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Abstract : The asymptotical behavior of a (1+1)-ES process on Rudolph's long k-paths is extensively investigated in this paper. First, in the case of k=l^alpha, we prove that the long $k$-path is a longpath for the (1+1)-ES, in the sense that the entire path has to be followed before convergence occurs. For alpha < 1/2, expected convergence time is still exponential but some shortcuts will occur meanwhile which speeds up the process.
Second, in the case of constant k, the statistical distribution of convergence time is calculated, and the influence of population size is investigated for different evolution strategies. Besides, the histogram of the first hitting time of the solution shows an anomalous peak close to zero, which corresponds to an exceptional set of events that speed up the expected convergence time with a factor of l^2. A direct consequence of this exceptional set is that performing independent (1+1)-ES processes proves to be more advantageous than any population based ES.
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