What is a differential evolution (DE) algorithm in C++ and how is it implemented?

Table of Contents

• Introduction
• Working Principles of Differential Evolution (DE)
  • Mutation
  • Crossover
  • Selection
  • Key Parameters
• Steps to Implement DE Algorithm in C++
  • 1. Initialization
  • 2. Mutation
  • 3. Crossover
  • 4. Fitness Function
  • 5. Selection
  • 6. Main DE Loop
  • 7. Conclusion

Introduction

The Differential Evolution (DE) algorithm is an evolutionary algorithm used for optimizing complex, multidimensional functions. It is particularly effective in global optimization problems, where traditional methods might struggle with local optima. DE uses a population-based approach, and its key mechanisms involve mutation, crossover, and selection, similar to genetic algorithms but with different strategies for generating new solutions.

In this guide, we will explore how the DE algorithm works and provide an example of its implementation in C++.

Working Principles of Differential Evolution (DE)

Mutation

In DE, new candidate solutions are created by combining existing solutions (vectors) in the population. The mutation step generates a "mutant vector" by adding the weighted difference between two random population vectors to a third vector.

Crossover

After mutation, a crossover step is applied to generate trial vectors by mixing the mutant vector with the original target vector. This helps in exploring different regions of the search space.

Selection

Finally, in the selection step, the trial vector is compared to the original target vector. If the trial vector has a better fitness value (based on the objective function), it replaces the target vector in the next generation.

Key Parameters

• Population Size (NP): The number of candidate solutions in the population.
• Mutation Factor (F): Controls the weight of the difference between vectors during mutation.
• Crossover Probability (CR): The probability of swapping elements during crossover.

Steps to Implement DE Algorithm in C++

1. Initialization

Initialize a population of candidate solutions randomly within the given search space. Each solution is represented as a vector of real numbers.

2. Mutation

For each individual in the population, select three distinct individuals at random and generate a mutant vector.

3. Crossover

Mix the mutant vector and the target vector to create a trial vector.

4. Fitness Function

Define a fitness function to evaluate the solutions. For example, you could use the Sphere function, which is commonly used in optimization problems.

5. Selection

Compare the fitness of the trial vector to the target vector and select the one with the better fitness for the next generation.

6. Main DE Loop

Run the DE algorithm by iterating through generations. Apply mutation, crossover, and selection to evolve the population.

7. Conclusion

The DE algorithm is an efficient, population-based optimization technique that combines mutation, crossover, and selection to iteratively improve candidate solutions. It can handle complex, nonlinear, and multimodal optimization problems. In C++, DE can be implemented by carefully managing the population of solutions, ensuring that fitness evaluations, mutation, and crossover are correctly applied to evolve the solutions over generations.