively improving them through:
|
||
|
||
1. **Mutation**: Create mutant vectors by combining existing solutions
|
||
2. **Crossover**: Mix mutant with target vector
|
||
3. **Selection**: Keep better solution (greedy selection)
|
||
|
||
### Key Parameters
|
||
|
||
- **Population Size** (`pop_size`): Number of candidate solutions (typically 10× problem dimension)
|
||
- **Mutation Factor** (`F`): Scale factor for difference vectors (0.5-1.0)
|
||
- **Crossover Rate** (`CR`): Probability of using mutant component (0.0-1.0)
|
||
- **Strategy**: Mutation/crossover strategy (see below)
|
||
|
||
## Strategies
|
||
|
||
OptimizR implements 5 DE strategies:
|
||
|
||
### 1. `rand/1/bin`
|
||
```
|
||
mutant = x_r1 + F * (x_r2 - x_r3)
|
||
```
|
||
Most explorative, good for diverse populations.
|
||
|
||
### 2. `best/1/bin`
|
||
```
|
||
mutant = x_best + F * (x_r1 - x_r2)
|
||
```
|
||
Exploitative, fast convergence but may get stuck.
|
||
|
||
### 3. `current-to-best/1/bin`
|
||
```
|
||
mutant = x_i + F * (x_best - x_i) + F * (x_r1 - x_r2)
|
||
```
|
||
Balanced exploration/exploitation.
|
||
|
||
### 4. `rand/2/bin`
|
||
```
|
||
mutant = x_r1 + F * (x_r2 - x_r3) + F * (x_r4 - x_r5)
|
||
```
|
||
More diversity through two difference vectors.
|
||
|
||
### 5. `best/2/bin`
|
||
```
|
||
mutant = x_best + F * (x_r1 - x_r2) + F * (x_r3 - x_r4)
|
||
```
|
||
Aggressive convergence to best solution.
|
||
|
||
## Usage Example
|
||
|
||
```python
|
||
import numpy as np
|
||
from optimizr import differential_evolution
|
||
|
||
def rastrigin(x):
|
||
A = 10
|
||
return A * len(x) + sum(x**2 - A * np.cos(2 * np.pi * x))
|
||
|
||
best_x, best_fx = differential_evolution(
|
||
objective_fn=rastrigin,
|
||
bounds=[(-5.12, 5.12)] * 10,
|
||
strategy="best1",
|
||
popsize=20,
|
||
maxiter=500,
|
||
adaptive=True,
|
||
)
|
||
|
||
print(f"Best fitness: {best_fx:.6f}")
|
||
print(f"Best solution: {best_x}")
|
||
```
|
||
|
||
## Advanced Features
|
||
|
||
### Adaptive jDE
|
||
|
||
Enable self-adaptive F and CR parameters:
|
||
|
||
```python
|
||
de = DifferentialEvolution(
|
||
bounds=[(-5, 5)] * 20,
|
||
adaptive=True, # Enable jDE
|
||
tau_F=0.1, # F adaptation rate
|
||
tau_CR=0.1 # CR adaptation rate
|
||
)
|
||
```
|
||
|
||
### Constraint Handling
|
||
|
||
For constrained optimization:
|
||
|
||
```python
|
||
def constraints(x):
|
||
"""Return array of constraint violations (> 0 means violated)"""
|
||
return np.array([
|
||
x[0]**2 + x[1]**2 - 1, # x0^2 + x1^2 <= 1
|
||
x[0] + x[1] - 2 # x0 + x1 <= 2
|
||
])
|
||
|
||
de = DifferentialEvolution(
|
||
bounds=[(-5, 5)] * 2,
|
||
constraints=constraints,
|
||
penalty_factor=1000
|
||
)
|
||
```
|
||
|
||
## Performance Tips
|
||
|
||
1. **Population Size**: Start with `10 × dim`, increase if stuck
|
||
2. **F parameter**:
|
||
- Low (0.4-0.6): Fine-tuning, local search
|
||
- High (0.8-1.0): Exploration, escape local minima
|
||
3. **CR parameter**:
|
||
- Low (0.1-0.3): Separable problems
|
||
- High (0.9-1.0): Non-separable, coupled variables
|
||
4. **Strategy Selection**:
|
||
- Unknown landscape → `rand/1/bin` or `rand/2/bin`
|
||
- Smooth, unimodal → `best/1/bin`
|
||
- Multimodal, deceptive → `current-to-best/1/bin`
|
||
|
||
## Benchmarks
|
||
|
||
Performance on standard test functions (10D, 500 iterations):
|
||
|
||
| Function | Success Rate | Avg Time | Best Fitness |
|
||
|----------|--------------|----------|--------------|
|
||
| Sphere | 100% | 12ms | 1e-12 |
|
||
| Rosenbrock | 98% | 18ms | 3e-6 |
|
||
| Rastrigin | 87% | 22ms | 0.02 |
|
||
| Ackley | 95% | 15ms | 2e-8 |
|
||
|
||
*Compared to SciPy `differential_evolution`: 50-80× faster*
|
||
|
||
## Mathematical Details
|
||
|
||
### Mutation Operator
|
||
|
||
For strategy `rand/1/bin`:
|
||
|
||
$$
|
||
\mathbf{v}_{i,g} = \mathbf{x}_{r_1,g} + F \cdot (\mathbf{x}_{r_2,g} - \mathbf{x}_{r_3,g})
|
||
$$
|
||
|
||
Where:
|
||
- $\mathbf{v}_{i,g}$: Mutant vector for individual $i$ at generation $g$
|
||
- $\mathbf{x}_{r_j,g}$: Randomly selected individuals ($r_1 \neq r_2 \neq r_3 \neq i$)
|
||
- $F \in [0, 2]$: Mutation scaling factor
|
||
|
||
### Crossover Operator
|
||
|
||
Binomial crossover:
|
||
|
||
$$
|
||
u_{i,j,g} = \begin{cases}
|
||
v_{i,j,g} & \text{if } \text{rand}(0,1) < CR \text{ or } j = j_{rand} \\\\
|
||
x_{i,j,g} & \text{otherwise}
|
||
\end{cases}
|
||
$$
|
||
|
||
Ensures at least one component from mutant.
|
||
|
||
### Selection Operator
|
||
|
||
Greedy selection:
|
||
|
||
$$
|
||
\mathbf{x}_{i,g+1} = \begin{cases}
|
||
\mathbf{u}_{i,g} & \text{if } f(\mathbf{u}_{i,g}) \leq f(\mathbf{x}_{i,g}) \\\\
|
||
\mathbf{x}_{i,g} & \text{otherwise}
|
||
\end{cases}
|
||
$$
|
||
|
||
## References
|
||
|
||
1. Storn, R., & Price, K. (1997). *Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces*. Journal of global optimization, 11(4), 341-359.
|
||
|
||
2. Das, S., & Suganthan, P. N. (2011). *Differential evolution: A survey of the state-of-the-art*. IEEE transactions on evolutionary computation, 15(1), 4-31.
|
||
|
||
3. Brest, J., et al. (2006). *Self-adapting control parameters in differential evolution: A comparative study on numerical benchmark problems*. IEEE transactions on evolutionary computation, 10(6), 646-657.
|
||
|
||
## See Also
|
||
|
||
- [API Reference](../api/differential_evolution.md)
|
||
- [Jupyter Tutorial](https://github.com/ThotDjehuty/optimiz-r/blob/main/examples/01_differential_evolution_tutorial.ipynb)
|
||
- [Benchmarks](../benchmarks.md)
|