A GRATIS theorem for relational optimization

We are studying the NFL Theorems with regard to relational optimization. In relational optimization, we represent the optimization problem by a formal relation, and the solution by the set of maximal (or non-dominated) elements of this relation. This appears to be a natural extension of standard optimization, and covers other notions of optimality as well. It will be shown that in this case, the NFL Theorems do not hold and that there are pairs of algorithms and a performance measure, where one always outperforms the other with regard to this performance measure if averaged over all possible relations. More specifically, the outperforming algorithm is an algorithm, where elements are checked one by one if they are dominated by any other element, while the outperformed algorithm is an algorithm, where elements are checked one by one if they dominate any other element. The proof is accompanied by a complete analysis of a special case, where also other performance measures are shown to be better when using the former algorithm.