How to Read Many-Objective Solution Sets in Parallel Coordinates

• Scatter plots become difficult to comprehend in higher-dimensional spaces, limiting a familiar way to inspect solution sets.The scatter plot may be readily drawn only in 2D or 3D Cartesian space.Source: §1 · p. 1
• Parallel coordinates map multidimensional vectors to polylines across axes in a 2D graph and scale to many objectives.Many-objective optimization is defined here as optimizing more than three objectives.Source: §1 · p. 1
• Because this 2D mapping loses information, the plot is less straightforward than a scatter plot for presenting a solution set.Source: §1 · p. 2
• The paper asks whether parallel coordinates reveal solution-set quality, shape and distribution, and relations among objectives, including the effect of axis order.Source: §1 · p. 2
• The authors offer observations intended as guidelines for using parallel coordinates in evolutionary multiobjective optimization.Source: §1 · p. 2

• Each point in m-dimensional space becomes a polyline across m parallel axes, with each vertex's position encoding its value on that dimension.Adding dimensions requires adding axes, so the representation scales with data dimensionality.Source: §2 · p. 2

3 Quality Measuring

• A solution set’s range is directly visible, while convergence, coverage, and uniformity require more cautious interpretation.The paper describes extensity as a feature parallel coordinates can reveal directly; it frames the other quality aspects as questions for examination.Source: §3 · pp. 2–3

3.1 Convergence

• When polylines are not overcrowded, parallel coordinates show Pareto dominance and can be clearer if dominated solutions are removed.Dominance is a convergence criterion; the paper suggests removing dominated solutions when only nondominated ones matter, especially for comparing sets.Source: §3.1 · p. 3
• On 10-objective DTLZ2, NSGA-II spans 0–around 3.5 against a Pareto-front range of 0–1, while GrEA matches the front’s range and converges well.The passage says the GD+ results shown in Figure 2 confirm these observations.Source: §3.1 · p. 3, Figure 2 · p. 3, §3 · p. 3
• Matching the Pareto-front range does not ensure comparable convergence: on 10-objective WFG7, NSGA-II and GrEA reach virtually the same range, but NSGA-II has significantly worse GD+.The paper gives two possible explanations: NSGA-II may not actually be close to the front, or most solutions may converge while a small portion remains far away within the front’s range.Source: §3.1 · p. 4, Figure 3 · p. 4

3.2 Coverage

• Parallel coordinates show unreached objective regions and can suggest distribution patterns, such as IBEA’s boundary concentration and the 0–0.2 gaps for IBEA and SMS-EMOA.The examples include IBEA failing to cover the first six objectives on 10-objective DTLZ2, and IBEA and SMS-EMOA missing the 0–0.2 region on all three objectives in the Figure 5 sets.Source: §3.2 · p. 4, Figure 4 · p. 4, Figure 5 · p. 5
• Different solution sets can have identical parallel-coordinate plots: set B has better coverage than set A, which contains four duplicate solutions, and changing objective order can distinguish them.The example is shown in Cartesian coordinates and in parallel coordinates; the two plots match under the original ordering.Source: §3.2 · pp. 4, 6, Figure 7 · p. 5, Figure 8 · p. 6
• A 2D parallel-coordinate view cannot fully reflect coverage in the original objective space; on WFG7, NSGA-II appears better in the plot, but GrEA has better DCI.For real-world problems with an unknown Pareto front, the paper says one cannot tell whether a set reaches the optimal objective regions.Source: §3.2 · pp. 4, 6, Figure 9 · p. 6

3.3 Uniformity

• Uniformly distributed polylines often indicate uniformity, but irregular line patterns need not mean poor solution distribution; MOEA/D and BCE-MOEA/D illustrate this caveat.MOEA/D’s systematic weight vectors yield equivalent values across objectives, whereas BCE-MOEA/D’s niching can spread solutions across each objective’s range; BCE-MOEA/D has better SP, which is not apparent in the parallel plot. Overlaid lines can also obscure distribution, as in the NSGA-II and SPEA2 example.Source: §3.3 · pp. 6–7, Figure 10 · p. 7, Figure 11 · p. 7

4 Solution Set Distributions

• More intersections between lines joining adjacent objectives indicate stronger conflict, while no intersections indicate harmonious objectives.The passage illustrates these patterns with objective pairs in Figure 16.Source: §4 · p. 8
• When all lines between two objectives intersect at one point, the objectives are negatively linearly dependent; the four-objective ML-DMP example has f1 = −f3 and f2 = −f4.The corresponding pairs intersect at one point in Figure 17(b).Source: §4 · p. 9, Figure 17 · p. 10
• Cartesian lines map to intersection points in parallel coordinates, whose horizontal position depends on slope and whose vertical position depends on the line’s intercept.For negative slopes, the intersection lies between the axes; a slope of −1 places it midway.Source: §4 · p. 9
• Lines intersecting at the same horizontal position but different vertical positions represent Cartesian point-pairs with the same slope, a pattern often seen for absolutely uniformly distributed points.The passage identifies this pattern in Figure 19(b).Source: §4 · p. 11

5 Objective Order in Parallel Coordinates

• Changing objective order can change the apparent conflict between adjacent objectives, as the two orders in Figure 20 show weak versus intense adjacent conflict.The orders are f1, f2, f3, f4, f5 and f1, f3, f5, f2, f4, respectively.Source: §5 · p. 11
• A plot of m objectives displays only m − 1 relationships at a time, so its axis order exposes only a small portion of all objective relationships.Figures 21 and 22 show patterns, including linear dependencies, after some objectives are swapped.Source: §5 · p. 11
• Finding a good axis order is nontrivial; prior approaches cited include clutter reduction and placing harmonious objectives together.The passage also notes that the most harmonious objectives have been arranged in a row in multiobjective optimization.Source: §5 · p. 11

6 How to Draw A Parallel Coordinate Plot

• The plotting instructions cover MS Excel, MATLAB, LaTeX, and Origin, with steps or code provided for each tool.Tables 1–4 contain the respective procedures.Source: §6 · p. 11, Table 1 · p. 13, Table 2 · p. 13
• Figures 20–22 show solution sets with different objective orders, including examples for 5-objective ML-DMP, 5-objective DTLZ5(I,M), and 4-objective ML-DMP.The DTLZ5(I,M) example specifies I = 3.Source: Figure 20 · p. 12, Figure 21 · p. 12, Figure 22 · p. 12
• Origin was used to draw all parallel-coordinate graphs of data examples in the paper.Source: §6 · p. 13

7 Conclusions

• Parallel coordinates can partly reflect convergence, coverage, and uniformity, but serve as an assistant tool rather than a replacement for quality metrics.Source: §7 · p. 13
• Overlapping polygonal lines can reduce clarity, even as parallel coordinates turn some geometric features of many-objective solution sets into visible 2D patterns.Source: §7 · p. 13
• Objective-axis order matters, and rearranging axes according to the solution-set features under consideration may better present objective relationships.Source: §7 · p. 13
• Future work is directed at easing interpretation, including arranging highly conflicting or harmonious objectives together and considering line coverage between objectives.The passage gives swapping f1 and f2 as an example of reducing line coverage.Source: §7 · pp. 13, 15