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> and every solution not in the set is outperformed by at least one solution in the Pareto front in every objective Is that trying to say: "for every solution

> and every solution not in the set is outperformed by at least one solution in the Pareto front in every objective Is that trying to say: "for every solution not in the set, there exists at least one objective such that at least one solution in the Pareto set beats that solution in that objective" i.e. every non-Pareto-front solution is beaten in some objective(s) by a Pareto-front solution, however it may be unbeaten in other objectives. Or is it: "for every objective in the system, every solution that is not in the set is beaten in that objective by one or more Pareto-set solutions." Or is it: "For every solution not in the set, there exists at least one Pareto solution which beats it in every objective."

speedstyle 2mo ago
The latter. Every solution not in the set is 'dominated' (outperformed in every objective) by some specific point in the set.
- kazinator 2mo ago
  The way it's written is misleading. The Pareto front solutions are all dominators of the non-solutions, but not dominators of each other. For A to dominate B, A has to be at least as good (i.e. no worse) than B in every objective under consideration and A has to be strictly better than B in at least one objective. Every solution in the Pareto front set dominates every solution not in that set: is at least as good in all optimization parameters and strictly better in at least one. Among the front set, there is no mutual dominance: if we pick any pair out of the set, one may be better than the other in one or more parameters, but worse in one or more. If it were not worse in one or more than the other, that other would not belong in the front set due to being dominated. Consider a space where we have two solutions. One is no worse than the other in every objective, and strictly better in one objective. Here, our Pareto front set contains that one solution and the other one is not in the set. Yet, the one not in the set is not beaten in every objective, just in that objective where the dominator is strictly better.
  - speedstyle 2mo ago
    Yes sorry, dominant means strictly better in some sense, but may be equal in many objectives. > Every solution in the Pareto front set dominates every solution not in that set Not quite – for each solution not in the set, there exists a solution in the set that dominates it. For instance a front with (0,2) (1,1) (2,0) would not have (1,0). While (0,2) doesn't dominate (1,0), something else does.
    - kazinator 2mo ago
      Yes; it is not relevant whether the dominating solution is in the set or not. It is a fact that the solutions in the front dominate all those not in the front, but the weakest statement we can make is that if a for a given solution B, we find another one A which is a dominant of B, then B is not in the front (regardless of whether or not A is). Anyway, a couple of hours ago, I fixed up the informal wording in the article, which had misleading interpretations.

speedstyle 2mo ago

The latter. Every solution not in the set is 'dominated' (outperformed in every objective) by some specific point in the set.

kazinator 2mo ago
The way it's written is misleading. The Pareto front solutions are all dominators of the non-solutions, but not dominators of each other. For A to dominate B, A has to be at least as good (i.e. no worse) than B in every objective under consideration and A has to be strictly better than B in at least one objective. Every solution in the Pareto front set dominates every solution not in that set: is at least as good in all optimization parameters and strictly better in at least one. Among the front set, there is no mutual dominance: if we pick any pair out of the set, one may be better than the other in one or more parameters, but worse in one or more. If it were not worse in one or more than the other, that other would not belong in the front set due to being dominated. Consider a space where we have two solutions. One is no worse than the other in every objective, and strictly better in one objective. Here, our Pareto front set contains that one solution and the other one is not in the set. Yet, the one not in the set is not beaten in every objective, just in that objective where the dominator is strictly better.
- speedstyle 2mo ago
  Yes sorry, dominant means strictly better in some sense, but may be equal in many objectives. > Every solution in the Pareto front set dominates every solution not in that set Not quite – for each solution not in the set, there exists a solution in the set that dominates it. For instance a front with (0,2) (1,1) (2,0) would not have (1,0). While (0,2) doesn't dominate (1,0), something else does.
  - kazinator 2mo ago
    Yes; it is not relevant whether the dominating solution is in the set or not. It is a fact that the solutions in the front dominate all those not in the front, but the weakest statement we can make is that if a for a given solution B, we find another one A which is a dominant of B, then B is not in the front (regardless of whether or not A is). Anyway, a couple of hours ago, I fixed up the informal wording in the article, which had misleading interpretations.