Some techniques for analyzing time complexity of evolutionary algorithms

In this paper we introduce some techniques for the analysis of time complexity of evolutionary algorithms (EAs) based on a finite search space. The Markov property and the decomposition of a matrix are employed for the exact analytic expressions of the mean first hitting times that EAs reach the optimal solutions (FHT-OS). Dynkin’s formula, one-step increment analysis and some other probability methods are adopted for the lower and upper bounds of the mean FHT-OS. The theory of a non-negative matrix, including the Perron–Frobenius theorem, the Jordan standard form, etc., are applied for the convergence of EAs. In addition, some techniques for time complexity and convergence of general adaptive EAs are also involved in this paper. Finally, the theoretical results obtained in this paper are verified effectively by applying them to the analysis of a typical evolutionary algorithm, (1 + 1)-EA.

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