An Alternative Preference Relation to Deal with Many-Objective Optimization Problems Antonio López-Jaimes Japan Aerospace Exploration Agency Institute of Space and Astronautical Science December 17th, 2012
Outline of the Presentation Motivation The Chebyshev Achievement Function Approximating the Entire Pareto Front Experimental Results Conclusions December 17th, 2012 A. López 2/36
Outline of the Presentation Motivation The Chebyshev Achievement Function Approximating the Entire Pareto Front Experimental Results Conclusions December 17th, 2012 A. López 3/36
Difficulties of Many-objective Problems (1/3) f 2 � Number of nondominated vectors quickly increases z with the number of objectives: ( 2 k − 2 ) / ( 2 k ) . f 1 � Number of dominance resistant solutions (DRSs) increases with the objectives: � Solutions with good values in most objectives, but a poor value in at least one objective. December 17th, 2012 A. López 4/36
Motivation and Proposal (2/3) � Need for techniques with good scalability with respect to the number of objectives. � Both in terms of convergence, � and computational time. December 17th, 2012 A. López 5/36
Motivation and Proposal (2/3) � Need for techniques with good scalability with respect to the number of objectives. � Both in terms of convergence, � and computational time. � On the other hand, some researchers (Schütze, Lara, Coello 2011) have found that in some problems the difficulty does not significantly increases with the number of objectives. December 17th, 2012 A. López 5/36
Motivation and Proposal (3/3) Therefore... 1. We propose an alternative preference relation to deal with Many-objective problems. � Good convergence without sacrificing distribution. � Efficient and suitable for parallel implementation. � Easy to use it just by replacing the Pareto dominance relation. 2. We also study the effect of DRSs in the widely used DTLZ test problem suite and also in some WFG problems. December 17th, 2012 A. López 6/36
Outline of the Presentation Motivation The Chebyshev Achievement Function Approximating the Entire Pareto Front Experimental Results Conclusions December 17th, 2012 A. López 7/36
Chebyshev Achievement Function � The new preference relation is based on the Chebyshev achievement function . � An achievement function is a special scalarizing function , s : R k → R , parameterized by a reference point, z ref . z 1 ( z 1 − z ref 1 ) 2 i } � Unlike Euclidean distance, max { z i − z ref 2 ) 2 2 ) 2 ( z 2 − z ref z 2 ( z 2 − z ref ( z 1 − z ref 1 ) 2 z ref December 17th, 2012 A. López 8/36
Chebyshev Achievement Function � The new preference relation is based on the Chebyshev achievement function . � An achievement function is a special scalarizing function , s : R k → R , parameterized by a reference point, z ref . z 1 max { z i − z ref i } i } � Unlike Euclidean distance, max { z i − z ref z 2 � the Chebyshev function takes only the maximum difference between a vector z and a reference point z ref . z ref December 17th, 2012 A. López 8/36
Augmented Chebyshev Achievement Function Definition (Wierzbicki 1980, Ehrgott 2005) The augmented Chebyshev achievement function is defined by s ( z | z ref ) = i = 1,..., k { λ i ( z i − z ref max i ) } � The reference point: aspiration levels for each objective. � The weight vector, λ , normalize the objective values. December 17th, 2012 A. López 9/36
Augmented Chebyshev Achievement Function Definition (Wierzbicki 1980, Ehrgott 2005) The augmented Chebyshev achievement function is defined by k s ( z | z ref ) = i = 1,..., k { λ i ( z i − z ref � λ i ( z i − z ref max i ) , i ) } + ρ i = 1 � The reference point: aspiration levels for each objective. � The weight vector, λ , normalize the objective values. � The second term helps to remove weakly Pareto optimal solutions . December 17th, 2012 A. López 9/36
Augmented Chebyshev Achievement Function Definition (Wierzbicki 1980, Ehrgott 2005) The augmented Chebyshev achievement function is defined by k s ( z | z ref ) = i = 1,..., k { λ i ( z i − z ref � λ i ( z i − z ref max i ) , i ) } + ρ i = 1 � The reference point: aspiration levels for each objective. � The weight vector, λ , normalize the objective values. � The second term helps to remove weakly Pareto optimal solutions . � Any solution of the Pareto front can be generated by s ( z | z ref ) . December 17th, 2012 A. López 9/36
The Chebyshev Preference Relation � Combines Pareto dominance + Chebyshev function . � Defines a Region of Interest (RoI) around a reference point. solutions with s ( z | z ref ) � s min + δ , best achievement size of the RoI e b y s h h C e v � Solutions outside RoI are compared with Chebyshev value. z 1 Pareto � Solutions in RoI are compared z min z 2 using Pareto dominance. z ref � Solutions in RoI dominate RoI solutions outside RoI . s min δ December 17th, 2012 A. López 10/36
Application of the Chebyshev Relation Unfeasible reference point 1 Feasible space 0.9 0.8 Optimal PF NSGA − II 0.7 0.6 * = (0.7057, 0.5055) 0.5 z ∞ 0.4 z ref = ( 0.5 , 0.3) 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 December 17th, 2012 A. López 11/36
Application of the Chebyshev Relation Feasible reference point December 17th, 2012 A. López 12/36
Outline of the Presentation Motivation The Chebyshev Achievement Function Approximating the Entire Pareto Front Experimental Results Conclusions December 17th, 2012 A. López 13/36
Approximating the Entire Pareto Front To approximate the entire Pareto front we used as reference point the approximation of the ideal point maintained by the Chebyshev relation. Points always kept � Use a stringent criterion for solutions far from the Pareto front for guiding the solutions towards the ideal point, z min � and Pareto dominance for solutions near the Pareto front z ⋆ for covering the entire Pareto front. s min δ = 0.9 December 17th, 2012 A. López 14/36
Replacing Pareto Dominance by Another Relation In order to improve efficiency... � Instead of the Pareto dominance, another secondary preference relation can be used instead. � Here we used a preference relation based on the binary ǫ -indicator. December 17th, 2012 A. López 15/36
Replacing Pareto Dominance by ǫ -indicator Relation Epsilon Indicator, I ǫ I ǫ ( z 2 , z 1 ) = min ǫ ∈ R { z 2 i � ǫ + z 1 i for i = 1, . . . , k } “Extent” by which a solution dominates another one: ǫ < 0 ǫ > 0 z 2 ≺ z 1 dominates by ǫ = 0 needs to dominate z 1 � Using a fitness function, F ǫ ( z ) , based on the ǫ -indicator we can define a preference relation: Epsilon relation, R ǫ A solution z 1 “dominates” z 2 wrt the relation R ǫ if and only if: F ǫ ( z 1 ) > F ǫ ( z 2 ) . December 17th, 2012 A. López 16/36
Replacing Pareto Dominance by ǫ -indicator Relation Two examples of fitness functions ( P is the current population): Maximim fitness function (Balling 2003) F min ǫ ( z 1 ) = z 2 ∈ P \{ z 1 } I ǫ ( z 2 , z 1 ) min Sum of I ǫ values (Zitzler and Künzli 2004) F sum ( z 1 ) = � − I ǫ ( { z 2 } , { z 1 } ) � � − exp ǫ z 2 ∈ P \{ z 1 } ǫ < 0 ǫ > 0 z 2 ≺ z 1 dominates by ǫ = 0 needs to dominate z 1 December 17th, 2012 A. López 17/36
Outline of the Presentation Motivation The Chebyshev Achievement Function Approximating the Entire Pareto Front Experimental Results Analysis of Dominance Resistant Solutions Conclusions December 17th, 2012 A. López 18/36
Algorithms and Parameter Settings (1/2) � NSGA2 used as baseline optimizer: � Pareto dominance. � Chebyshev relation with I sum . ǫ � Chebyshev relation with I min ǫ . � Test Problems: Problem Features DTLZ1, DTLZ3 Multiple local Pareto fronts. DTLZ4 Nonuniform solution density. DTLZ7, WFG2 Disconnected PF opt . WFG6 Nonseparable MOP. � Objectives were varied from 3 to 14 objectives. � Number of distance-related variables: 20 (5 for DTLZ1). � Number of position-related variables: k − 1. December 17th, 2012 A. López 19/36
Algorithms and Parameter Settings (2/2) � Performance Indicators: Generational Distance Problem GD = ( � z � / | P | ) − 0.5 DTLZ1 GD = ( � z � 2 / | P | ) − 1 DTLZ2-4 � GD g = g ( x ) − 1 DTLZ7 GD x in design space WFG2, WFG6 � Inverted Generational distance. � Epsilon Indicator. � Other parameters: Parameter Value Population size 200 Generations 200 Crossover rate 0.9 Mutation rate 1 /n Crossover index 20 Mutation index 20 December 17th, 2012 A. López 20/36
Generational Distance GD values for DTLZ1 and WFG6 using from 3 to 14 objectives. 450 0.95 NSGA2 400 NSGA2 0.9 NSGA2−I sum NSGA2−I sum Generational distance (average) ε Generational Distance (average) 350 WFG6 ε NSGA2−I min 0.85 NSGA2−I min DTLZ1 ε 300 ε 0.8 250 0.05 200 0.04 0.75 0.03 150 0.7 0.02 100 0.01 0.65 50 0 3 4 6 8 10 12 14 0 3 4 6 8 10 12 14 3 4 6 8 10 12 14 Num. of objectives Num. of objectives Remarkable improvement Marginal improvement December 17th, 2012 A. López 21/36
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