Constructing Low Star Discrepancy Point Sets with Genetic Algorithms

The recently active research area of black-box complexity revealed that for many optimization problems the best possible black-box optimization algorithm is significantly faster than all known evolutionary approaches. While it is not to be expected that a general-purpose heuristic competes with a problem-tailored algorithm, it still makes sense to look for the reasons for this discrepancy. In this work, we exhibit one possible reason---most optimal black-box algorithms profit also from solutions that are inferior to the previous-best one, whereas evolutionary approaches guided by the ``survival of the fittest'' paradigm often ignore such solutions. Trying to overcome this shortcoming, we design a simple genetic algorithm that first creates λ offspring from a single parent by mutation with a mutation probability that is k times larger than the usual one. From the best of these offspring (which often is worse than the parent) and the parent itself, we generate further offspring via a uniform crossover operator that takes bits from the winner offspring with probability 1/k only. A rigorous runtime analysis proves that our new algorithm for suitable parameter choices on the \onemax test function class is asymptotically faster (in terms of the number of fitness evaluations) than what has been shown for (μ \stackrel+}{, λ) EAs. This is the first time that crossover is shown to give an advantage for the \onemax class that is larger than a constant factor. Using a fitness-dependent choice of k and~λ, the optimization time can be reduced further to linear in~n. Our experimental analysis on several test function classes shows advantages already for small problem sizes and broad parameter ranges. Also, a simple self-adaptive choice of these parameters gives surprisingly good results