title: On convergence of differential evolution over a class of continuous functions with unique global optimum.
abstract: Differential evolution (DE) is arguably one of the most powerful stochastic real-parameter optimization algorithms of current interest. Since its inception in the mid 1990s, DE has been finding many successful applications in real-world optimization problems from diverse domains of science and engineering. This paper takes a first significant step toward the convergence analysis of a canonical DE (DE/rand/1/bin) algorithm. It first deduces a time-recursive relationship for the probability density function (PDF) of the trial solutions, taking into consideration the DE-type mutation, crossover, and selection mechanisms. Then, by applying the concepts of Lyapunov stability theorems, it shows that as time approaches infinity, the PDF of the trial solutions concentrates narrowly around the global optimum of the objective function, assuming the shape of a Dirac delta distribution. Asymptotic convergence behavior of the population PDF is established by constructing a Lyapunov functional based on the PDF and showing that it monotonically decreases with time. The analysis is applicable to a class of continuous and real-valued objective functions that possesses a unique global optimum (but may have multiple local optima). Theoretical results have been substantiated with relevant computer simulations.
authors.affiliation: Indian Institute of Science Bangalore, Bangalore 560 012, India. sayan88tito@gmail.com
authors.forename: Sayan
authors.initials: S
authors.lastname: Ghosh
authors.forename: Swagatam
authors.initials: S
authors.lastname: Das
authors.forename: Athanasios V
authors.initials: AV
authors.lastname: Vasilakos
authors.forename: Kaushik
authors.initials: K
authors.lastname: Suresh
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D015233:Models, Statistical; D013269:Stochastic Processes