Continuous Optimization Tutorial

This tutorial covers optimizing functions with real-valued variables. You'll learn how to set up, run, and analyze continuous optimization problems.

The Sphere Function

We'll start with the Sphere function, a simple unimodal benchmark:

f(x) = Σxᵢ² = x₁² + x₂² + ... + xₙ²

Properties:

Global minimum: 0 at origin (all zeros)
Unimodal: Single optimum, no local minima
Convex: Any local minimum is global
Separable: Each variable is independent

Complete Example

{{#include ../../../examples/sphere_optimization.rs}}

Source: examples/sphere_optimization.rs

Running the Example

cargo run --example sphere_optimization

Understanding the Components

Search Bounds

let bounds = MultiBounds::symmetric(5.12, DIM);

This creates bounds [-5.12, 5.12] for each of the 10 dimensions. The Sphere function is typically tested in this range.

Operators for Real-Valued Optimization

Simulated Binary Crossover (SBX)

SbxCrossover::new(20.0)

SBX creates offspring similar to parent values. The distribution index (eta = 20.0) controls spread:

Higher eta → offspring closer to parents (exploitation)
Lower eta → more diverse offspring (exploration)

Polynomial Mutation

PolynomialMutation::new(20.0)

Adds bounded perturbations to gene values. The distribution index controls mutation magnitude.

Fitness Function

let fitness = Sphere::new(DIM);

Built-in Sphere computes the sum of squares. Since GAs maximize by default, the function is negated internally.

Analyzing Results

println!("Best fitness: {:.6}", result.best_fitness);
println!("Generations:  {}", result.generations);
println!("Evaluations:  {}", result.evaluations);

Key metrics:

Best fitness: Should approach 0 (the global minimum)
Generations: How many evolutionary cycles completed
Evaluations: Total fitness function calls

Convergence Statistics

println!("{}", result.stats.summary());

The summary shows:

Mean and standard deviation per generation
Diversity metrics
Improvement trends

Parameter Tuning

Population Size

.population_size(100)

Size	Trade-off
Small (20-50)	Fast generations, less diversity
Medium (100-200)	Good balance
Large (500+)	More exploration, slower per generation

For Sphere (unimodal), smaller populations work well. Multimodal functions need larger populations.

Operator Parameters

SBX Distribution Index

.crossover(SbxCrossover::new(eta))

eta	Effect
1-5	High exploration
15-25	Balanced
50+	High exploitation

Mutation Probability

.mutation(PolynomialMutation::new(20.0).with_probability(0.1))

Default is 1/dimension. Increase for more exploration.

Elitism

.elitism(true)
.elite_count(2)

Elitism preserves the best individuals across generations, preventing loss of good solutions.

Common Issues

Premature Convergence

Symptoms: Population converges too quickly, stuck at suboptimal solution

Solutions:

Increase population size
Lower selection pressure (smaller tournament size)
Increase mutation probability

Slow Convergence

Symptoms: Fitness improves very slowly