Visualization in Multiobjective Optimization

Final version

Bogdan Filipič Tea Tušar Tutorial slides are available at CEC Tutorial, Donostia - San Sebastián, June 5, 2017 http://dis.ijs.si/tea/research.htm

Computational Intelligence Group Department of Intelligent Systems Jožef Stefan Institute Ljubljana, Slovenia

Contents

Introduction

A taxonomy of visualization methods

Visualizing single approximation sets Introduction

Visualizing repeated approximation sets

Summary

References

3 Introduction Introduction

Multiobjective optimization problem Visualization in multiobjective optimization Minimize Useful for different purposes [14] f: X → F • Analysis of solutions and solution sets f:(x ,..., x ) 7→ (f (x ,..., x ),..., f (x ,..., x )) 1 n 1 1 n m 1 n • Decision support in interactive optimization • Analysis of algorithm performance • X is an n-dimensional decision space ⊆ Rm ≥ • F is an m-dimensional objective space (m 2) Visualizing solution sets in the decision space • Problem-speciﬁc → Conﬂicting objectives a set of optimal solutions • If X ⊆ Rm, any method for visualizing multidimensional • Pareto set in the decision space solutions can be used • Pareto front in the objective space • Not the focus of this tutorial

4 5

Introduction Introduction

Visualization can be hard even in 2-D Stochastic optimization algorithms Visualizing solution sets in the objective space • Single run → single approximation set • Interested in sets of mutually nondominated solutions called → approximation sets • Multiple runs multiple approximation sets

• Different from ordinary multidimensional solution sets Single run Three runs Ten runs • The focus of this tutorial 1.2 1.2 1.2 1 1 1 0.8 0.8 0.8 Challenges 0.6 0.6 0.6 0.4 0.4 0.4 • High dimension and large number of solutions 0.2 0.2 0.2 0 0 0 • Limitations of computing and displaying technologies 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Run 1 Run 3 Run 5 Run 7 Run 9 • Cognitive limitations Run 2 Run 4 Run 6 Run 8 Run 10 The Empirical Attainment Function (EAF) [15] or the Average Runtime Attainment Function (aRTA) [3] can be used in such cases 6 7 Introduction

This tutorial is not about • Visualization of a few solutions for decision making purposes (see [29]) • Visualization in the decision space A taxonomy of visualization • General multidimensional visualization methods not previously used on approximation sets methods

This tutorial covers • Visualization of entire sets in the objective space • Single approximation sets [1] • Repeated approximation sets [2, 3]

A taxonomy of visualization methods

Visualization in multiobjective optimization

Single Repeated approximation sets approximation sets Visualizing single

Individual solutions Set properties Showing Showing (visualizing solutions (visualizing solutions performance performance approximation sets independently from dependently from at a time over time each other) each other)

Showing original Showing transformed Showing individual Showing aggregated values of solutions values of solutions solution properties properties

Not optimization Optimization based based

9 Methodology Benchmark approximation sets

Two different sets that can be instantiated in any dimension [1] Evaluating and comparing visualization methods • Linear with a uniform distribution of solutions • No existing methodology for evaluating or comparing • Spherical with a nonuniform distribution of solutions (more at visualization methods the corners and less at the center) • Propose benchmark approximation sets (analog to benchmark • Sets are intertwined problems in multiobjective optimization) • Visualize the sets using different methods Size of each set • Observe which set properties are distinguishable after • 2-D: 50 solutions visualization • 3-D: 500 solutions • 4-D: only 300 solutions since most methods cannot handle more

10 11

Benchmark approximation sets Benchmark approximation sets

These two sets are not sufﬁcient for all purposes! Missing: 1 Linear Linear Spherical Spherical 1 • A set with knees 0.8 0.8 • A set with different relations between objectives, temporarily 0.6 1 0.6 using [13]: f3 0.8 f Original 2 0.4 2 0.6 f 0.4 40 0.4 0.2 0.2 35 30 0 25 0.2 20 0.2 15 0.4 10 0.6 5 0 0 f1 0.8 f f f f f f f f f f f f 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 11 12 1 f Sorted 1 40 • A 35 sequence of sets mimicking convergence in time 30 25 • … 20 (possibly others) 15 10 12 5 13 0 f11 f3 f10 f1 f2 f4 f6 f8 f12 f9 f5 f7 Desired properties of visualization methods Visualizing single approximation sets

Demonstration on the 4-D linear and spherical sets

• Preservation of the Individual solutions (Visualizing solutions independently from each • Dominance relation between solutions other) → Showing original values of solutions • Front shape Visualization in multiobjective optimization • Objective range • Scatter plot matrix • Distribution of solutions • Bubble chart Single Repeated approximation sets approximation sets • Robustness • Parallel coordinates [19] Individual solutions Set properties Showing Showing (visualizing solutions (visualizing solutions • Handling of large sets performance performance independently from dependently from • Radar chart at a time over time • Simultaneous visualization of multiple sets each other) each other) • Chord diagram [22], TBA Showing original Showing transformed Showing individual Showing aggregated • Scalability in number of objectives values of solutions values of solutions solution properties properties • Heat maps [32] • Simplicity Not optimization Optimization • Interactive decision maps [26] based based Demonstration on the 12-D approximation set • Showing relations between objectives

14 15

Scatter plot matrix Scatter plot matrix

1 1 1 2 3 4 Most often f 0.5 f 0.5 f 0.5

• Scatter plot in a 2-D space 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 • Matrix of all possible combinations f1 f1 f1 1 1 → m(m−1) • m objectives 2 different combinations 3 4

f 0.5 f 0.5

Alternatively 0 0 0 0.5 1 0 0.5 1 • Scatter plot in a 3-D space f2 f2 1 → m(m−1)(m−2) • m objectives 6 different combinations 4

Linear f 0.5 Spherical

0 0 0.5 1

f3 16 17 Scatter plot matrix Bubble chart

Linear 1 1 Spherical

f4 0.5 f4 0.5

0 0

0.5 0.5 4-D objective space f 1 f 1 2 0.5 1 0.5 1 0 1 0 f3 f3 • Similar to a 3-D scatter plot 1 1 • Fourth objective visualized with point size

f4 0.5 f3 0.5

0 0 5-D objective space 0.5 0.5 f 1 f 1 1 0.5 1 0.5 1 0 1 0 • Fifth objective visualized with colors f2 f2

Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈ ≈

18 19

Bubble chart Parallel coordinates

0.8 Linear Spherical • m objectives → m parallel axes 0.6 f3 0.4 • Solution represented as a polyline with vertices on the axes 0.2 • Position of each vertex corresponds to that objective value 0 • No loss of information 0.2

0.4 1.0 f1 0.6 0.8 1 0.8 0.8 0.6 0.6 0.4 1 0.2 f2 0.4

0.2

Preservation of the 0.0 Handling of Simultaneous f1 f2 f3 f4 dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈ ≈

20 21 Parallel coordinates Parallel coordinates

Original Linear Spherical 40 1 1 35 30 0.8 0.8 25 20 15 0.6 0.6 10 5 0.4 0.4 0 f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12

0.2 0.2 Sorted 40 35 0 0 30 f1 f2 f3 f4 f1 f2 f3 f4 25 20 15 Preservation of the Handling of Simultaneous 10 dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization 5 ≈ ≈ 0 f11 f3 f10 f1 f2 f4 f6 f8 f12 f9 f5 f7

22 23

Radar chart Radar chart

Linear Spherical

f2 f2

• Similar to parallel coordinates

f3 0 0.25 0.5 0.75 1 f1 f3 0 0.25 0.5 0.75 1 f1 • Additionally connects the two extreme coordinates • m objectives → m radial axes • Also called a spider chart, polar chart, star plot,…

f4 f4

Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈

24 25 Radar chart Heat maps

Original Sorted

f4 f1 f5 f2 f3 f10

f6 f4 f2 f3 • m objectives → m columns • One solution per row f7 f6 0 10 20 30 40 f 0 10 20 30 40 f 1 11 • Each cell colored according to objective value • No loss of information

f12 f7 f8 f8

f11 f5 f9 f12 f10 f9

26 27

Heat maps Heat maps

Original Linear Spherical 40 1.0 0.8 35 0.9 0.7 30 0.8 25 0.6 0.7 20 15 0.6 0.5 10 0.5 0.4 5 0.4 0.3 0 f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 0.3 0.2 0.2 Sorted 0.1 40 0.1 35 0.0 0.0 30 f f f f f f f f 1 2 3 4 1 2 3 4 25 20 15 Preservation of the 10 Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization 5 0 f11 f3 f10 f1 f2 f4 f6 f8 f12 f9 f5 f7

28 29 Interactive decision maps Interactive decision maps

Linear Spherical 1 1 The Edgeworth-Pareto hull (EPH) of an approximation set A contains 0.8 0.8 all points in the objective space that are weakly dominated by any solution in A. 0.6 0.6 2 2 f f 0.4 0.4 Interactive decision maps 0.2 0.2 • Visualize the surface of the EPH, not the actual approximation 0 0 set 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 f f • Plot a number of axis-aligned sampling surfaces of the EPH 1 1

f3 f3 • Color used to denote third objective 0 0.5 1 0 0.5 1 f4 = 0.5 f4 = 0.5 • Fixed value of the forth objective

Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈ ≈ 30 31

Visualizing single approximation sets Radial coordinate visualization

Individual solutions (Visualizing solutions independently from each Also called RadViz other) → Showing transformed values of solutions • Inspired from physics f • Radial coordinate visualization 1 • Objectives treated as anchors, [17, 39] Visualization in multiobjective optimization equally spaced around the • 3-D Radial coordinate visualization Single Repeated approximation sets circumferenceapproximation of sets a unit circle [18], TBA f2 f4 Individual solutions Set properties • Solutions attached to anchors with Showing Showing (visualizing solutions (visualizing solutions • Tetrahedron coordinates model [6] performance performance independently from dependently from ‘springs’ at a time over time each other) each other) • Polar plots [16], TBA • Spring stiffness proportional to the Showing original Showing transformed Showing individual Showing aggregatedobjective value • Hyper-radial visualization [9] values of solutions values of solutions solution properties properties f3 • Solution placed where the spring • Level diagrams [7, 8] Not optimization Optimization based based forces are in equilibrium • Prosections [1]

32 33 Radial coordinate visualization Tetrahedron coordinates model

f 1 Linear Spherical • Only for four objectives f4 • Similar to RadViz • Objectives treated as anchors, placed at the vertices of a regular f3 f2 tetrahedron • Solutions attached to anchors with ‘springs’ f1 f2 • Spring ﬂexibility proportional to the objective value

f4 • Solution placed where the f3

Preservation of the forces are in equilibrium Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈ 34 35

Tetrahedron coordinates model Hyper-radial visualization

f4 Linear Spherical

• Solutions preserve distance (hyper-radius) to the ideal point

f1 • Distances are computed separately for two subsets of objectives f2 • Indifference curves denote points with the same preference

Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈

36 37 Hyper-radial visualization Level diagrams

0.8 Linear Spherical 0.7

0.6

0.5 4 f

3 0.4 f

0.3 • m objectives → m diagrams

0.2 • Plot solutions with objective fi on the x axis and distance to the

0.1 ideal point on the y axis

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

f1f2

Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈

38 39

Level diagrams Level diagrams with asymmetric norm

1 1

0.8 0.8

0.6 0.6 • Compute an asymmetric norm a (very similar to the Iε+ 0.4 0.4 indicator) between any solution and the reference point 0.2 0.2 Linear Linear a = 0 ⇒ Distance to the ideal point Spherical Distance to the ideal point Spherical • the solution dominates the reference point 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 • a > 0 ⇒ the solution needs to be moved by a to dominate the f1 f2 reference point 1 1 0.8 0.8 • Use on our benchmark approximation sets 0.6 0.6 • The spherical set is used as the reference set 0.4 0.4 • a = 0 ⇒ the solution from the linear set dominates a solution 0.2 0.2 Linear Linear

Distance to the ideal point Spherical Distance to the ideal point Spherical 0 0 from the spherical set 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ⇒ f3 f4 • a > 0 the solution from the linear set needs to be moved by at least a to dominate one solution from the spherical set Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈

40 41 Level diagrams with asymmetric norm Prosections

Linear 0 0.05 0.1 0.15 0.2 0.25 • Visualize only part of the objective space 1 1 • Dimensionality reduction by projection of solutions in a section 0.8 0.8 • Need to choose prosection plane, angle and section width 0.6 0.6

0.4 0.4 Linear 1 0.2 0.2 Spherical Linear Spherical 1 0.8 Distance to the ideal point Spherical Distance to the ideal point Spherical 0 0 0.8 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.6 f f 0.6 1 1 2 f 3 3 0.8 f 0.4 f2 0.6 0.4 0.4 1 1 0.2 0.2 0 0.2 0.8 0.8 0.2 0.6 0.6 0.4 0 f1 0.6 0 0.2 0.4 0.6 0.8 1 0.8 0.4 0.4 f1f2 1 0.2 0.2 Before prosection After prosection Distance to the ideal point Spherical Distance to the ideal point Spherical 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

f3 f4 42 43

Prosections Visualizing single approximation sets

300 solutions 3000 solutions Linear Linear Spherical Spherical 1 1 Set properties (Visualizing solutions dependently from each 0.8 0.8 → → other) Showing individual solution-based propertiesVisualization inNot multiobjective optimization 0.6 1 0.6 1 f4 0.8 f4 0.8 optimization based 0.4 f3 0.6 0.4 f3 0.6 Single Repeated 0.4 0.4 approximation sets approximation sets 0.2 0.2 0.2 0.2 • Distance and distribution charts [5] 0 0 Individual solutions Set properties Showing Showing (visualizing solutions (visualizing solutions 0.2 0.2 • Pareto shells [38] performance performance independently from dependently from 0.4 0.4 at a time over time each other) each other) 0.6 0.6 • Hyper-space diagonal counting [4] f1f2 0.8 f1f2 0.8 1 1 Showing original Showing transformed Showing individual Showing aggregated • Treemaps [40], TBAvalues of solutions values of solutions solution properties properties

• Trade-off region maps [31], TBA Not optimization Optimization based based Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈

44 45 Distance and distribution charts Distance and distribution charts

1 Linear 0.8 Spherical

• Plot solutions against their distance to the Pareto front and 0.6

distance to other solutions 0.4 Distance • Distance chart 0.2 0 • Plot distance to the nearest non-dominated solution 0 50 100 150 200 250 300

• Distribution chart 1 Linear • Sort solutions w.r.t. ﬁrst objective 0.8 Spherical • Plot distances between consecutive solutions 0.6 0.4

• For the ﬁrst/last solution, compute distance to ﬁrst/last Distribution non-dominated solution 0.2 0 • k solutions → k + 1 distances 0 50 100 150 200 250 300

• All distances normalized to [0, 1] Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈

46 47

Pareto shells Pareto shells

• Use nondominated sorting to split solutions to Pareto shells

• Represent solutions in a graph Shell 0 Shell 1 • Connect dominated solutions to those that dominate them (we

show only one arrow per dominated solution) Linear Spherical

Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization

48 49 Hyper-space diagonal counting Hyper-space diagonal counting

Linear • Inspired by Cantor’s proof that shows |N| = |N2| = |N3| ... Spherical

...

(1, 3) (2, 3)

(1, 2) (2, 2) (3, 2) 6 80 (1, 1) (2, 1) (3, 1) (4, 1) 4

2 60

0 40 • Discretize each objective (choose a number of bins) f3f4 bins Number of solutions 0 20 20 • In the 4-D case 40 f f bins 60 0 • Enumerate the bins for objectives f1 and f2 1 2 80

• Enumerate the bins for objectives f3 and f4 • Plot the number of solutions in each pair of bins Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈ 50 51

Visualizing single approximation sets Principal component analysis

Set properties (Visualizing solutions dependently from each other) → Showing individual solution-based properties → Optimization based

• Principal component analysis [42] Visualization in multiobjective optimization

• Sammon mapping [33, 36] Single Repeated approximation sets approximation sets • Principal components are linear combinations of objectives that • Neuroscale [27, 11] maximize variance (and are uncorrelated with already chosen Individual solutions Set properties Showing Showing (visualizing solutions (visualizing solutions performance performancecomponents) • Multidimensional scalingindependently[39] from dependently from at a time over time each other) each other) • Isomap [34, 24] • They are the eigenvectors with the highest eigenvalues of the Showing original Showing transformed Showing individual Showing aggregated covariance matrix • Seriated heatmapsvalues[39 of solutions] values of solutions solution properties properties

Not optimization Optimization • Two-stage mapping [23] based based • Distance-based and dominance-based mappings [12]

52 53 Principal component analysis Sammon mapping

0.8

0.4 • A non-linear mapping • Aims to preserve distances between solutions ∗ 0 • dij distance between solutions xi and xj in the objective space

• dij distance between solutions xi and xj in the visualized space

-0.4 • Stress function to be minimized Second principal component ∑ ∗ − 2 (dij dij) Linear S = Spherical dij -0.8 i

Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈

54 55

Sammon mapping Sammon mapping

0.8 Linear Spherical

0.4 0.8

0.4

0 Third 0 coordinate -0.4 0.8

Second coordinate 0.4 -0.8 -0.8 0 -0.4 -0.4 Second 0 -0.4 coordinate First 0.4 -0.8 coordinate 0.8 Linear Spherical -0.8 -0.8 -0.4 0 0.4 0.8 Preservation of the Handling of Simultaneous First coordinate dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈

56 57 Neuroscale Neuroscale

1.2

0.8

0.4 • A non-linear mapping • Aims to minimize the same stress function as Sammon mapping 0

• Uses a radial basis function neural network to model the -0.4 projection Second coordinate

-0.8 Linear Spherical -1.2 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 First coordinate

58 59

Neuroscale Multidimensional scaling

Linear Spherical • Classical multidimensional scaling aims at preserving similarities between solutions

0.8 • Here, dominance distance is used to measure similarity

0.4 • Two solutions are similar if they share dominance relationships with a third solution Third 0 coordinate 1 ∑m [ -0.4 S(a, b; z) = I((a < z ) ∧ (b < z )) 0.8 m i i i i 0.4 i=1 -0.8 -0.8 0 -0.4 Second ∧ 0 -0.4 + I((ai = zi) (bi = zi)) 0.4 coordinate ] First 0.8 -0.8 coordinate 1.2 + I((ai > zi) ∧ (bi > zi)) 1 ∑ D(a, b) = (1 − S(a, b; z)) k − 2 Preservation of the ∈{ } Handling of Simultaneous z/ a,b dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈

60 61 Multidimensional scaling Isomap

0.4

0.2 • Assumes solutions lie on some low-dimensional manifold and the distances along this manifold should be preserved

0 • Creates a graph of solutions, where only the neighboring solutions are linked

Second coordinate -0.2 • The geodesic distance between any two solutions is calculated as the sum of Euclidean distances on the shortest path between -0.4 linear the two solutions spherical -0.4 -0.2 0 0.2 0.4 • Uses multidimensional scaling to perform the mapping based First coordinate on these distances

Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈

62 63

Isomap Isomap

0.8 Linear Spherical

0.8 0.4 0.4

Third 0 coordinate 0 -0.4 0.8 0.4 -0.8 -0.8 0

Second coordinate -0.4 Second 0 -0.4 coordinate -0.4 First 0.4 -0.8 coordinate 0.8

Linear Spherical -0.8 Preservation of the Handling of Simultaneous -0.8 -0.4 0 0.4 0.8 dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization First coordinate ≈ ≈ ≈

64 65 Seriated heatmaps Seriated heatmaps

Linear Spherical 300 300

250 250 • Heatmaps with rearranged objectives and solutions 200 200 • Similar objectives and similar solutions are placed together 150 150 • Ranks are used instead of actual objective values for a more uniform color usage 100 100

• Similarity can be computed using 50 50 • Euclidean distance 0 0 • Spearman’s footrule f1 f4 f2 f3 f1 f4 f2 f3 • Kendall’s τ metric

Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈

66 67

Two-stage mapping Two-stage mapping

0.6 Linear Steps Spherical

• Split solutions to nondominated and dominated solutions 0.5

• Compute r as the average norm of nondominated solutions 0.4 • Find a permutation of nondominated solutions that minimizes implicit dominance errors and sum of distances between 0.3 consecutive solutions 0.2 • First stage: distribute nondominated solutions on the circumference of a quarter-circle with radius r in the order of 0.1 the permutation and with distances proportional to their 0 distances in the objective space 0 0.1 0.2 0.3 0.4 0.5 0.6 • Second stage: map each dominated solution to the minimal Preservation of the Handling of Simultaneous point of all nondominated solutions that dominate it dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ ≈

68 69 Distance- and dominance-based mappings Distance- and dominance-based mappings

Both mappings Distance-based mapping Dominance-based mapping 2 1.6 Linear Linear • Use nondominated sorting to split solutions to Pareto shells 1.8 Spherical Spherical 1.4 • Project solutions onto the circumference of circles (with circle 1.6 1.2 radius proportional to front number) 1.4 1 1.2 Distance-based mapping Dominance-based mapping 1 0.8 0.8 • Tries to preserve closeness of • Aims at preserving 0.6 0.6 solutions dominance relations among 0.4 0.4 solutions 0.2 • Similarity between solutions 0.2

deﬁned as dominance • All x ≺ y can be shown 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 similarity correctly Preservation of the Handling of Simultaneous • Solution ordering using • Tries to minimize cases where dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization spectral seriation x ⊀ y is not shown correctly / / ≈ ≈

70 71

Visualizing single approximation sets Self-organizing maps

Set properties (Visualizing solutions dependently from each other) • Self-organizing maps (SOMs) are neural networks Visualization in multiobjective optimization → Showing aggregated properties Single Repeated • Nearby solutions are mapped to nearby neurons in the SOM approximation sets approximation sets • Self-organizing maps [21, 30] Individual solutions Set properties • A SOM can be visualized using the uniﬁed distance matrix Showing Showing (visualizing solutions (visualizing solutions performance performance independently from dependently from at a time over time • Aggregation trees [13] each other) each other) • Distance between adjacent neurons is denoted with color → Showing original Showing transformed Showing individual Showing aggregated • Similar neurons light color • MoGrams [35], TBA values of solutions values of solutions solution properties properties • Different neurons (cluster boundaries) → dark color Not optimization Optimization based based

72 73 Self-organizing maps Aggregation trees

• Binary trees that show relationships between objectives Linear Spherical • Iterative clustering of objectives based on their harmony • Computation of different types of conflict • Percentages quantify the conflict between objectives • Colors used to show type of conflict • global conflict (black) • local conflict on ’good’ values (red) • local conflict on ’bad’ values (blue)

• Can be used to sort objectives in other representations (parallel coordinates, radial charts, heat maps)

74 75

Aggregation trees Aggregation trees

f10 + f8 + f6 + f4 + f2 + f1 + f12 + f9 + f5 + f7 + f11 + f3 − 100%

Linear Spherical f10 + f8 + f6 + f4 + f2 + f1 + f12 + f9 + f5 + f7 − 25% f11 + f3 − 25%

f4 + f1 + f3 + f2 − 96.2711% f4 + f1 + f3 + f2 − 9692.2711%3733% f10 + f8 + f6 + f4 + f2 + f1 + f12 + f9 + f5 − 25% f7 f11 f3

Original − 40 f10 + f8 + f6 + f4 + f2 + f1 + f12 + f9 5% 35 f5 30 25 f4 + f1 − 72.5244% f3 + f2 − 70.0489% f4 + f1 − 7275.5244%9911% f3 + f2 − 7074.0489%0533% 20 15 f10 + f8 + f6 + f4 + f2 + f1 + f12 − 0% f9 10 5 0 f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12

Sorted f10 + f8 + f6 + f4 − 0% f2 + f1 + f12 − 0% 40 35 30 f4 f1 f3 f2 f4 f1 f3 f2 25 20 f10 + f8 − 0% f6 + f4 − 0% f2 + f1 − 0% f12 15 10 5 0 f11 f3 f10 f1 f2 f4 f6 f8 f12 f9 f5 f7 f10 f8 f6 f4 f2 f1

76 77 Visualizing repeated approximation sets

Visualization in multiobjective optimization Showing performance at a time Single Repeated Visualizing repeated • Empiricalapproximation Attainment sets Function (EAF) [15] approximation sets

Individual solutions Set properties approximation sets Showing Showing (visualizing solutions (visualizing solutions Showing performance over time performance performance independently from dependently from at a time over time each other) each other) • Average Runtime Attainment Function Showing original Showing(aRTA) transformed [3] Showing individual Showing aggregated values of solutions values of solutions solution properties properties

Not optimization Optimization based based

Empirical attainment function Empirical attainment function

Goal-attainment EAF values [15] • Approximation set A • Algorithm A, approximation sets A1, A2,..., Ar • A point in the objective space z is attained by A when z is • EAF of z is the frequency of attaining z by A , A ,..., A weakly dominated by at least one solution from A 1 2 r • Summary (or k%-) attainment surfaces 1.2 1.2 1.2 1 1 1 0.8 0.8 0.8 1

0.6 0.6 0.6 2/3 1/3 0.4 0.4 0.4 0 0.2 1st run 0.2 best 2nd run median 0.2 3rd run worst 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0 0.2 0.4 0.6 0.8 1 1.2

79 80 Empirical attainment function Visualization of 3-D EAF

Need to compute and visualize a large number (over 10 000) of Differences in EAF values [25] points/cuboids • Algorithm A, approximation sets A , A ,..., A 1 2 r Exact case • Algorithm B, approximation sets B1, B2,..., Br • Attainment surfaces: Visualization of facets • Visualize differences between EAF values • EAF values: Slicing [2] 1.2 1.2 • EAF differences: Slicing, Maximum intensity projection [41, 2] 1 1

1 0.8 0.8 1/2 Approximated case

0.6 0.6 0

0.4 0.4 -1/2 • Attainment surfaces: Grid-based sampling [20] 1st run of A 25%-att. surf. -1 0.2 2nd run of A 0.2 50%-att. surf. 1st run of B 75%-att. surf. • EAF values: Slicing, Direct volume rendering [10, 2] 2nd run of B 100%-att. surf. 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 • EAF differences: Slicing, Maximum intensity projection, Direct volume rendering

81 82

Benchmark approximation sets Exact 3-D EAF values and differences

Sets of approximation sets Slicing • 5 linear approximation sets with a uniform distribution of solutions (100 solutions in each) • Visualize cuboids intersecting the slicing plane • 5 spherical approximation sets with a nonuniform distribution • Need to choose coordinate and angle of solutions (100 solutions in each)

o 1 Linear ′ Spherical o 0.8 f3 0.6

0.4 ′ 0.2 z z 1 f2 0 0 1 ϕ 0.2 0.8 r 0.4 0.6 f1 0.6 0.4 0.8 0.2 1 0

83 84 Exact 3-D EAF values and differences Exact 3-D EAF differences

Slicing Maximum intensity projection 1.2 1.2 1.2 1 1 1 • Volume rendering method for spatial data represented by voxels 0.8 0.8 0.8

0.6 0.6 0.6 • Simple and efﬁcient 0.4 0.4 0.4 • No sense of depth, cannot distinguish between front and back 0.2 0.2 0.2

0 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Slice of Lin Slice of Sph Slice of Lin-Sph and ◦ ◦ ◦ at φ = 5 at φ = 5 Sph-Lin at φ = 5

1.2 1.2 1.2

1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4 Projection plane Viewpoint 0.2 0.2 0.2

0 0 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 © Christian Lackas Slice of Lin Slice of Sph Slice of Lin-Sph and ◦ ◦ ◦ at φ = 45 at φ = 45 Sph-Lin at φ = 45 85 86

Exact 3-D EAF differences Approximated attainment surfaces

Grid-based sampling Maximum intensity projection Repeat for all fifj, i < j (i.e. f1f2, f1f3 and f2f3): • Suitable for visualizing EAF differences (focus on large k × k f f differences) • Construct a grid on the plane i j • Sorting w.r.t. EAF differences (smaller to larger) • Compute intersections between the attainment surface and the axis-aligned lines on the grid • Plot on top of previous ones Median attainment surfaces

1 1

0.8 0.8

0.6 0.6 f3 f3 0.4 0.4

0.2 0.2 0 0 0 0 0 0 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.6 0.6 f1 0.6 0.6 f1 f2 0.8 0.8 f2 0.8 0.8 1 1 1 1

Linear Spherical 87 88 Approximated EAF values and differences Approximated 3-D EAF differences

Discretization into voxels Maximum intensity projection • Discretization of cuboids • Plots produced using Voreen [28, 37] • Discretization from the space of EAF values/differences • Some loss of information Slicing

1 1 1 1

0.8 0.8 0.8 0.8

0.6 0.6 0.6 0.6

0.4 0.4 0.4 0.4

0.2 0.2 0.2 0.2

0 0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Exact 643 voxels 1283 voxels 2563 voxels

89 90

Approximated 3-D EAF values and differences Approximated 3-D EAF differences

Direct volume rendering of Lin-Sph

Direct volume rendering • Volume rendering method for spatial data represented by voxels • A transfer function assigns color and opacity to voxel values 1/5 2/5 3/5 • Enables to see “inside the volume” • Requires the deﬁnition of the transfer function

4/5 5/5 1/5 and 5/5

91 92 Approximated 3-D EAF differences Approximated 3-D EAF values

Direct volume rendering of Sph-Lin

Direct volume rendering of Sph

1/5 2/5 3/5

1/5 and 5/5

4/5 5/5 1/5 and 5/5

93 94

Average Runtime Attainment Function Approximated aRTA values

aRTA value Two algorithms on the sphere-sphere problem [3] • Algorithm A run r times A B • All solutions that are nondominated at creation are recorded Algorithm Algorithm • aRTA(z) is the average number of evaluations needed to attain z

aRTA ratio • Algorithms A and B • Compute ratio between aRTA(z) values for A and B •

Visualization using grid-based sampling [3]

95 96 Approximated aRTA ratios

aRTA ratio between Algorithms A and B [3]

Summary

Summary Summary

Visualization in multiobjective optimization

Single Repeated approximation sets approximation sets

Individual solutions Set properties Showing Showing (visualizing solutions (visualizing solutions performance performance independently from dependently from at a time over time each other) each other)

Showing original Showing transformed Showing individual Showing aggregated values of solutions values of solutions solution properties properties

Not optimization Optimization based based

98 99 Summary Index

Methods for visualizing single approximation sets (page) • Visualization in multiobjective optimization useful for various • Aggregation trees (75) • Pareto shells (48) purposes • Bubble chart (19) • Polar plots (TBA) • Chord diagram (TBA) • Customized methods are needed to address the peculiarities of • Principal component analysis (53) • Distance and distribution charts (46) approximation set visualization • Prosections (43) • Distance- and dominance-based • Radar chart (24) • Many new approaches in the last years mappings (70) • Radial coordinate visualization (33) • Heat maps (27) • Sammon mapping (55) Cited papers on visualization in multiobjective optimization • Hyper-radial visualization (37) 7 • Scatter plot matrix (16) • Hyper-space diagonal counting (50) 6 • Self-organizing maps (73) • Interactive decision maps (30) 5 • Seriated heatmaps (66) • Isomap (63) 4 • Tetrahedron coordinates model (35) 3 • Level diagrams (39) • Trade-off region maps (TBA) 2 • MoGrams (TBA) Number of publications • Treemaps (TBA) 1 • Multidimensional scaling (61) • Two-stage mapping (68) 0 • Neuroscale (58) 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 • 3-D Radial coordinate visualization (TBA) Year • Parallel coordinates (21)

100 101

Index Acknowledgement

Methods for visualizing repeated approximation sets (page) The authors acknowledge the ﬁnancial support from the Slovenian Research Agency (research core funding No. P2-0209 and project No. Z2-8177 Incorporating real-world • Direct volume rendering • Slicing problems into the benchmarking of multiobjective optimizers). • Approximated EAF values (94) • Exact EAF values (85) • Approximated EAF differences • Approximated EAF values and This work is part of a project that has received funding from (92) differences (89) the European Union’s Horizon 2020 research and innovation • Grid-based sampling • Maximum intensity projection program under grant agreement No. 692286. • Approximated attainment • Exact EAF differences (87) surfaces (88) SYNERGY • Approximated EAF differences • Approximated aRTA values (96) Synergy for Smart Multi-Objective Optimization (90) • Approximated aRTA ratios (97) www.synergy-twinning.eu

102 103 References i

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