Piecewise Linear Topology, Evolutionary Algorithms, and Optimization Problems

Schemata theory, Markov chains, and statistical mechanics have been used to explain how evolutionary algorithms (EAs) work. Incremental success has been achieved with all of these methods, but each has been stymied by limitations related to its less-thanglobal view. We show that moving the question into topological space helps in the understanding of why EAs work. Purpose. In this paper we use piecewise linear topology to attempt a global explanation of why EAs work. We show that when an EA solves a convex optimization problem, piecewise linear-mappings (PL-mappings) are the simplicial mappings of the original polytopes. We also demonstrate that, since a quotient space exists, the union of the simplicial vertices generates the convex hull of the convex optimization problem the EA is seeking to solve. The unique convex hull is found via tight triangulations, which are a form of combinatorial topology. Methods. Piecewise linear topology, basic optimization theory, Borel algebra, algebraic geometry, and tight triangulations are used to study why EAs work. Results. It is demonstrated that the use of piecewise linear topology in a combinatorial form helps explain why EAs work. Conclusions. Moving the “why EAs work” question into topological space is found to be helpful in understanding this basic EA question. The basic topological conditions are married to a Borel algebra in order to account for EA operators such as mutation and selection, and combinatorial topology demonstrates the EAs’ ability to find the unique convex hull.