Is Algebraic Differential Evolution Really a Differential Evolution Scheme?

The Algebraic Differential Evolution (ADE) is a recently proposed combinatorial evolutionary scheme which mimics the behaviour of the classical Differential Evolution (DE) in discrete search spaces which can be represented as finitely generated groups. ADE has been successfully applied to both permutation and binary optimization problems. However, in the previous works, the relationship between ADE and the classical continuous DE has been only intuitively sketched without any theoretical or experimental proof. Here, we fill this gap by providing both theoretical and experimental justifications proving that ADE is a full-fledged generalization of DE which works across different search spaces. First, we formally prove that there exists a concrete implementation of ADE's algebraic operations converging to the classical vector operations of DE, then we propose a real-vector implementation of ADE and we experimentally prove that its behaviour is statistically equivalent to DE. As conclusion, we also pave the way for further applications of the original DE idea to mixed discrete/continuous search spaces.