Theoretical evidence suggests that non-elitist evolutionary algorithms (EAs) with non-linear selection mechanisms can efficiently overcome broad classes of local optima where elitist EAs fail. However, the analysis assumes a weak selective pressure and mutation rates carefully chosen close to the "error threshold", above which they cease to be efficient. On problems easier for hill-climbing, the populations may slow down these algorithms, leading to worse runtime compared with variants of the elitist (1+1) EA.

Most discrete evolutionary algorithms (EAs) implement elitism, meaning that they make the biologically implausible assumption that the fittest individuals never die. While elitism favours exploitation and ensures that the best seen solutions are not lost, it has been widely conjectured that non-elitism is necessary to explore promising fitness valleys without getting stuck in local optima. Determining when non-elitist EAs outperform elitist EAs has been one of the most fundamental open problems in evolutionary computation. A recent analysis of a non-elitist EA shows that this algorithm does not outperform its elitist counterparts on the benchmark problem JUMP. We solve this open problem through rigorous runtime analysis of elitist and non-elitist population-based EAs on a class of multi-modal problems. We show that with 3-tournament selection and appropriate mutation rates, the non-elitist EA optimises the multi-modal problem in expected polynomial time, while an elitist EA requires exponential time with overwhelmingly high probability. A key insight in our analysis is the non-linear selection profile of the tournament selection mechanism which, with appropriate mutation rates, allows a small sub-population to reside on the local optimum while the rest of the population explores the fitness valley. In contrast, we show that the comma-selection mechanism which does not have this non-linear profile, fails to optimise this problem in polynomial time. The theoretical analysis is complemented with an empirical investigation on instances of the set cover problem, showing that non-elitist EAs can perform better than the elitist ones. We also provide examples where usage of mutation rates close to the error thresholds is beneficial when employing non-elitist population-based EAs.

It 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.

In this article, we consider a fitness-level model of a non-elitist mutation-only evolutionary algorithm (EA) with tournament selection. The model provides upper and lower bounds for the expected proportion of the individuals with fitness above given thresholds. In the case of so-called monotone mutation, the obtained bounds imply that increasing the tournament size improves the EA performance. As corollaries, we obtain an exponentially vanishing tail bound for the Randomized Local Search on unimodal functions and polynomial upper bounds on the runtime of EAs on the 2-SAT problem and on a family of Set Cover problems proposed by E. Balas.

Understanding how the time complexity of evolutionary algorithms (EAs) depend on their parameter settings and characteristics of fitness landscapes is a fundamental problem in evolutionary computation. Most rigorous results were derived using a handful of key analytic techniques, including drift analysis. However, since few of these techniques apply effortlessly to population-based EAs, most time complexity results concern simple EAs, such as the (1+1) EA. We present the level-based theorem, a new technique tailored to population-based processes. It applies to any nonelitist process where offspring are sampled independently from a distribution depending only on the current population. Given conditions on this distribution, our technique provides upper bounds on the expected time until the process reaches a target state. The technique is demonstrated on pseudo-Boolean functions, the sorting problem, and approximation of optimal solutions in combinatorial optimization. The conditions of the theorem are often straightforward to verify, even for genetic algorithms and estimation of distribution algorithms which were considered highly nontrivial to analyze. The proofs for the example applications are available in the supplementary materials. Finally, we prove that the theorem is nearly optimal for the processes considered. Given the information the theorem requires about the process, a much tighter bound cannot be proved.

Recently, it has been proven that evolutionary algorithms produce good results for a wide range of combinatorial optimization problems. Some of the considered problems are tackled by evolutionary algorithms that use a representation, which enables them to construct solutions in a dynamic programming fashion. We take a general approach and relate the construction of such algorithms to the development of algorithms using dynamic programming techniques. Thereby, we give general guidelines on how to develop evolutionary algorithms that have the additional ability of carrying out dynamic programming steps.

We consider the optimization problem of finding the best possible offspring as a result of a recombination operator in an evolutionary algorithm, given two parent solutions. The optimal recombination is studied in the case where a vector of binary variables is used as a solution encoding. By means of efficient reductions of the optimal recombination problems (ORPs) we show the polynomial solvability of the ORPs for the maximum weight set packing problem, the minimum weight set partition problem, and for linear Boolean programming problems with at most two variables per inequality, and some other problems. We also identify several NP-hard cases of optimal recombination: the Boolean linear programming problems with three variables per inequality, the knapsack, the set covering, the p-median, and some other problems.