2.1 Multiobjective Optimization

For a vector valued function f:X⊂ℝn→ℝℓ, the multiobjective optimization problem (MOP), under consideration requires the simultaneous minimization of its ℓ component functions f1,…,fℓ. A solution is optimal when the elements of the image Y=f(X), are nondominated in the sense of Pareto [¹¹], which derives from a partial order in ℝℓ, whose definition we recall below for the convenience of the reader.

Let (Y,≼) be a subset Y⊂ℝℓ equipped with the partial order ≼ defined for y,y′∈Y by:

An element y∈Y is said to be dominated byy′∈Y and denoted y′≺y, if y′≼y and y≠y′. Moreover, y∈Y is dominated byY′⊂Y, written Y′≼y, if there exists some y′∈Y′ such that y′≼y, otherwise it is said to be nondominated by it, Y′≼y. A subset Y′⊂Y is dominated by a subset Y″⊂Y, and written Y″≼Y′, if for every y′∈Y′ there exist some y″∈Y″ such that y″≼y′. If this is not the case Y′ is said to be nondominated by Y″ and denoted Y′≼Y″.

If Y=f(X) is the objective space of some MOP with decision space X⊂ℝn and objective function f:X→ℝℓ, its Pareto front is defined as the set Y*:={y∈Y|∃ y′∈Y:y′≺y} of nondominated elements. An element x∈X is called Pareto-optimal if its image is nondominated, i.e., f(x)∈Y*, and the set X* of all Pareto-optimal points is called Pareto set.

Finally, if Y⊂ℝℓ, we say that a performance indicator given by a function ℐ:𝒫(Y)→ℝ is Pareto-compliant if for subsets A,B⊂Y the strict dominance condition A≼B and B≼A implies the relation ℐ(A)≤ℐ(B) (or in a stronger version ℐ(A)<ℐ(B)). We refer the reader to [¹⁸] for details. In section 3.2 we provide a brief discussion of the compliance of the indicator associated to Δp,q with Pareto-related optimality criteria.

2.2 Power Means

For a finite set X={xi}i=1N⊂[0,∞) and a non-zero real q, the q-average or the q-power mean of X is given by:

Remark. When the set of values taken by an indexing quantity has been explicitly specified, e.g. i∈I:={1,…,N}, for convenience, the following abbreviated notation will be used:

A comprehensive reference on the theory and properties of means is e.g. [⁵], where proofs of the statements presented in this section can be found.

It is well-known that limit cases of power means recover familiar quantities, for example:

lim⁡q→0ℳq(X)=(∏i=1Nxi)1N, is the standard geometric mean of X, additionally:

The special case q=−1 corresponds to the harmonic mean and it is also of our interest:

Now, we can define the q-average of a finite set for any q in the extended real line ℝ¯:=[−∞,∞]. Let Y:={yi}i∈I be a finite subset of [0,∞). The following properties hold for power means:

1. If xi≤yi for all indices i∈I, then for any q∈ℝ¯:

2. For p,q∈ℝ¯, if p≤q, then:

3. If A=(aij) denotes an array of indexed positive elements with i∈I and j∈J, then their p-average satisfies:

4. For p≥1 it follows from Minkowski inequality that:

5. The harmonic mean admits the bound:

2.3 Averaged Hausdorff Distance

Suppose that A and B belong to the family of finite subsets of ℝn, denoted by 𝒫0(ℝn), and that p≥1. Recall that the “modified” generational distance (GDp) and the inverted generational distance (IGDp) are defined by power means as follows (see [¹³]):

and:

where d(⋅,⋅) stands for the standard Euclidean metric. Let us denote by Δp:𝒫0(ℝn)×𝒫0(ℝn)→ℝ the so-called averaged Hausdorff distance, i.e.:

It has been established in [¹³] that Δp does not satisfy the triangle inequality but a weaker version given by:

where N≥1 is a constant with |A|,|B|,|C|≤N. Because of this, Δp is not a proper metric but a semimetric with relaxed triangle inequality. This notion has appeared with several conflicting names in the literature, among which inframetric is probably the one with less friction with pre-existing terminology.

There are two related conditions that relax the triangle inequality for a function d:X×X→[0,∞). Namely, the existence of a constant C>0 such that for any points a,b,c∈X one of the following properties hold:

1. The C-relaxed triangle inequality:

2. The C-inframetric inequality:

Since condition (2) implies (1), with the same constant C and, reciprocally, the C-relaxed triangle inequality implies the 2C-inframetric one, it is clear that both conditions are equivalent for appropriate choice of constants. Hence, a semimetric satisfying anyone of these conditions will be called an inframetric.

3.1 Metric Properties

Now we study the behavior of the distances GDp,q and Δp,q from a metric perspective. Since the elements of the distance matrix do not satisfy a relation of the kind DAC=DAB+DBC (and in general these matrices are of different sizes), their properties are not immediate from (3.3) and the triangle inequality for ℓp,q-norms. Here, we provide self-contained proofs for future reference and completeness.

Theorem 3.3. For parametersp,q∈[1,∞)the generational(p,q)-distanceGDp,qsatisfies the triangle inequality, namely:

for any finite setsA,B,C⊂ℝn.

Proof. Let assume that A={ai}i=1NA, B=(bj)j=1NB, and C=(ck)k=1NC. From the triangle inequality for the Euclidean metric d(⋅,⋅) on ℝn we know that for arbitrary values of the indices i∈I:={1,…,NA}, j∈J:={1,…,NB}, and k∈K:={1,…,NC} it holds that:

Let us abbreviate these quantities by δik:=d(ai,ck), δij:=d(ai,bj), and δjk:=d(bj,ck), where the indices i,j,k will be understood to take values in the sets I, J, and K defined above, respectively. Then, for any of them:

By (2.1), we can take the q-average over all k∈K at both sides of the previous inequality to obtain:

where the last line follows by using Minkowski inequality (2.4) for q≥1. Since the averaged quantities in the first term are independent of k, we have for all i∈I and j∈J that:

Here, we consider two cases for the parameters p,q∈[1,∞) independently.

Casep≤q: Under this assumption, we take at both sides of (3.4) the p-average over all i∈I, and using (2.4) for p≥1 at the RHS (right-hand side), we get:

Due to the independence on i∈I of the p-averaged quantity in the last term at the RHS, this simplifies to:

the LHS (left-hand side) is precisely GDp,q(A,C). Now, we take at both sides the p-average over all j∈J, use that the LHS is independent of j, and employ (2.4) for p≥1 at the RHS, to find:

where the interchange at the last line is allowed by property (2.3). Now, property (2.2) for p≤q ensures that in the first term at the RHS we can replace the inner ℳp by ℳq to get an equal or larger quantity, therefore:

which proves the claim.

Caseq≤p: Let us take at both sides of (3.4) the q-average over all j∈J. Using that the LHS is independent of j, and (2.4) for q≥1 at the RHS, we obtain:

where the change of ℳq by ℳp in the last-term of the RHS is justified by property (2.2) for q≤p.

Finally, we take at both sides the p-average over all i∈I , employ (2.4) for p≥1 at the RHS and use that the last term is independent of i∈I, to write:

The corollary below states that the indicator Δp,q is a semimetric that becomes a metric if p,q≥1. When this is not the case but still |p|,|q|≥1, the theorem that follows assures that Δp,q is at least inframetric and provides the associated constants.

Corollary 3.4. Forp,q∈ℝ¯the(p,q)-averaged distanceΔp,qis a semimetric on disjoint members of the family𝒫0(ℝn) 𝒫0(ℝn)of finite subsets ofℝn. Moreover, forp,q∈[1,∞), the(p,q)-averaged distance is a proper metric on disjoint subsets ofA,B∈𝒫0(ℝn).

Proof. From Definition 3.2, it is easy to see that Δp,q(A,B)≥0 as well as:

for every A,B∈𝒫0(ℝn) and all p,q∈ℝ¯. Moreover, it is also clear from Definition 3.1 that GDp,q(A,B\A)=0 if and only if A=∅ or B⊆A, hence from Definition 3.2 we find, for A,B≠∅, that:

These properties show that Δp,q is a semimetric for any p,q∈ℝ¯.

Since the maximum of two functions satisfying the triangle inequality also satisfies it, Theorem 3.3 shows that Δp,q satisfies the triangle inequality for all p,q∈[1,∞) and the cases p or q equal to ∞ follow by taking the appropriate limits.

Theorem 3.5. For anyp,q∈ℝwith|p|,|q| >1the generational(p,q)-distanceGDp,qsatisfies a relaxed triangle inequality. Explicitly:

for allA,B,C∈𝒫0(ℝn), any constantN≥1such that|A|,|B|,|C|≤Nand whereris given by:

Proof. For arbitrary p≠0, let us assume that q<0, so that |q|=−q. We can write:

which combined with the property (2.5) gives us:

An analogous inequality holds for arbitrary q≠0 if we assume that p<0. Therefore, we can write in general:

where NA,NB≤N and:

Notice that in both cases r can be chosen to take the second value, i.e., 1r:=1|p|+1|q|, when the sharpness of the constants does not matter. Finally, when |p|,|q| ≥1 we employ the triangle inequality for GD|p|,|q| to onclude that:

which finishes the proof.

From Corollary 3.4 we know that Δp,q is a metric for p,q≥1. More generally, the following corollary states that Δp,q is an inframetric when p,q≥1.

Corollary 3.6. For anyp,q∈ℝwith|p|,|q| ≥1the(p,q)-averaged distanceΔp,qsatisfies the following relaxed triangle inequality on disjoint subsetsA,B,C∈𝒫0(ℝn):

for any constantN≥1such that|A|,|B|,|C| ≤Nand where1r:=1|p|+1|q|.

Proof. The corollary follows immediately from Theorem 3.5 and Definition 3.2.

Theorem 3.7. LetA,B∈𝒫0(ℝn)and suppose thatp≤p′,q≤q′, then:

Proof. If follows easily from Definition 3.2 and two applications of property (2.2).

Corollary 3.8. LetA,B∈𝒫0(ℝn)and suppose thatp≤p′, then:

Proof. We obtain the corollary by using (3.2) and taking the limit q→−∞ in Theorem 3.7. For the convenience of the reader we present here an alternative, self-contained proof useful in the continuous case. Let X be a measure space with finite μ-measure. For any function f:X→ℂ in the Lebesgue space Lr(X) with r≥1, a simple modification of the Hölder inequality tells us that:

For any p,p′∈ℝ with 1≤p≤p′ we have p′p≥1, so with r=p′p and f=gp in (3.5) we obtain:

Taking as X the set A={ai}i=1N, as g:ℝn→ℝ the function given by g(x):=d(x,B), and as μ the discrete measure μd on A:

the inequality (3.6) becomes:

That is GDp(A,B)≤GDp′(A,B), which easily implies Δp(A,B)≤Δp′(A,B).

3.2 Pareto-Compliance

A discussion of the Pareto-compliance for the averaged GDp, IGDp and Δp-indicators appeared in [^{13, Sec. III}]. Similar observations can be made for the corresponding (p,q)-indicators introduced in this work. Here we will concentrate only in providing a complete proof of a result (analogous to Proposition 3 there) that describes the behavior of the GDp,q-indicator. The assumptions required are stronger than the compliance notion defined in Section 2.1 but they are useful to understand what will be needed in a very general situation.

If a MOP problem has an associated objective function f:X⊂ℝn→ℝℓ whose objective space Y=f(X) has a Pareto front Y*, and A⊂Y denotes an approximating subset, the explicit GDp,q and Δp,q-performance indicators assigned to A are given, respectively, by:

For convenience, given x,y∈ℝℓ and q∈ℝ¯ we will abbreviate δq(x):=ℳy∈Y*q d(x,y).

Theorem 3.9. Letp,q∈ℝand suppose that a pair of distinct finite subsetsA,B⊂Ysatisfy:

thenℐp,qGD(A)<ℐp,qGD(B).

Proof. Let us assume that A={ai}i=1NA, where its elements are arranged in a nonincreasing order with respect to δq(ai), this means that:

By conditions 1 and 2, we can decompose B into a partition B=B1∪⋯∪Bm of subsets defined recursively for i=1,…,m (with 1≤m≤NA), as:

with B0:=∅. Let Ni denote the size of Bi satisfying 1≤Ni≤NB=∑i=1mNi and arrange the elements of each subset Bi={bk(i)}k=1Ni in a nondecreasing order with respect to δq(bk(i)), i.e., for each fixed i:

Note that from the construction and condition 4, for i=1,…,m, we have that:

In particular, for the first element b1(i) of each Bi, which minimizes the set {δq(b)|b∈Bi} we have δq(ai)≤δq(b1(i)). Moreover, condition 2 necessarily implies:

Due to the ordering of A and this observation,

But the inequality still holds if, for any given value of j=1,…,m, we replace at the RHS of (3.8) the element b1(j) in the j-th term by any other bk(j)∈Bj (with k=2,…,Nj) while keeping all the remaining terms fixed. Therefore, there are Nj possible choices for this element, and in consequence Nj different inequalities for any given j=1,…,m:

where kj=1,…,Nj. When varying j=1,…,m, this procedure yields a total of ∑j=1mNj=NB inequalities with the same LHS, and using (2.1) we can take the average of all of them. Since the LHS remains the same, we obtain:

Notice that conditions 3 and 4 imply that the previous inequality has to be strict since the LHS contains all the elements of A and the RHS all the elements of B.

Rearranging and counting the terms, we get that:

which after a reordering in the first term becomes:

but using (3.7) we obtain the inequality:

Thus, ℐp,qGD(A)<ℐp,qGD(B) as expected.

The behavior of ℐp,qΔ is more delicate due to the definition of Δp,q, but for disjoint subsets not intersecting Y* similar arguments can be employed. We remark also that condition 3 is only needed to ensure an strict inequality in the conclusion. We would have obtained ℐp,qGD(B)p≤ℐp,qGD(B)p by removing condition 3 and weakening 4 to the condition:

4’. ∀a∈A, ∀b∈B:a≼b⇒δq(a)≤δq(b).