The progress rate of the (1 +λ)-ES (Evolution Strategy) is analyzed on the parabolic ridge test function. A differentprogress behavior is observed for the (1, λ)-ES than for the sphere model test function. The characteristics of the progress rate picture for the plus strategy differs little from the one obtained for the sphere model, but this strategy has drastically worse progress rate values than those obtained for the comma strategy. The dynamics of the distance to the progress axis is also investigated. A theoretical formula is derived to estimate the change in this distance over generations. This formula is used to derive the expected value of the problem-specific distance to the ridge axis. The correctness of the formulae is supported by simulation results.

The progress behavior of evolution strategies (ES) using recombination is analyzed in this paper on the parabolic ridge. This test function represents landscapes far from the optimum. The ES algorithms with intermediate and dominant recombination are considered in the analysis. The derivations are presented for intermediate recombination. Thereafter, the formulae for dominant recombination are obtained using the so-called surrogate mutation model. In the analysis, the formulae are derived for the progress rate φ and for the stationary distance R (∞) to the ridge axis. As a result, it will be shown that the progress rate φ can increase if recombination is applied. Simulations are used to show the appropriateness of the formulae derived.