Introduction Analysis and Tempering Summary/Conclusions/What next? Getting Lost or Getting Trapped: On the Effect of Moves to Incomparable Points in Multiobjective Hillclimbing Workshop on Theoretical Aspects of Evolutionary Multiobjective Optimization - Current Status and Future Trends GECCO Michael Emmerich, Andr´ e Deutz, Johannes Kruisselbrink, Rui Li July 8, 2010 Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Getting Lost or Getting Trapped: On the Effect of Moves to Incomparable Points in Multiobjective Hillclimbing Workshop on Theoretical Aspects of Evolutionary Multiobjective Optimization - Current Status and Future Trends GECCO Michael Emmerich, Andr´ e Deutz, Johannes Kruisselbrink, Rui Li July 8, 2010 Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Table of Contents Introduction: EA Moves to Incomparable Solutions Analysis of Algorithms and Taming Single Point Schemes (1+1)-IMEA restricting incomparibility, tempering divergence (1+1)-LIMEA Population Based Schemes divergent behavior (2+2)-IMEA Restricting Incomparability, Tempering Divergence for Population Based EAs commonly used EAs Summary/Conclusions/What next? Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Search Algorithms on (Strict) Partial Orders ◮ In approximating Pareto Fronts EAs very often allow moves to incomparable solutions ◮ Incomparability relationship between solutions is not transitive ◮ Study questions associated with moves to incomparable solutions ◮ Divergent or cycling behavior of algorithms with incomparable moves ◮ Can occur in elitist schemes which disallow moves to dominated solutions ◮ Measures to counteract this? ◮ Exploitation versus exploration; tension between divergence and exploitation Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Edgeworth-Pareto Dominance ◮ (Minimization) The point p (Pareto) dominates any green point I.e., it is strictly better in at least one of the two objectives ◮ Aka: p strictly dominates any point in the green area. ◮ Notation: ≺ . ◮ Is a strict partial order (irreflexive, asymmetric and transitive). Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Weak Dominance ◮ p weakly dominates any green point ◮ I.e., (( p strictly dominates any green point except itself ) and ( p dominates itself)) ◮ I.e., p is better or equal to any green point in all objectives. ◮ Notation: � . ◮ Is a partial order (reflexive, antisymmetric, and transitive Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Incomparibility The point p and any blue point are incomparable – is not transitive. Notation: � Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Dominated Any grey point (and the point p ) (weakly) dominates p Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Summary ◮ p weakly dominates any green point and itself ◮ Any grey point (and the point p ) (weakly) dominates p ◮ Weak dominance is a p.o. ◮ Any blue point and p are incomparable (is intransitive) Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Remark on terminology Aside: Strict dominance and weak dominance are terms used by J.Bader, D.Brockhoff, S.Welten, and E.Zitzler in On Using Populations of Sets in Multiobjective Optimization , EMO 2009, LNCS5467 Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? dominance induced by a map and partial order on the codomain of the map f : A ⊆ R n → R m then f induces an important binary relation which is reflexive and transitive (and in general not antisymmetric) on R n as follows x , x ′ ∈ A : x � x ′ ⇔ f ( x ) � f ( x ) (For that matter any partial order on a set B and a map f : A → B , give rise to a binary relation which is reflexive and transitive on A . (In general this induced order is not a partial order (no anti-symmetry).)) Can define in the usual way Pareto Front , Efficient Set etc etc Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Getting worse in case of moves to incomparable solutions I Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Getting worse in case of moves to incomparable solutions II Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Getting worse in case of moves to incomparable solutions III Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Zooming in on Algorithms and Taming Divergence ◮ Single point analysis: ◮ (1+1)-IMEA (Incomparable Move Evolutionary Algorithm) ◮ (1+1)-LIMEA (only certain incomparable moves are allowed via utility function) ◮ Population based methods: ◮ (2+2)-IMEA ◮ Tempering influence of incomparability for population based schemes ◮ NSGA-II ◮ SMS-EMOA Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? (The) Example Problem, Model Landscape We will use the following 2D multiobjective problem to study the behavior of (1+1)-IMEA, (1+1)-LIMEA, (2+2)-IMEA Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Single Point Schemes if q ≺ x t − 1 then Algorithm 1 (1 + 1)-IMEA x t = q input: x 0 , output: x t , t = else 1 , 2 , . . . if x t − 1 ≺ q then t = 1 x t = x t − 1 while t < T max do else q = mutate( x t − 1 ) if x t − 1 � q or x t − 1 ∼ q For Selection See Right then Column x t = output x t UniformRandom { q , x t − 1 } t = t + 1 end if end while end if end if Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Single Point Schemes Transition Matrix, K ,of the MC of the Example Problem and the (1+1)-IMEA x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 2 1 1 x 1 0 0 0 0 0 0 0 0 0 3 6 6 1 5 1 1 x 2 0 0 0 0 0 0 0 0 8 8 8 8 1 5 1 1 x 3 0 0 0 0 0 0 0 0 8 8 8 8 1 2 1 x 4 0 0 0 0 0 0 0 0 0 6 3 6 1 2 1 x 5 0 0 0 0 0 0 0 0 0 6 3 6 1 1 3 1 1 x 6 0 0 0 0 0 0 0 10 10 5 10 10 1 1 5 1 x 7 0 0 0 0 0 0 0 0 8 8 8 8 1 1 5 1 x 8 0 0 0 0 0 0 0 0 8 8 8 8 1 1 3 1 1 x 9 0 0 0 0 0 0 0 10 10 5 10 10 1 2 1 x 10 0 0 0 0 0 0 0 0 0 6 3 6 1 2 1 x 11 0 0 0 0 0 0 0 0 0 6 3 6 1 2 1 x 12 0 0 0 0 0 0 0 0 0 6 3 6 Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Single Point Schemes limit behavior of the (1+1)-IMEA on the example problem ◮ The MC specified by the transition matrix K is regular ◮ Fixed row tuple w of K (i.e, w such that w = w K ) is a strictly positive probability vector and theory tells you w j gives the probability of being in x j in the long run (where w j is the j-th entry of w ), and it is independent of the starting state. Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Single Point Schemes Hence, ending up in the Pareto Front has a chance of roughly 21% Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Single Point Schemes How to deal with incomparability? ◮ We have two extremes: disallow moves to incomparable solutions (if the child dominates the parent, it is chosen, otherwise choose the parent) and on the other hand if parent and child are incomparable can possibly choose the child. ◮ Of course, the first extreme entails that you cannot diverge from optimal solutions, but it has the well-known disadvantage of needing to overcome traps. ◮ Suggestion: middle of the road; allow for exploration, that is, allow for some moves to incomparable solutions Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Single Point Schemes How to deal with incomparability? I ◮ Work with a utility function with constant positive weights, w i , i = 1 , . . . , m ; w i > 0, that is, m m x ≺ w x ′ iff � � w i f i ( x ′ ) w i f i ( x ) < i =1 i =1 ◮ Respects the Pareto dominance x ≺ x ′ ⇒ x ≺ w x ′ Natural Computing Leiden University, The Netherlands
Introduction Analysis and Tempering Summary/Conclusions/What next? Single Point Schemes How to deal with incomparability? II Natural Computing Leiden University, The Netherlands
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