The $(1 + (\lambda, \lambda))$ GA Is Even Faster on Multimodal Problems

The $(1 + (\lambda,\lambda))$ genetic algorithm is a recent invention of the theory community. Rigorous runtime analyses on unimodal fitness functions showed that it can indeed be faster than classical evolutionary algorithms, though on these simple problems the gains were only moderate. In this work, we conduct the first runtime analysis of this algorithm on a multimodal problem class, the jump functions benchmark. We show that with the right parameters, the $(1 + (\lambda,\lambda))$ GA optimizes any jump function with jump size $2 \le k \le n/16$ in expected time $O(n^{(k+1)/2} e^{O(k)} k^{-k/2})$, which significantly and already for constant $k$ outperforms standard mutation-based algorithms with their $\Theta(n^k)$ runtime and standard crossover-based algorithms with their $O(n^{k-1})$ runtime. Our work suggests some general advice on how to set the parameters of the $(1 + (\lambda,\lambda))$, which might ease the further use of this algorithm.