Explain the Differential Evolution method

Question

Can someone please explain the Differential Evolution method? The Wikipedia definition is extremely technical.

A dumbed-down explanation followed by a simple example wou

Answer 1

Answering my own question...

Overview

• The principal difference between Genetic Algorithms and Differential Evolution (DE) is that Genetic Algorithms rely on crossover while evolutionary strategies use mutation as the primary search mechanism.
• DE generates new candidates by adding a weighted difference between two population members to a third member (more on this below).
• If the resulting candidate is superior to the candidate with which it was compared, it replaces it; otherwise, the original candidate remains unchanged.

Definitions

• The population is made up of NP candidates.
• Xi = A parent candidate at index i (indexes range from 0 to NP-1) from the current generation. Also known as the target vector.
• Each candidate contains D parameters.
• Xi(j) = The jth parameter in candidate Xi.
• Xa, Xb, Xc = three random parent candidates.
• Difference vector = (Xb - Xa)
• F = A weight that determines the rate of the population's evolution.
  • Ideal values: [0.5, 1.0]
• CR = The probability of crossover taking place.
  • Range: [0, 1]
• Xc` = A mutant vector obtained through the differential mutation operation. Also known as the donor vector.
• Xt = The child of Xi and Xc`. Also known as the trial vector.

Algorithm

1. For each candidate in the population
   • for (int i = 0; i
2. Choose three distinct parents at random (they must differ from each other and i)

do
{
  a = random.nextInt(NP);
} while (a == i)
do
{
  b = random.nextInt(NP);
} while (b == i || b == a);
do
{
  c = random.nextInt(NP);
} while (c == i || c == b || c == a);

3. (Mutation step) Add a weighted difference vector between two population members to a third member
   • Xc` = Xc + F * (Xb - Xa)
4. (Crossover step) For every variable in Xi, apply uniform crossover with probability CR to inherit from Xc`; otherwise, inherit from Xi. At least one variable must be inherited from Xc`

int R = random.nextInt(D);
for (int j=0; j < D; ++j)
{
  double probability = random.nextDouble();
  if (probability < CR || j == R)
    Xt[j] = Xc`[j]
  else
    Xt[j] = Xi[j]
}

5. (Selection step) If Xt is superior to Xi then Xt replaces Xi in the next generation. Otherwise, Xi is kept unmodified.

Resources

• See this for an overview of the terminology
• See Optimization Using Differential Evolution by Vasan Arunachalam for an explanation of the Differential Evolution algorithm
• See Evolution: A Survey of the State-of-the-Art by Swagatam Das and Ponnuthurai Nagaratnam Suganthan for different variants of the Differential Evolution algorithm
• See Differential Evolution Optimization from Scratch with Python for a detailed description of an implementation of a DE algorithm in python.