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...Our proposal: Automatic configuration of anytime algorithms by means of the hypervolume measure As mentioned above, directly using the attainment function as the basis of an automatic configuration tool for anytime algorithms seems difficult. Instead, we identify the hypervolume measure as the best available choice for this task. The main reasons are its high discriminatory power, being Pareto-compliant, and the possibility of incorporating user preferences into the automatic configuration process. There is also substantial and ongoing research on the theoretical properties of the hypervolume [7]. 4.1. Hypervolume measure of performance profiles As a first step, let us define the classical Pareto-dominance relation on performance profiles: Definition 4 (Better in terms of Pareto-optimality, C). Given two performance profiles Pi and P j, which are the result of running two algorithms Ai and A j on the same problem instance π, we say that Pi is better, in terms of Pareto-optimality, than P j (Pi C P j) iff Pi , P j, and ∀(t j, q j) ∈ P j, ∃(ti, qi) ∈ Pi, such that (ti, qi) ≤ (t j, q j). 1For brevity, we do not explain the attainment function approach here and its application to the anal...

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...hin Theorem 6 of [2] does not depend on μ but on a lower bound on the reference point to ensure that both extremes are included in the optimal μ-distribution. Recently, a limit result has been proven =-=[3]-=- which shows that the lower bound of [2, Theorem 6] converges to the nadir point2 if μ goes to infinity but the result does not state how fast (in μ) the nadir point is approached. Clearly, choosing t...

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... For the optimum sets of size µ, the term optimal µ-distribution was introduced and such sets have already been characterized theoretically for the standard, as well as for the weighted, HV indicator =-=[1, 2]-=-. Before we investigate the concrete optimal µ-distributions on specific test functions also for the R2 indicator in Sec. 4, we first prove some general theoretical statements about the R2 indicator. ...

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...dicator values with respect to the best achievable value becomes possible. For the hypervolume indicator, several results on these optimal µ-distributions are known [63], [62], [75], [69], [70], [61] =-=[64]-=-, but the understanding of the optimization goal for other indicators is less developed. Recently, we started to investigate the optimal µ-distributions, both theoretically and numerically, for the so...

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...e outcome expected from these algorithms, the preferences and bias introduced by the choice of the indicator have to be understood. For the HV indicator, empirical studies [6] and theoretical results =-=[1, 2]-=- do already exist. With respect to the R2 indicator, we showed that the indicator tends to place the points more concentrated in the central region of the Pareto front than the HV and that the optimal...

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...ques. Differential Evolution is used as underlying evolutionary technique. Scalarization techniques behavior is observed. As basis for the comparison we use standard test problem: WFG, DTLZ, and ZDT [=-=Auger et al., 2011-=-]. All problems are real valued unconstrained and require the minimization of objective functions. The hypervolume and inverted generational distance are computed for the obtained solution sets. Numer...

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...ition based MOEAs [21], [20], [54], [8], [7], [30], [56], [58], [55], [1], [34], [32], [39], [42], [41], [26], [25], [35], [37], [33], [41]), and Indicator Based algorithms [63], [5], [3], [22], [4], =-=[2]-=-, [14]. They mainly emphasize three conflicting goals: firstly, the final approximate Pareto front (PF) should be as close as possible to the true PF; secondly, the final set of Pareto optimal solutio...

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...e outcome expected from these algorithms, the preferences and bias introduced by the choice of the indicator have to be understood. For the HV indicator, empirical studies [6] and theoretical results =-=[1, 2]-=- do already exist. With respect to the R2 indicator, we showed that the indicator tends to place the points more concentrated in the central region of the Pareto front than the HV and that the optimal...

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...ifficult problem for the Decision Maker. Palliative solutions for those problems are (i) considering more relaxed dominance relations to facilitate discrimination [3], (ii) modifying fitness measures =-=[10, 2]-=-, (iii) optimizing scalarized single-objective functions [33], and (iv) dimensionality reduction through identification of redundant objectives [13]. Still in the category of objective reduction, the ...