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Visualization in Multiobjective Optimization

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Visualization in Multiobjective Optimization Final version Visualization in Multiobjective Optimization 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 2 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] 8 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 • Objective range • Scatter plot matrix • Distribution of solutions • Bubble chart • Robustness • Parallel coordinates [19] • Handling of large sets • Radar chart • Simultaneous visualization of multiple sets • Chord diagram [22], TBA • Scalability in number of objectives • Heat maps [32] • Simplicity • Interactive decision maps [26] 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 Visualization in multiobjective optimization • Scatter plot in a 2-D space 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 Single Repeated • Matrix of all possibleapproximation combinations sets approximation sets f1 f1 f1 1 1 m(m−1) Individual solutions ! Set properties • m objectives different combinations Showing Showing (visualizing solutions 2 (visualizing solutions performance performance 3 4 independently from dependently from f 0.5 f 0.5 at a time over time each other) each other) Alternatively 0 0 Showing original Showing transformed Showing individual Showing aggregated 0 0.5 1 0 0.5 1 values of solutions values of solutions solution properties properties • Scatter plot in a 3-D space f2 f2 1 − − ! Notm(m optimization1)(m 2) Optimization • m objectives based6 differentbased 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 5 ≈ 3 ≈ 3 ≈ 3 5 3 18 19 Bubble chart Parallel coordinates 1 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 5 ≈ 3 ≈ 3 ≈ 3 5 3 20 21 Parallel coordinates Parallel coordinates Linear Spherical 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 f1 f2 f3 f4 f1 f2 f3 f4 Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ 5 3 ≈ 3 5 5 3 3 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 Original • Additionally connects the two extreme coordinates 40 35 m ! m 30 • objectives radial axes 25 20 • Also called a spider chart, polar chart, star plot,… 15 10 5 0 f1 f2 f3 f4 f4 f5 f6 f7 f8 f9 f10 f11 f12f4 Sorted 40 35 Preservation of the Handling of Simultaneous dominance 30 front shape objective distribution Robustness Scalability Simplicity large sets visualization relation 25 range of solutions ≈ 20 5 3 ≈ 3 5 5 3 3 15 10 24 5 25 0 f11 f3 f10 f1 f2 f4 f6 f8 f12 f9 f5 f7 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 5 5 3 5 3 5 5 3 3 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.
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