On Steady-State Evolutionary Algorithms and Selective Pressure: Why Inverse Rank-Based Allocation of Reproductive Trials is Best
This work addresses optimization efficiency in evolutionary algorithms, providing theoretical and practical insights for researchers in computational optimization, though it is incremental as it builds on existing selective pressure concepts.
The paper tackles the problem of selective pressure in steady-state evolutionary algorithms (EAs) for global optimization, proving that uniform parent selection leads to exponential runtimes on benchmark functions like TwoMax, while selecting the worst individual or using inverse tournament selection enables efficient optimization with high probability, as shown through rigorous proofs and experimental evidence on functions like TruncatedTwoMax and standard benchmarks.
We analyse the impact of the selective pressure for the global optimisation capabilities of steady-state EAs. For the standard bimodal benchmark function \twomax we rigorously prove that using uniform parent selection leads to exponential runtimes with high probability to locate both optima for the standard ($μ$+1)~EA and ($μ$+1)~RLS with any polynomial population sizes. On the other hand, we prove that selecting the worst individual as parent leads to efficient global optimisation with overwhelming probability for reasonable population sizes. Since always selecting the worst individual may have detrimental effects for escaping from local optima, we consider the performance of stochastic parent selection operators with low selective pressure for a function class called \textsc{TruncatedTwoMax} where one slope is shorter than the other. An experimental analysis shows that the EAs equipped with inverse tournament selection, where the loser is selected for reproduction and small tournament sizes, globally optimise \textsc{TwoMax} efficiently and effectively escape from local optima of \textsc{TruncatedTwoMax} with high probability. Thus they identify both optima efficiently while uniform (or stronger) selection fails in theory and in practice. We then show the power of inverse selection on function classes from the literature where populations are essential by providing rigorous proofs or experimental evidence that it outperforms uniform selection equipped with or without a restart strategy. We conclude the paper by confirming our theoretical insights with an empirical analysis of the different selective pressures on standard benchmarks of the classical MaxSat and Multidimensional Knapsack Problems.