Overview: Genetic algorithms (GAs) are optimization methods inspired by natural selection and genetics. They are used to solve optimization problems by evolving a population of candidate solutions.

Key concepts:

• Population: Set of possible solutions.
• Fitness: A measure of how good each solution is.
• Selection: Choosing individuals for reproduction.
• Crossover: Combining parts of two solutions to create a new solution.
• Mutation: Randomly altering a solution.

Objective: We aim to optimize the quadratic function \(f(x) = x^2 - 4x + 4\), minimizing the value of \(f(x)\).

Function: \[ f(x) = x^2 - 4x + 4 \]

We begin by creating an initial population of three individuals: \(x_1 = 1\), \(x_2 = 3\), and \(x_3 = 0\).

population <- c(1, 3, 0)
population

## [1] 1 3 0

Fitness function: The fitness of an individual is how well it minimizes the function \(f(x)\).

For each individual, calculate \(f(x)\): \[ f(x) = x^2 - 4x + 4 \]

f <- function(x) {
    x^2 - 4*x + 4
}

# we calculate fitness
fitness <- f(population)
fitness

## [1] 1 1 4

Let’s plot the quadratic function \(f(x)\) to visualize the optimization landscape.

We select individuals for crossover based on fitness. Lower fitness values are better in this case, so we select \(x_1 = 1\) and \(x_2 = 3\).

# w select the parents with the best fitness (lowest values)
selected_parents <- population[order(fitness)][1:2]
selected_parents

## [1] 1 3

Crossover: Combines the genes (values) of two parents to create offspring.

Mutation: Introduces random changes to avoid local minima.

# perform crossover (simple example: no change in this step)
offspring <- selected_parents

# introduce mutation with a small probability (no mutation for this example)
mutated_offspring <- offspring
mutated_offspring

## [1] 1 3

We replace one of the individuals in the population with the offspring to form a new population.

# replace the least fit individual in the population with one of the offspring
new_population <- c(selected_parents, population[which.min(fitness)])
new_population

## [1] 1 3 1

Theoretical optimal point:

The minimum of the quadratic function occurs at: \[ x = \frac{-b}{2a} = 2 \] The corresponding function value is \(f(2) = 0\).

# calculate the optimal point
optimal_x <- -(-4) / (2 * 1)
optimal_y <- f(optimal_x)
c(optimal_x, optimal_y)

## [1] 2 0

Key Takeaways:

• Genetic algorithms can be applied to solve optimization problems.
• Through selection, crossover, and mutation, a population evolves to find the optimal solution.
• This approach is effective for problems where traditional optimization methods might struggle.

Further Applications:

• Neural network optimization.
• Solving NP-hard problems like the traveling salesman problem.
• Automated design and engineering.

Any questions?