Bounded Pareto Archiving: Theory and Practice
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We consider algorithms for the sequential storage, or ‘archiving’, of solutions discovered during a Pareto optimization procedure. We focus particularly on the case when an a priori hard bound, N,is placed on the capacity of the archive of solutions. We show that for general sequences of points, no algorithm that respects the bound can maintain an ideal minimumnumber of points in the archive. One consequence of this, for example, is that a strictly bounded archive cannot be expected in general to contain the Pareto front of the points generated so far (where this set is smaller than the archive bound). Using the notion of an ideal ε-approximation set—the subset (of size ≤ N)of a whole sequence of points which minimizes ε—we also show that no archiving algorithm can attain this ideal for general sequences. This means that in general no archiving algorithm can be expected to maintain an ‘optimal’ representation of the Pareto front when the size of that set is larger than the archive bound. Furthermore, if the ranges of the PF of the sequence are not known a priori, no algorithm that certifies (using its own internal epsilon parameter ε arc ) that it maintains an epsilon-approximate set of the sequence, can maintain ε arc )within any fixed multiple of the minimal (ideal) ε value. In a case study we go on to demonstrate several scenarios where e-based archiving algorithms proposed by Laumanns et al. — which perform well when the archive’s capacity is nota priori bounded—perform poorly when ε is adapted ‘on the fly’ in order to respect a capacity constraint. For each scenario we demonstrate that the performance of an adaptive grid archiving (AGA) algorithm (which does notassure a formally guaranteed approximation level) performs comparatively well, in practice.
KeywordsPareto Front Input Sequence Pareto Optimal Solution Objective Space Pareto Optimal Point
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