Theory of (1 + 1) ES on the RIDGE

Previous research proposed the uniform mutation inside the sphere as a new mutation operator for evolution strategies (continuous evolutionary algorithms), with a case study of the elitist algorithm on the SPHERE. For that landscape, one-step success probability and expected progress were estimated analytically, and further proved to converge, as space dimension increases, to the corresponding asymptotics of the algorithm with normal mutation. This article takes the analysis further by considering the RIDGE, an asymmetric landscape almost uncovered in the literature. For the elitist algorithm, estimates of expected progress along the radial and longitudinal axes are derived, then tested numerically against the real behavior of the algorithm on several functions from this class. The global behavior of the algorithm is predicted correctly by iterating the one-step analytical formulas. Moreover, experiments show identical mean value dynamics for the algorithms with uniform and normal mutation, which implies that the derived formulas apply also to the normal case. Essential to the whole analysis is
$\theta $
, the inclination angle of the RIDGE. The behavior of the algorithm on the SPHERE and HYPERPLANE is also obtained, at the limits of the
$\theta $
interval (0°, 90°].