Petr Pošík

Hi, Eduard,

1] Regarding the "Pareto optimal" and "non-dominated" distinction... I think I understand you. But there are two situations that should be distinguished:

a] We have very large (possibly infinite) space of candidate solutions, X, and their fitness values, F. This set X cantains the set of Pareto-optimal solutions, X*, X* \in X, which may be finite, but very often is infinite. The set X* is Pareto-optimal and of course it is also a set of non-dominated solutions.

b] On the other hand, in multiobjective optimization, you usually do not get the (whole theorethically possible) Pareto-optimal solutions; what you get is a finite set Xnd of solutions which are not dominated by any of solutions you came across during the optimization. The solutions in Xs are only approximation of the whole set X* of Pareto-optimal solutions. Sets X* and Xnd might be (and often are) different and IMHO this distinction is important.

If you consider the matrix of scores that is given to your function as the whole universe of possible solution evaluations, then you are right - the Pareto-optimal and non-dominated solutions are the same.

It is also true, that for many practical applications this distinction is quite subtle and not very important. I prefer to call the output of your function the non-dominated set.

2] Regarding the fact if two solutions which have the same fitness value can be part of the non-dominated (Pareto-optimal) set: on the page you pointed to, there is the following definition:

"A point x* is said to be (glob ally) Pareto optimal or a (globally) efficient solution or a non-dominated or a non-inferior point for (MOP) if and only if there is no x such that fi(x)<=fi(x*) for all i, with at least one strict inequality."

This definition is correct, but your interpretation is wrong. You say "The definition of Pareto points demands at least one strict inequality". But I would rather say "The definition of Pareto points demands NONEXISTENCE of other points with at least one strict inequality." Example:

Let's have a set of 2 points x1 and x2, and both points have the same evaluation, i.e. fi(x1)=fi(x2). Then the points do not dominate each other, right? Since, all their fitness scores are equal, there is no score in which the inequality is strict, right?

Consider the question if x1 is Pareto-optimal: The definition says that x1 is Pareto-optimal if there is no other point that would dominate x1. SInce there is only one other point x2 and since x2 does not dominate x1, then x1 is non-dominated and Pareto-optimal.

The similar holds for x2. So that both points are Pareto-optimal.

I hope we now understand each other.

Cheers, Petr.

I'll admit, this is a simple thing. Nothing dramatic here, with only a couple of lines of simple code. But it might make your own code just a wee bit simpler to write and to read later on. And simple, modular code is always good. If holds on plots are something you do often, then you may like this utility.

I like the help. An H1 line, etc. The 5 rating is a tip of my hat to the author.