Abstract
Theoretical analysis of the dynamics of evolutionary algorithms is believed to be very important to understand the search behavior of evolutionary algorithms and to develop more efficient algorithms. In this chapter, we illustrate the dynamics of a canonical Differential Evolution (DE) algorithm with DE/rand/1 type mutation and binomial crossover. The chapter proposes a simple mathematical model of the underlying evolutionary dynamics of a one-dimensional DE-population. The model shows that the fundamental dynamics of each agent (parameter vector) in DE employs the gradient-descent type search strategy, with a learning rate parameter that depends on control parameters like scale factor F and crossover rate CR of DE. The stability and convergence-behavior of the proposed dynamics is analyzed in the light of Lyapunov’s stability theorems. The mathematical model developed in this Chapter, provides important insights into the search mechanism of DE in a near neighborhood of an isolated optimum. Empirical studies over simple objective functions are conducted in order to validate the theoretical analysis.
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