# Differential Evolution

Differential Evolution (DE) is a robust, simple, and efficient global optimization algorithm that has been widely used in many areas of scientific research, engineering, and computational statistics. It is a population-based, stochastic search algorithm that uses concepts of mutation, crossover, and selection to evolve a population of candidate solutions towards an optimal solution.

## Overview

Differential Evolution was first introduced by R. Storn and K. Price in 1997. It is particularly well-suited for optimization problems that are non-differentiable, non-linear, and multimodal. DE operates through simple arithmetic operations and follows a simple scheme: it initializes a population of candidate solutions and then improves these solutions through iterative processes of mutation, crossover, and selection.

## How Differential Evolution Works

1. Initialization: DE starts with a randomly generated population of candidate solutions. Each individual in the population is a potential solution to the optimization problem.

2. Mutation: In the mutation phase, DE creates a donor vector by adding the weighted difference between two population vectors to a third vector. This process introduces variability into the population and helps in exploring the search space.

3. Crossover: The crossover phase involves mixing the donor vector and a target vector to produce a trial vector. This is done by either taking components from the donor vector or the target vector, promoting diversity in the population.

4. Selection: In the selection phase, the trial vector competes with the target vector. If the trial vector yields a better fitness value (i.e., it is a better solution to the problem), it replaces the target vector in the next generation. Otherwise, the target vector survives.

## Applications of Differential Evolution

Differential Evolution has been successfully applied in various fields such as machine learning, image processing, neural networks, and many others. It is particularly effective in solving complex optimization problems where the objective function is non-differentiable, noisy, discontinuous, or multimodal.