Theoretical analysis of evolutionary computation on continuously differentiable functions

Authors:
Youhei Akimoto;Yuichi Nagata;Isao Ono;Shigenobu Kobayashi
Affiliations:
Tokyo Institute of Technology, Yokohama-shi, Japan;Tokyo Institute of Technology, Yokohama-shi, Japan;Tokyo Institute of Technology, Yokohama-shi, Japan;Tokyo Institute of Technology, Yokohama-shi, Japan
Venue:
Proceedings of the 12th annual conference on Genetic and evolutionary computation
Year:
2010

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Abstract

This paper investigates theoretically the convergence properties of the stochastic algorithms of a class including both CMAESs and EDAs on constrained minimization of continuously differentiable functions. We are interested in algorithms that do not get stuck on a slope of the function, but converge only to local optimal points. Convergence to a point that is neither a stationary point of the function nor a boundary point is evidence that the convergence properties are not well behaved. We investigate what properties are necessary/sufficient for the algorithm to avoid this type of behavior, i.e., what properties are necessary for the algorithm to converge only to local optimal points of the function. We also investigate the analogous conditions on the parameters of two variants of modern EC-based stochastic algorithms, namely, a CMAES employing rank-μ update and an EDA known as EMNAglobal. The comparison between the apparently similar two systems shows that they have significantly different theoretical behaviors. This result presents us with an insight into the way we design well-behaved optimization algorithms.