Computing path similarities is an approach to identify areas in the decision space that are already covered by a majority of the population members. However, in contrast to computing a distance matrix and finding the minimum or median values, as was done in [WM21b, WM22a], imposing an order on a set of curves would open the possibility of finding neighbourhood relationships. With this approach, it is possible to find less covered areas with less computational effort.
When the crowding distance mechanism [DPAM02] is used as an example, it is evident that computing the degree of isolation for a solution – by analysing the distance to its neighbours – can yield benefits. In the crowding distance mechanism, solutions are ordered concerning the objectives being optimised.
Then, the distance between the two neighbours is computed per objective and summated. This methodology allows a more focused niching, since less crowded sections are found and can be explored more. In the current analysis, we apply this concept to a set of paths in decision space. However, there is no natural total order on a set of paths or curves in a metric space [WM21a].
To order a set of paths, one can impose a binary relation on it. A relation≤ on a set Mis a total order if the following requirements are fulfilled for all a,b,c∈M:
a≤b∧b≤a⇒a=b(Antisymmetric) a≤b∧b≤c⇒a≤c(Transitive)
a≤b∨b≤a(Connected)
(6.6)
Our approach to impose a total order on a set of curves is to contract every curve to a single point and then compute the signed length of an orthogonal vector from a predefined line to these contraction points. Using acontraction metric fcm, we then transfer each path from its metric space into thecontracted space, where each path is represented as a single point [WM21a].
6.5.1 Contraction Metrics
Definition 1(Contraction Metric). Acontraction metriccan be, but is not lim-itedistad to, a measure of central tendency on a set ofmpoints{x1,x2,· · ·,xm}, where eachxi∈Rn. It can be used to contract a path, defined by its set of nodes to a single point, which can be used for further computations. LetPbe a set ofk pathsP={p1,p2,· · ·,pk}, where each path pi= (x1,x2,· · ·,xm)is of a variable lengthm∈N,m≥1, where eachxi∈Rn. Then a contraction metric fcmmaps each path to a point, which serves as the path’s representative, in the previously mentionedcontracted spaceRn; hence, fcm:P→Rn. This space has the same number of dimensions as do the points in the respective path.
For the contraction, the following metrics (measures of central tendency) can be used. We employ the set notation because all these metrics are based on sets of points and not onn-tuples. It should be noted that this list is not exhaustive, since other measurements can be used.
Definition 2(Centroid). The centroidCof a setXofmpoints{x1,x2,· · ·,xm} with eachxi∈Rnis defined by:
C(X) =x1+x2+· · ·+xm
m (6.7)
The centroid, also known as thecentre of massof a polygon, is the point where the polygon can be balanced when placed on the tip of a needle.
Definition 3(Geometric Median). The geometric median of a set ofmpoints {x1,x2,· · ·,xm}with eachxi∈Rnis defined by:
arg min
y∈Rn m
∑
i=1
∥xi−y∥2 (6.8)
Here, a pointyis to be found which minimises the sum to allxi in the set of nodes. This problem is also known as theFermat-Weber problem[CLM+16].
We useWeiszfeld’salgorithm, which iteratively computes the geometric me-dian [Wei37].
6.5.2 Imposing an Order
After computing the contraction pointxcm= fcm(P)for each path pi∈Pand pi = (x1,· · ·,xm), we can impose an order on the resulting set of points. As follows, we use the notation ofxy#»for a vector from pointxto pointy. Hence, xy#»=
y0−x0
... yn−xn
for two pointsx,y∈Rn. We take the vectorv# »se=x# »1xmfrom the start to the endpoint and find an orthogonal vectorw# »se, so that⟨v# »se,w# »se⟩=0,
⟨#»a,#»
b⟩denotes the dot product. This equation holds ifv# »se= x
y
andw# »se= y
−x
. We construct the orthogonal vector accordingly. Then, we compute the vector fromxcm tox1, i.e., #»r =x# »cmx1. To determine the signed distance dcm = fdistCm(xcm) of the point xcm to the line from x1 to xm, we compute the dot product of−r# »and ˆwse, where ˆwseis the normalised vectorw# »se, hence fdistCm(xcm) =⟨−r,# » wˆse⟩. This value is also known as the scalar projection of the vector−r# » in the direction of the vector ˆwse. In other words, it is the signed length of the projection of−r# »ontow# »se, hence−r# »w# »se. This value can be negative or positive, depending on which side of the vectorv# »sethe point lies. In Figure 6.5, the used vectors and their relations are shown in a two-dimensional metric space.
Proof. Let−→
r =−−−−−→
pcmpstart, andvse⊥wse. So, cosθ=∥rwse∥
∥r∥
cosθ∥r∥=∥rwse∥ cosθ∥r∥ ∥wse∥=∥rwse∥ ∥wse∥
⟨wse,r⟩=∥rwse∥ ∥wse∥
⟨wse,r⟩
∥wse∥ =∥rwse∥
⟨wse,r⟩=∥rwse∥
After computing fdistCmfor every contraction point of each path pi ∈P, we order the values in ascending or descending order. We thus impose a total order on the set of paths by ordering their respective fdistCm-values. Figure 6.6 shows an example map, in which different markers denote the different paths. Further-more, the respective contraction points are shown using the same markers. In addition, the contraction point’s distance is colour coded. Paths on the left side of the dashed centre line have a negative distance, whereas paths on the right side have positive distances. In this example, we used the geometric median as the contraction metric [WM21a].
This methodology is independent of the path’s orthogonal coordinate system and can be applied to any set of paths. The reason behind this is that both the centroid
Figure 6.5: Vectors used for measuring the distance between xcmand the vectorvse. The dashed line represents an example path. [WM21a]
0 2 4 6 8 10 12 14
X-Coordinate -2
0 2 4 6 8 10 12 14
Y-Coordinate
Vectors in Decision Space
and the geometric median are equivariant under Euclidean transformations. Only pairwise distances account for a change in their values [Eft15, Kim94].
6.5.3 Path Density-Based NSGA-II
In [WM21b], the authors used the well-known NSGA-II algorithm [DPAM02]
and changed the replacement mechanism by employing theirFréchet sorting method instead of using crowding distance. When the population cannot be filled with the following front during the replacement process, the solutions are usually ordered using crowding distance to find solutions in less dense areas. The authors in [WM21b] instead computed the distance matrix of all solutions and then assigned the respective minimum value of the distances to the respective solution. Through this approach, they performed the niching in decision space instead of the objective space. The solution with the highest minimum distance was then brought into the next generation. They applied the same technique during the selection process [WM21a].
In our novel approach, instead of computing the distances to all other paths in the respective front and taking the minimum, we compute the distance between the two neighbours of a specific solution, which reduces computational cost and increases the niching. The solutions are ordered according to their respective contraction points, and the Fréchet density values are computed by taking the average distance to the two neighbours of a solution. The two outer solutions are assigned with an infinite distance, as is done in crowding distance. In Figure 6.7, a small example is shown. Three paths are depicted in white, while the paths’
respective couplings are shown in black. The middle path is the one whose degree of isolation is computed. The algorithm determines the distances to its two neighbours, and the average is then assigned as its isolation value. For
Figure 6.6: Paths and their respective contraction points (geometric median, denoted by the same marker as in the paths) [WM21a]
1 2 3 4 5 6 7 8 9
X-Coordinate 1
2 3 4 5 6 7 8 9
Y-Coordinate
DistancesofContractionPoints
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3
Signed distance of Contraction Point to center line
Figure 6.7: Three paths, in white.
Black: The paths’
couplings. [WM21a]
Fréchet Density Example
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
X-Coordinate 0.5
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
Y-Coordinate
a path pi in an ordered set of paths P={p1,· · ·,pk}, and using one of the above-mentioned contraction metrics, theFréchet density valueis defined by:
FDV(pi) =∑1j=−1δdF(pi,pi+j)
2 (6.9)
whereδdF(pi,pj)is the discrete Fréchet distance between the pathspiandpj. It should be noted that other similarity metrics can be used here as well.
For the sake of clarity, the algorithm from [WM21b] is denoted as NSGA-II-CRFD, whereas our approach is denoted asNSGA-II-DEFDXX. HereXX refers to the contraction metric used;CTmeanscentroid,GMmeansgeometric median,DEmeansdensity, andFDmeansFréchet distance. The solutions with a higher Fréchet density value are thus brought into the next generation and selected if they are non-dominated during the selection phase.