Generalized jump functions

Jump functions are the most studied non-unimodal benchmark in the theory of evolutionary algorithms (EAs). They have significantly improved our understanding of how EAs escape from local optima. However, their particular structure - to leave the local optimum the EA can only jump directly to the global optimum - raises the question of how representative the recent findings are. For this reason, we propose an extended class Jumpk,δ of jump functions that incorporate a valley of low fitness of width δ starting at distance k from the global optimum. We prove that several previous results extend to this more general class: for all k = o (n1/3) and δ ≤ k, the optimal mutation rate for the (1 + 1) EA is [EQUATION], and the fast (1 + 1) EA runs faster than the classical (1 + 1) EA by a factor super-exponential in δ. However, we also observe that some known results do not generalize: the randomized local search algorithm with stagnation detection, which is faster than the fast (1 + 1) EA by a factor polynomial in k on Jumpk, is slower by a factor polynomial in n on some Jumpk,δ instances. Computationally, the new class allows experiments with wider fitness valleys, especially when they lie further away from the global optimum.