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Evolution strategies with cumulative step length adaptation on the noisy parabolic ridge. Technical Report CS-200602, Faculty of Computer Science, Dalhousie University, 2006. shtml. [2] D. V. Arnold and A. MacLeod. Hierarchically organised evolution strategies on the parabolic ridge. In M. , editors, Proceedings of the 2006 Genetic and Evolutionary Computation Conference. ACM Press, New York, 2006. -G. Beyer. On the performance of (1, λ)-evolution strategies for the ridge function class. IEEE Transactions on Evolutionary Computation, 5(3):218–235, 2001.

Hooke and T. A. Jeeves. Direct search solution of numerical and statistical problems. Journal of the ACM, Vol. 8, pp. 212-229, 1961. 8. J. Jagerskupper and C. Witt. Runtime analysis of a (mu+1)es for the sphere function. Technical report, 2005. 9. J. Nelder and R. Mead. A simplex method for function minimization. Computer Journal 7, pages 308–311, 1965. 10. G. Rudolph. Convergence rates of evolutionary algorithms for a class of convex objective functions. Control and Cybernetics, 26(3):375–390, 1997.

Averaging on xn (which can be random if the algorithm is stochastic) leads to Ef it Exn 1{xn ∈S } ≤ 1/N ( ). Step 2: summing on all possible (ri )i∈[[1,n]] n Ef it sup Exn 1{xn ∈S rn } ≤ Ef it Exn 1{xn ∈S rn } ≤ Ef it Exn 1{xn ∈S rn } ≤ |Ki | N( ) i=1 (thanks to step 1). Note for short Kn = n ni=1 |Ki |. Then, P (d(xn , f it∗ ) ≤ ) ≥ n 1 − δ implies Kn /N ( ) ≥ 1 − δ. This implies that n ≥ log(1 − δ)/ log(Kn ) + log(N ( ))/ log(Kn ). Note that for many algorithms, Kn is constant, and therefore Kn is constant.

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