Some Multiobjective Optimizers are Better than Others

The No-Free-Lunch (NFL) theorems hold for general multiobjective fitness spaces, in the sense that, over a space of problems which is closed under permutation, any two algorithms will produce the same set of multiobjective samples. However, there are salient ways in which NFL does not generally hold in multiobjective optimization. Previously we have shown that a ‘free lunch’ can arise when comparative metrics (rather than absolute metrics) are used for performance measurement. Here we show that NFL does not generally apply in multiobjective optimization when absolute performance metrics are used. This is because multiobjective optimizers usually combine a generator with an archiver. The generator corresponds to the ‘algorithm’ in the NFL sense, but the archiver filters the sample generated by the algorithm in a way that undermines the NFL assumptions. Essentially, if two multiobjective approaches have different archivers, their average performance may differ. We prove this, and hence show that we can say, without qualification, that some multiobjective approaches are better than others.