Measure-theoretic analysis of performance in evolutionary algorithms

The performance of evolutionary algorithms has been studied extensively, but it has been difficult to answer many basic theoretical questions using the existing theoretical frameworks and approaches. In this paper, the performance of evolutionary algorithms is studied from a measure-theoretic point of view, and a framework is offered that can address some difficult theoretical questions in an abstract and general setting. It is proven that the performance of continuous optimizers is in general non linear and continuous for finitely determined performance criteria. Since most common optimizers are continuous, it follows that in general there is substantial reason to expect that mixtures of optimization algorithms can outperform pure algorithms on many if not most problems. The methodology demonstrated in this paper rigorously connects performance analysis of evolutionary algorithms and other optimization methods to functional analysis, which is expected to enable new and important theoretical results by leveraging prior work in these fields.