# Multimodal Optimization Tutorial
This tutorial tackles the **Rastrigin function**, a challenging multimodal benchmark with many local optima. You'll learn strategies for escaping local optima and finding global solutions.
## The Rastrigin Function
The Rastrigin function is defined as:
```text
f(x) = 10n + Σ[xᵢ² - 10cos(2πxᵢ)]
```
**Properties:**
- Global minimum: 0 at origin
- **Highly multimodal**: ~10ⁿ local minima!
- Non-separable: Variables interact through cosine terms
- Deceptive: Local minima look similar to global minimum
<div class="fugue-explorable" data-viz="ga-anatomy" data-landscape="rastrigin" data-seed="11"></div>
*Rastrigin’s trap, live: many basins, one global. Watch where selection concentrates the population — and how mutation (violet streaks) occasionally kicks a lineage over a ridge. Raise the mutation probability and the population stays scattered; lower it and it commits early.*
## The Challenge
With 20 dimensions, the Rastrigin function has approximately 10²⁰ local minima. A naive optimization will likely get trapped in one of these local minima.
## Complete Example
```rust,ignore
{{#include ../../../examples/rastrigin_benchmark.rs}}
```
> **Source**: [`examples/rastrigin_benchmark.rs`](https://github.com/fugue-evo/fugue-evo/blob/main/examples/rastrigin_benchmark.rs)
## Running the Example
```bash
cargo run --example rastrigin_benchmark
```
<div class="fugue-explorable fv-inline" data-viz="mini-diversity" data-seed="19" data-k="3" data-caption="Diversity is the budget you spend to escape local optima. Drag k to trade exploration against exploitation."></div>
## Key Strategies
### 1. Larger Population
```rust,ignore
.population_size(200)
```
More individuals means more parallel exploration of the search space. For multimodal functions, this helps maintain diversity and cover more basins of attraction.
### 2. Higher Selection Pressure
```rust,ignore
.selection(TournamentSelection::new(5))
```
Tournament size of 5 (vs. 3 for Sphere) increases selection pressure, helping the population converge faster on good regions once found.
### 3. More Exploration in Crossover
```rust,ignore
.crossover(SbxCrossover::new(15.0))
```
Lower distribution index (15 vs. 20) creates more diverse offspring, helping explore new regions.
### 4. Elitism
```rust,ignore
.elitism(true)
.elite_count(2)
```
Critical for multimodal optimization! Without elitism, the best solution can be lost due to selection randomness.
### 5. More Generations
```rust,ignore
.max_generations(500)
```
Multimodal problems need more time to find and refine global optima.
## Understanding Results
### Tracking Progress
```rust,ignore
let history = result.stats.best_fitness_history();
for (i, fitness) in history.iter().enumerate() {
if i % 50 == 0 {
println!(" Gen {:4}: {:.6}", i, fitness);
}
}
```
Watch for:
- **Rapid early improvement**: Finding good basins
- **Plateaus**: Stuck in local optima
- **Jumps**: Escaping to better regions
### Solution Quality
```rust,ignore
let max_deviation = result
.best_genome
.genes()
.iter()
.map(|x| x.abs())
.fold(0.0f64, |a, b| a.max(b));
println!("Max deviation from origin: {:.6}", max_deviation);
```
For Rastrigin, each gene should be close to 0. Large deviations indicate the solution is in a local optimum.
## Alternative Approaches
### Island Model
For heavily multimodal problems, consider the Island Model:
```rust,ignore
// Multiple populations with periodic migration
let result = IslandModelBuilder::<RealVector, _, _, _, _, f64>::new()
.num_islands(4)
.island_population_size(50)
.migration_interval(25)
.migration_policy(MigrationPolicy::Best(2))
// ... operators
.build(&mut rng)?
.run(200, &mut rng)?;
```
See [Island Model Tutorial](./island-model.md) for details.
### Restart Strategy
Manual restarts can help escape deep local optima:
```rust,ignore
let mut best_overall = f64::NEG_INFINITY;
let mut best_genome = None;
for restart in 0..5 {
let result = SimpleGABuilder::<RealVector, f64, _, _, _, _, _>::new()
// ... configuration
.build()?
.run(&mut rng)?;
if result.best_fitness > best_overall {
best_overall = result.best_fitness;
best_genome = Some(result.best_genome);
}
println!("Restart {}: {:.6}", restart, result.best_fitness);
}
```
## Parameter Guidelines for Multimodal Functions
| Population | 50-100 | 150-300 |
| Tournament size | 2-3 | 4-7 |
| SBX eta | 15-25 | 10-15 |
| Mutation probability | 1/n | 1.5/n - 2/n |
| Generations | 100-300 | 300-1000 |
| Elitism | Optional | Essential |
## Diagnosing Problems
### Stuck in Local Optima
**Symptoms**:
- Fitness plateaus early
- Solution values are multiples of π (Rastrigin local minima)
**Solutions**:
1. Increase mutation probability
2. Use Island Model
3. Try different random seeds
4. Add restarts
### Loss of Diversity
**Symptoms**:
- All individuals become similar
- No improvement despite many generations
**Solutions**:
1. Increase population size
2. Lower selection pressure
3. Add diversity maintenance (niching)
## Exercises
1. **Vary dimensions**: Compare results for DIM = 10, 20, 30
2. **Compare seeds**: Run with 5 different seeds and analyze variance
3. **Population study**: Try populations of 50, 100, 200, 400
## Next Steps
- [Island Model Tutorial](./island-model.md) - Parallel populations for multimodal optimization
- [CMA-ES Tutorial](./cmaes.md) - Alternative algorithm for continuous optimization