A Topologically Consistent Visualization of HighDimensional Pareto-front for Multi-Criteria

Decision MakingAKM Khaled Ahsan Talukder

Department of Computer Science and EngineeringMichigan State University

East Lansing, USAtalukde1@msu.edu

Kalyanmoy DebDepartment of Electrical and Computer Engineering

Michigan State UniversityEast Lansing, USAkdeb@egr.msu.edu

COIN Report Number 2018006www.coin-laboratory.com

Abstract—There are a good number of different algorithmsto solve multi- and many-objective optimization problems andthe final outcome of these algorithms is a set of trade-offsolutions that are expected to span the entire Pareto-front.Visualization of a Pareto-front is vital for an initial decision-making task, as it provides a number of useful information,such as closeness of one solution to another, trade-off amongconflicting objectives, localized shape of the Pareto-front vis-a-vis the entire front, and others. Two and three-dimensionalPareto-fronts are trivial to visualize and allow all the aboveanalysis to be done comprehensively. However, for four or moreobjectives, visualization for extracting above decision-makinginformation gets challenging and new and innovative methodsare long overdue. Not only does a trivial visualization becomesdifficult, the number of points needed to represent a higher-dimensional front increase exponentially. The existing high-dimensional visualization techniques, such as parallel coordinateplots, scatter plots, RadVis, etc., do not offer a clear and naturalview of the Pareto-front in terms of trade-off and other vitallocalized information needed for a convenient decision-makingtask. In this paper, we propose a novel way to map a high-dimensional Pareto-front in two and three dimensions. Theproposed method tries to capture some of the basic topologicalproperties of the Pareto points and retain them in the mappedlower dimensional space. Therefore, the proposed method canproduce faithful representation of the topological primitives ofthe high-dimensional data points in terms of the basic shape(and structure) of the Pareto-front, its boundary, and visualclassification of the relative trade-offs of the solutions. As aproof-of-principle demonstration, we apply our proposed palettevisualization method to a few problems.

Index Terms—Many-objective Optimization, High-dimensionalPareto-front, Visualization, RadVis, Parallel coordinate plot,Decision Making.

I. INTRODUCTION

With the recent advancement in many-objective optimiza-tion (i.e. problems with four or more conflicting objectivefunctions), there have been a number of post-optimizationissues that need to be addressed on an immediate basis, if thesemethods have to be routinely used in practice. One of the mostchallenging issues is to comprehend the obtained trade-offsolutions produced by a many-objective optimization (MOOP)solver, so that one or more preferred solutions can be chosen

in a systematic manner by the decision makers. In an MOOPproblem, if the target problem is consisted of two or threeobjectives, a 2D or 3D scatter plot is the most intuitive way tovisualize the results. Decision-makers (DM) can easily locateand isolate critical points that correspond to most beneficialtrade-offs among objectives in terms of least gain to mostsacrifice in migrating to a neighboring solution. DMs can alsoconsider other criteria, such as, robustness or reliability of asolution (i.e. risk assessment) or some other utility functionsrepresenting the practical usefulness of a solution.

Naturally, it is not possible for DMs to comprehend four ormore spatial dimensions visually in a straightforward manner.but a number of different visualization methods exists and arepopularly used: (i) parallel coordinate plot (PCP) [1], RadVis[2], multiple scatter plots [3], and others. While most of thesemethods allow high-dimensional data to be plotted in a two-dimensional plot, they often do not implicate the inherenttrade-off information against their neighbors. In fact, an ideaof who is one’s neighbor is also not clearly demonstrated bythese plots. While such visualization methods may be useful inother contexts, for multi-criterion decision making purposes,they certainly do not provide a clear picture. The goal ofthis paper is to highlight the need for specialized visualiza-tion methods for trade-off data solely for the purpose of aconvenient and easier decision making task. We propose onesuch technique and provide our initial results to demonstrateits working principle. This is an important topic to the fieldof evolutionary many-objective optimization (EMO), whichis currently handicapped without such tools in making theotherwise efficient optimization algorithms [4], [5], [6] lessmotivating to be applied in practice.

The paper is organized as follows: in Section II, we brieflydescribe a few popularly used existing visualization techniquesand discuss their limitations thereafter. In the section III,we discuss a number of key decision-making considerationswhich DMs may be interested in applying to a set of trade-off solutions. Then, in Section IV, we describe our proposedtechnique in details. In Section V, we present some resultsand analysis of the proposed method and conclude the paper

in Section VI with some future directions of our current work.

II. EXISTING VISUALIZATION METHODS

In terms of high-dimensional and multivariate data analysis,there have been a number of different visualization methodssuggested. Some of which have been applied to the visualiza-tion of MOOP solution sets. As we have already mentioned,one of the most widely used methods is the PCP [1], inwhich every solution is represented with a line marking theindividual objective values on multiple parallel vertical axes.Thus, if the lines for two solutions cross each other betweentwo objective axes, these two solutions constitute a trade-offbetween these two specific objectives. However a PCP plotdoes not provide a clear indication of the trade-off betweenone solution with its neighbor in a comprehensive manner.Another challenging task is to arrange the parallel vertical axesoptimally. Without the optimal arrangement of objective axes,a meaningful comprehension of the trade-off points becomesdifficult.

A ‘Heatmap’ is another approach [7], in which all objectivevalues (and variables and constraint values) of every trade-off solution are stacked in a row but colored based on theirrelative values. If an objective value is closer to its individualminimal value (assuming minimization of that objective), adarker red marker is placed. Thus, two rows (representing twotrade-off solutions) having dissimilar colors at its objectiveplaces will indicate widely different solutions, but a keeneye enabling detection of changes of colors at two objectivelocations between two solutions can only understand the trade-off that the two solutions possess.

Despite the loss of important information relevant to deci-sion making, dimension-reduction methods often offer a usefulvisualization tool and it is especially appealing since theyare able to represent solutions as a point-clouds in the two-dimensional plane, since humans are more adept at interpretingplanar diagrams. Along with PCA, there are sophisticateddimension-reduction methods such as self-organizing maps(SOM) [8], decision tree and Sammon Mapping [9], which canmap high-dimensional data points onto two-dimensional plane.More recently proposed Stochastic Neighborhood Embedding(SNE) [10] techniques are gaining popularity in the high-dimensional data visualization community. The interestingaspect about SNE based techniques that given a similaritymetric, they can infer clusters in the data points. Hencesuch technique is capable of showing similar data points ina separate point cloud, which is not easily identifiable justlooking at the data values.

A. Limitations of Existing MOOP Visualization Techniques

Despite the fact that there are a number of multivariatedata visualization techniques, we do not foresee their directapplication in the evolutionary multi-objective optimization(EMO) and multi-criterion decision-making (MCDM) context.However, we discuss a few recent methods which are promis-ing.

Fig. 1. A RadVis plot of two concentric 5-dimensional hyperspheres. Theblue points enclose the red points in the original high-dimensional space, butsuch a structure is not at all visible in two-dimensional RadVis plot – theyare all superimposed on one another.

In RadVis [2], each dimension is represented by a di-mensional anchor, and each dimensional anchor is distributedevenly on a unit circle, as shown in Figure 1. Each line in thedata set corresponds to a point in the projection, that is linkedto every dimensional anchor by an imaginary spring. Eachspring’s stiffness corresponds to the value for that particulardata point in that particular dimension. The position of thepoint is defined as the point in the 2D space where the allsprings are in equlibrium. Although the procedure allows ahigher-dimensional data to be mapped in a two dimensionalspace, RadVis fails to represent even the most basic underlyingstructure of the data points. For example, if the data points arelaid out in an n-dimensional hypersphere, and if we want tosee which points lie on the surface and which lie inside thehypersphere, RadVis can not present them is a distinguishableway. This situation is presented in Figure 1. We find anothervisualization method called 3D-RadVis [11] which maps n-dimensional data points onto a three-dimensional space, forwhich the third axis represents the extent of non-linearityof the Pareto front. However, the proposed 3D-RadVis hasa similar problem to that of the original 2D-RadVis in thatit is not able to maintain the geometric relationships amongsolutions from higher dimension to 3D.

Nevertheless, there have been some recent research reportsthat address this issue from EMO perspective. For example, in[12], the authors present an idea called “Prosection Method”.The technique is to project the whole set of solutions to theorthogonal plane where only the solutions from the chosensection are projected. Because multiple planes can be selectedfor the projection (as in the scatter plot matrix), a prosectionmatrix is used to visualize all the orthogonal prosections si-multaneously. In addition, color coding can be used for distin-guishing between feasible and infeasible solutions. However,as this approach depends on the construction of orthogonal

plane from multiple objectives to be used as an axis in the finalplot, it still suffers from similar limitations as other methods,specially when the dimension is more than four.

We can find other examples in [13] where the authorspresent a multiple ways to represent a high-dimensional PF.The paper addresses a common problem with the well-knownHeatmap visualization. Since an arbitrary ordering of rowsand columns renders the Heatmap unclear, the method usesspectral seriation to rearrange the solutions (along with theobjective values) and thus enhance the clarity of the Heatmap.A multi-objective evolutionary optimizer is used to further en-hance the simultaneous visualization of solutions in objectiveand parameter space.

In another study [14], the authors proposed a variant of Rad-Vis method that maps data points from a high-dimensional ob-jective space into a 2D polar coordinate plot while preservingPareto dominance relationship, retaining shape and locationof the PF, and maintaining distribution of individuals. Theconvergence of the approximate front is measured by radialvalues of all population members on that front. Meanwhile,the diversity performance is mainly determined by niche countof each subregion in a high-dimensional objective space.

In terms of lower dimensional embedding techniques,Stochastic Neighborhood Embedding (SNE) [15] [10] hasgaining popularity in the recent years. SNE is a probabilis-tic approach that can place data points, described by high-dimensional vectors or by pairwise dissimilarities, in a low-dimensional space in a way that preserves neighborhoodrelations. A Gaussian is centered on each solution in thehigh-dimensional space and the densities under this Gaussian(or the given dissimilarities) are used to define a probabilitydistribution over all the potential neighbors of the solution.The aim of the embedding is to approximate this distributionas well as possible when the same operation is performed onthe low-dimensional “images” of the data points. A naturalcost function is a sum of Kullback-Leibler divergences, oneper solution, which leads to a simple gradient for adjustingthe positions of the low-dimensional images. Unlike otherdimensionality reduction methods, this probabilistic frame-work makes it easy to represent each point by a mixture ofwidely separated low-dimensional images. Although SNE canconstruct reasonably good visualizations, it is limited by a costfunction that is difficult to optimize and also it suffers fromthe so called crowding problem. Moreover, TSNE/SNE likemethod is completely oblivious to the topological primitives ofthe high-dimensional data points. This limitation is presentedin Figure 2.

All the existing methods for MOOP solution visualizationmainly focus on the visualization of convergence of the PF.Although visualization of the movement of solutions in thesearch space might be useful for an algorithm designer, DMsare generally not interested in it. Reducing the number ofsolution choices and the position of infeasible/robust solutions(with respect to others) are more important to a DM. Alsohuman eye is more used to spatial representation of data orfunction in two and three dimensions. Spatial representation

(a) (b)

Fig. 2. The left figure is the actual input data point, where the center bluepoints are surrounded by yellow points. The right figure is the TSNE mappingof the input point into another space after 5,000 iterations with perplexity value[15] of 2. The topological relation of the data points are lost.

enables us to navigate the space and make decisions accord-ingly. Therefore, reduction of the cardinality of Pareto solutionchoices and their simplest comprehension through a smartlower-dimensional visualization method is the main goal ofthis study.

III. VISUALIZATION FROM A DM’S PERSPECTIVE:THE PALETTE VISUALIZATION

Our goal of the visualization method differs from theexisting techniques in a number of interesting ways:

• Topological Consistency: We are specifically interestedin how close our visualization preserves the topology ofthe solutions from the original objective space to 2D or3D.

• Spatial Navigability: Human mind is trained to navigatein a spatial dimension. Hence, two and three dimensionalscatter plots are the most intuitive and intelligible repre-sentation of data. We want to keep this property of ‘spatialnavigability’ in the visualization.

• Solution Properties: We are also not interested in howthe solution converges to the true PF in this study, orvisualization of the search trajectory. We are interestedin a representative scheme of trade-off Pareto data sothat a multi/many-criteria based decision making can beperformed on them either quantitatively or qualitatively.

From our previous experiences with industry collaborations,we have seen that a decision making procedure requires a com-pletely different set of considerations than an EMO developerconsiders within an optimization algorithm. The visualizationprocedure which we propose here is mainly followed fromthe RadVis method [2], but extended in a three-dimensionalformat, which we call here as the Palette Visualization. Theentire plot is consisted of multiple layers two dimensionalRadVis plots where each layer comes from the different layer-wise division of higher-dimensional Pareto solutions fromboundary to the core. In the following subsections, we describethe procedure stepwise.

(a) (b)

Fig. 3. The left figure is a three-dimensional spherical Pareto-front in the first quadrant. The right figure shows the corresponding proposed palette visualization.The points are colored according to the layers of the data points (that is, the top-most ring the is outer-most (or boundary) layer in the sphere). The pointsclose to the centroid has the lowest z-axis value. Since there is no point with a very large trade-off in the Pareto-front, all points are marked with almostsame size markers.

A. Topological Consistency

As we have seen in Section II-A, in a high-dimensionalspace, it’s difficult to preserve the very basic topologicalprimitives when we map the points onto lower dimensionalspace. Even the inside-outside relations of the point clouddo not conform to their original relation after the mapping.We solve this problem by decomposing the point cloud intolayers of convex hulls and each convex hull layer is plottedon a three-dimensional RadVis plot where the position of eachlayer is defined on the z-axis. Generally outer layer of thepoint cloud will have the highest z-axis value and inner layershave smaller z-axis value as inner the layers come to thevicinity of the centroid of the point cloud. In this way, wecan ensure that outer layer of the original point cloud staysoutward on the RadVis plot so that the data points do notfall over each other. Moreover, all the points on the layersmaintain a similar distance from the centroid as in the originaldata points in the high-dimensional space. This arrangementcan at least maintain two basic topological properties of thehigh-dimensional data points: a) distances from the centroidof the data points and b)inner-outer relations of the originaldata points. An example of such plot is presented in Figure 3.

B. Spatial Navigability

Basically, the three-dimensional layer-wise RadVis plotgives the spatial navigability on the high-dimensional space.Since Pareto optimal objective function values only stay in oneof the high-dimensional quadrant (i.e. empty 3D-sphere in thefirst quadrant if the Pareto front is minimizing all non-negativeobjectives.), the shape of the point cloud is not complete. Asa result, the top portion of the shape of the point cloud is notgoing to be superimposed on the points on the bottom portion

of the point cloud (in the case of three-dimension). Therefore,each layer of the three-dimensional RadVis comes from uniqueboundary solutions of the original data points. This actuallyprovides the spatial navigability property to the visualization.A DM can now traverse the high-dimensional data pointswithout confusing the relative neighborhood relationships ofthe original data points. A DM can also understand whichpoints lie on the boundary of the original high-dimensionalspace and navigate and visualize the entire Pareto-front as ifhe/she is looking at a two/three dimensional scatter plot.

C. Pareto Solution Properties

Once we have the skeleton found from the three-dimensional RadVis plot, we can ask that for a given setof Pareto-optimal solutions (over 4 dimensions), what aspectsof the solution set is more relevant to look at? Perhaps, themost desirable aspect of the Pareto-front solutions are thosethat offer most trade-off among all the points in a solutionset. In literature they are known as Knee points [16]. Ifwe assume that we do not have any knowledge about theuser‘s preferences, it can be justified that the region aroundthat knee is most likely to be interesting for the DM. Sincethe knee solutions are characterized by the fact that a smallimprovement in either objective will cause a large loss inthe other objective, which makes such movement in eitherdirection not very useful.

Formally, the knee solutions are those, from which, movingtowards any of the objective axes will cause comparativelyhigher amount of deterioration in one or more other objectives.In this paper, we will follow the definition of knee points dis-cussed in [17]. Let us consider the simple Pareto-optimal frontdepicted in Figure 4, with two objectives to be minimized. Thisfront has a clearly visible bulge in the middle. If we assume

Fig. 4. A simple 3-D Pareto-optimal front of DEBMDK problem with a singleknee [18]. The axes are f1, f2 and f3. The knee solutions (i.e. solutions withbetter trade-off) are presented by circles with increasing radius.

linear preference functions, and (due to the lack of any otherinformation) furthermore assume that each preference functionis equally likely, the solutions at the knee are most likely tobe the most desirable choice of the DM. Note that in Figure4, due to the concavity at the edges, similar reasoning holdsfor the extreme solutions (edges), which is why these shouldbe considered knees as well. The goal of this paper is to howwe should represent such knee solutions if the Pareto-front isconstructed in a high dimensional space. Along with trade-offs,we can also represent distance from the centroid or constraintviolation value of each of the solutions in our visualizationmethod.

IV. PROPOSED PALETTE VISUALIZATION PROCEDURE

The Palette visualization is consisted of mainly of threechronological steps – a) Finding multiple clusters of points inthe original high-dimensional objective space, b) Layer-wisedecomposition (boundary to core) of points of each cluster,and c) Graphical representation of different relevant decisionmaking information for trade-off points in each cluster. Eachof these steps is described in the following subsections.

A. Clustering

The complete Pareto-optimal front may be constituted witha number of disconnected regions. In order to not confuseproperties of each disconnected region independently, ourproposed Palette visualization works for a single cluster at atime. Therefore, we first apply a suitable clustering approachin the original high-dimensional objective space to separatetrade-off points into separate clusters. In this study, we usean unsupervised algorithm, such as DBSCAN [19], for thispurpose, but any other clustering method can also be used.

B. Layer-wise Decomposition

This is the main crux of the proposed palette approach. Foreach cluster of data point, we further sub-divide points into

multiple layers, starting from boundary layer which is com-posed of all boundary points in the high-dimension. Thereafter,intermediate layers are extracted one by one until all pointsare classified into different layers. This procedure requires aboundary identification method which can be applied repeat-edly starting with the complete set of points and then deletingpoints which have already been identified with a layer. Thenumber of total layers will depend on the placement of thepoints with respect to each other and also on the definition ofboundary identification algorithm. In this study, we implementa simple Jarvis-March [20] based algorithm which is describedin Algorithm 1.

The algorithm first finds the centroid µ of the clustered datapoints. All points are then sorted according largest distancefrom the centroid to shortest. The point q having the largestdistance is declared as one of the boundary points in theboundary set Q. Then, for every point p from the sorted order,the procedure finds the angle θ between two vectors: (p− µ)and (p − q), for every member q ∈ Q. If 0 < θ ≤ θt forany of the already identified boundary point q, then p is not aboundary point and the next point in the sorted list is checked.In our study, we set the angle threshold θt to π/60. In case aboundary point is identified, it is included in the boundary setQ. Since in higher dimension, wrapping around is not possible,our algorithms runs in O(n2) time.

Although the above method is not the most-efficient pro-cedure, but it is one of the ways to identify boundary points.Other methods, like Quick-hull [21], can also be used and weare investigating other more efficient methods.

The above layer-wise division of points may end up pro-ducing a large number of layers for our next analysis. Tomake the procedure amenable for the final decision-makingpurpose, multiple layers can be combined together and only ahandful (say, 3-5) of layers can be formed. For this purpose,we combine layers so that almost an equal number of pointsare included in each finally decomposed layer.

C. Graphical Representation of Layers

Once the layer-wise decomposition is performed, pointsfrom each layer are then represented in a 2-D RadVis plot. Al-though we have used RadVis plot here, other many-dimensionto 2-D mapping can also be used. Stochastic NeighborhoodEmbedding (SNE) [10] is one other representation schemewhich can be used. It is important to note that the outer-most (boundary) layer in higher-dimension need not map asboundary points on a 2-D plot (such as RadVis or SNE); thepoints can cover the entire 2-D mapping region inside theRadVis polygon. For this purpose, we plot the subsequent layerof points staggered in a third dimension (vertical), in which thetop-most 2-D RadVis plot corresponds to the boundary layer,the next one below corresponds to the next to boundary layerand so on. The bottom-most 2-D RadVis plot correspond tothe inner-most layer in the original high-dimensional space.

Once the layer-wise points are mapped in 2-D RadVis plot,for decision-making purposes, we can represent the pointsaccording to various preferential features. Here, we compute

Algorithm 1 Decompose Pareto-front into boundary layers.Require: a cluster of data points C found from applying

DBSCAN on the Pareto-front in original dimension1: B ← ∅2: µ← centroid of C3: sort all points p ∈ C in descending order of ‖p− µ‖4: repeat5: i← 06: Li ← ∅7: for each point p ∈ C do8: if flag(p) is not set then9: u← (p− µ)/‖p− µ‖

10: for each q ∈ C do11: v← (p− q)/‖p− q‖12: θ ← cos−1 ((u · v)/(‖u‖‖v‖))13: if 0 < θ ≤ θt then14: flag(p)← non-boundary15: end if16: end for17: if flag(p) 6= non-boundary then18: flag(p)← boundary19: Li ← Li ∪ p20: end if21: end if22: end for23: B ← B ∪ Li

24: C ← C − Li

25: i← i+ 126: until C 6= ∅27: return B

the trade-off information of each point in the original high-dimensional space and mark a point with a larger circle, if thepoint corresponds to a higher trade-off.

1) Computation of Trade-off Value: In our case we areinterested in the relative trade-off of solutions known asknees. For the identification of the knee solutions, we willfollow the method presented in [17]. The definition of Pareto-optimality is a non-reflexive, transitive, and antisymmetricbinary relation, i.e., Pareto dominance. Pareto dominance isdefined in the following manner:

Given a set of objective functions F = [f1, f2, . . . , fM ] tobe minimized, the vector F(xi) is said to dominate anothervector F(xj), denoted F(xi) ≺ F(xj), if and only if fk(xi) ≤fk(xj) for all k ∈ {1, 2, . . . ,M} and fm(xi) < fm(xj) forsome m ∈ {1, 2, . . . ,M}. A point x∗ ∈ S is said to beglobally Pareto optimal or a globally efficient for a multi-objective optimization problem (MOP) if and only if theredoes not exist x ∈ S satisfying F(x) ≺ F(x∗). F(x∗) is thencalled globally non-dominated solution.

Trade-off value can be computed over a pair of non-dominated objective vectors and may be defined as the net gainof improvement in some objectives offset by the accompanyingdeterioration in other objectives as a result of substituting oneobjective vector with another non-dominated objective vector.

Algorithm 2 Palette Visualization1: compute trade-off value of each Pareto point2: take all the Pareto-points, use DBSCAN algorithm and

find n clusters {C1, C2, . . . , Cn}3: take a cluster Ci4: invoke Algorithm 1 to find the layer decomposition B ={L1, L2, . . . Lk} of Ci.

5: use RadVis to display each layer Li ∈ B on a two-and-half dimensional space in which the sequence of layers isstaggered along the z-axis.

6: Pareto points with higher trade-off value are highlighted.

Mathematical definition of trade-off is commonly given forevery pair of objective functions. The equation to computethis quantity has been borrowed from [17]:

T (xi,xj) =

∑1≤m≤M max[0, fm(xj)− fm(xi)]∑1≤m≤M max[0, fm(xi)− fm(xj)]

(1)

In the definition of T (xi,xj), the numerator evaluates thetotal improvement gained by exchanging xj with xi whilethe denominator evaluates the deterioration caused by theexchange. The actual metric to evaluate the worth of a solutionxi ∈ R ⊂ S (where, R is an ε neighborhood around xi)in terms of performance trade-off, is given in the followingequation:

µ(xi, R) = min∀j:xj∈R,xi 6≺xj ,xj 6≺xi

T (xi,xj) (2)

A solution with a larger value of the quantity µ(xi, R), withina neighborhood of R, signifies if the solution belong to theknee region. If solution far from the knee region the µ valuewill be smaller.

An overall algorithm for the entire procedure is summarizedin Algorithm 2. For multiple clusters, we can generate multiplepalette for an easy visualization of different clusters. If the totalnumber of layers are too many, we can also merge multiplelayers into one layer to reduce the total number of layers whichdecision makers need to analyze. An example plot is presentedin Figure 3 for a 3-D Pareto-front, where each of five layers onthe palette is actually composed of multiple individual layers.

The above procedure makes a systematic division of theoriginal high-dimensional data into separate clusters basedon their relative location in the entire Pareto-front and theninto different layers based on the centrality of points in theirown cluster. This 2 1

2 -D representation of the points allowsa decision-maker with a bottom-up approach to look at thepoints in a lower and manageable dimension, and associatethem with their true location in the original high dimensionalspace. Then, the relative trade-off value information of eachpoint in each layer and in each cluster will allow DMs toanalyze the Pareto-front further by only looking at a fewcritical points to finally select a single preferred solution.

Fig. 5. The figure is the palette visualization of the five-dimensionalDEBMDK problem [18]. There is only one knee in the Pareto-front whichcomes up as the largest circle in the plot. The top-most layer is the outer-mostboundary of the point cloud.

V. VISUALIZATION RESULTS

To demonstrate the working of the proposed palette visual-ization concept, first, we use a benchmark test problem from[18], called DEBMDK. The problem has a parameter that canbe changed to introduce multiple knees, however we choosethe problem with a single knee for an easier demonstrationof our palette visualization approach. In all cases, we havemarked the point size according to the trade-off values so thata bigger marker means a solution with higher trade-off value.Also, the solutions that are farthest from the centroid is coloredin red and closer ones are colored in blue color. The resultsfor the five-dimensional DEBMDK problem is presented inFigure 5. It is interesting to note that in addition to the actualknee point, the trade-off metric value given by equation 2is relatively large for the boundary points, we can see somepoints on the top-most layer with bigger radius.

Next, for the spherical Pareto-front problem shown in Fig-ure 3, all Pareto points are drawn with a similar marker size,indicating that in this problem, there is no knee-like point. Insuch cases, DM may be tempted to select a boundary point –from the top-most layer in our palette visualization plot, unlessthere are other more compelling practicalities to consider.

Finally, we apply our method to the vehicle crash-worthiness (CARCRASH) problem from [22]. This is a three-objective optimization problem, however the Pareto-front inthis problem is composed of three separate clusters (markedin the figure as clusters A, B and C). Our proposed palettevisualization can show three clusters independently. The entirePareto-front of the CARCRASH problem is presented inFigure 6.

The top cluster (A) and its palette visualization are presentedin Figure 7. Since all points in this cluster are drawn with asimilar marker size, there is a smooth variation of points inthis cluster. There is one knee-like point (E). Either point Eor a boundary point may be of interest to a DM and our top-most RadVis plot, representing boundary points, allows a DM

Fig. 6. The three-dimensional Pareto-front for the CARCRASH problem [22].

Fig. 7. Palette visualization of Cluster A for CARCRASH problem [22] inFigure 6. There is no outlier but one point E with a large trade-off value.Most other points have similar size. The top-most layer contains boundarypoints, which can also be considered as preferred candidate solutions.

to easily focus on the filtered boundary points for a preferredsolution.

The palette visualization for the second and third clusters to-gether are presented in Figure 8. A point with a relatively largetrade-off is observed in this cluster. This point corresponds topoint K marked in Figure 6, which staying in the intersectionof two parts of the Pareto-front makes an interesting choicefor the DM.

The most interesting part of this visualization is that ir-respective of the dimension of the Pareto-front, a DM caneasily navigate through the critical points in the objectivespace and make a decision about which solutions to choose.This visualization method thus helps to reduce the overheadassociated with solution assessment and provides a great aidto the human-computer interaction during the decision makingprocess.

Fig. 8. Palette visualization of Clusters B and C. One point (K) with a largetrade-off value is observed and can be considered as a preferred choice.

VI. CONCLUSIONS AND FUTURE WORK

In this paper, we have demonstrated an alternative visu-alization approach to address the issue of high dimensionalPareto-front visualization. Our approach filters out a fewcritical points from the entire non-dominated set of solutionsdiscovered by a multi-objective optimization algorithm. Theidentification of critical points is limited in this study onlyto knee points having a large local trade-off and boundarypoints which are important in their own rights. The system-atic filtering of solutions into clusters in the original high-dimensional space, then into layers according to their centralitywithin each cluster and finally into different levels of trade-off values allow a decision-maker to look at only a handful ofcritical solutions for an assisted decision-making task. Anotherimportant contribution is that the proposed palette visualizationcan be generically applied to any large dimension of trade-offdata.

Our approach has mainly based on the RadVis mapping, butany other approach that tries to retain the global topologicalrelations among the data points from high to low dimensioncan also be used. In the future we would like to improvethe current boundary identification procedure by adopting an‘alpha-shape’ [23] based non-convex hull finding algorithm.

Along with the trade-off values of the solutions, we canalso color the points according to the constraint violationvalues for constrained MOOPs. This will help a DM to havea clear idea of the extent of feasibility of their selected points.Our approach can also be helpful for visualization in otherscientific domains, specially our method might be useful tothe data-mining community.

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