The majority of empirical comparisons of multi-objective evolutionary algorithms (MOEAs) are performed on synthetic benchmark functions. One of the advantages of synthetic test functions is the a-priori knowledge of the optimal Pareto front. This allows measuring the proximity to the optimal front for the solution sets returned by the different MOEAs. Such a comparison is only meaningful if the cardinality of all solution sets is bounded by some fixed k. In order to compare MOEAs to the theoretical optimum achievable with k solutions, we determine best possible epsilon-indicator values achievable with solution sets of size k, up to an error of delta. We present a new algorithm with runtime O (k . log(2) (delta(-1))), which is an exponential improvement regarding the dependence on the error delta compared to all previous work. We show mathematical correctness of our algorithm and determine optimal solution sets for sets of cardinality k 2 {2, 3, 4, 5, 10, 20, 50, 100, 1000} for the well known test suits DTLZ, ZDT, WFG and LZ09 up to error delta = 10(-25).

Most biobjective evolutionary algorithms maintain a population of fixed size μ and return the final population at termination. During the optimization process many solutions are considered, but most are discarded. We present two generic postprocessing algorithms which utilize the archive of all non-dominated solutions evaluated during the search. We choose the best μ solutions from the archive such that the hypervolume or ε-indicator is maximized. This postprocessing costs no additional fitness function evaluations and has negligible runtime compared to most EMOAs. We experimentally examine our postprocessing for four standard algorithms (NSGA-II, SPEA2, SMS-EMOA, IBEA) on ten standard test functions (DTLZ 1–2,7, ZDT 1–3, WFG 3–6) and measure the average quality improvement. The median decrease of the distance to the optimal ε-indicator is 95

The goal of bi-objective optimization is to find a small set of good compromise solutions. A common problem for bi-objective evolutionary algorithms is the following subset selection problem (SSP): Given n solutions P ⊂ R2 in the objective space, select k solutions P* from P that optimize an indicator function. In the hypervolume SSP we want to select k points P* that maximize the hypervolume indicator IHYP(P*, r) for some reference point r ∈ R2. Similarly, the ε-indicator SSP aims at selecting k~points P* that minimize the ε-indicator Iε(P*,R) for some reference set R ⊂ R2 of size m (which can be R=P). We first present a new algorithm for the hypervolume SSP with runtime O(n (k + log n)). Our second main result is a new algorithm for the ε-indicator SSP with runtime O(n log n + m log m). Both results improve the current state of the art runtimes by a factor of (nearly) $n$ and make the problems tractable for new applications. Preliminary experiments confirm that the theoretical results translate into substantial empirical runtime improvements.