Lecture 1 - Genetic Algorithms (GAs) for function optimization"

In genetic.algo.optimizeR: Genetic Algorithm Optimization

knitr::opts_chunk$set(echo = TRUE)

Let's get started

• What are the genetic algorithms (GAs)?

Introduction to GAs

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.

Problem to solve

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

Function: $$ f(x) = x^2 - 4x + 4 $$

Initialize population

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

Evaluate fitness

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

Visualizing the fitness function

Let's plot the quadratic function $f(x)$ to visualize the optimization landscape.

library(ggplot2)

x <- seq(from = 0, to = 4, length.out = 100)
y <- f(x)

ggplot(data.frame(x, y), aes(x, y)) +
  geom_line(color = "blue") +
  labs(title = "Quadratic function: f(x)", x = "x", y = "f(x)") +
  theme_minimal()

Selection

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

Crossover and mutation

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

Replacement

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

Iteration and convergence

The process is repeated over multiple generations. The population evolves over time, approaching the optimal solution, to eventually reach it.

Finding the optimal solution

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)

Conclusion

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.

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Thank You!

Any questions?

genetic.algo.optimizeR documentation built on Oct. 10, 2024, 5:07 p.m.