Non-elitist evolutionary algorithms excel in fitness landscapes with sparse deceptive regions and dense valleys

ABSTRACTIt is largely unknown how the runtime of evolutionary algorithms depends on fitness landscape characteristics for broad classes of problems. Runtime guarantees for complex and multi-modal problems where EAs are typically applied are rarely available. We present a parameterised problem class SparseLocalOptα,ε where the class with parameters α, ϵ ∈ [0, 1] contains all fitness landscapes with deceptive regions of sparsity ε and fitness valleys of density α. We study how the runtime of EAs depends on these fitness landscape parameters. We find that for any constant density and sparsity α, ε ∈ (0, 1), SparseLocalOptα,ε has exponential elitist (μ + λ) black-box complexity, implying that a wide range of elitist EAs fail even for mildly deceptive and multi-modal landscapes. In contrast, we derive a set of sufficient conditions for non-elitist EAs to optimise any problem in SparseLocalOptα,ε in expected polynomial time for broad values of α and ε. These conditions can be satisfied for tournament selection and linear ranking selection, but not for (μ, λ)-selection.