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Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveFitness Lands apes and Graphs:Multimodularity, Ruggedness and NeutralitySébastien VerelDOLPHIN team - INRIA Lille-Nord EuropeI3S laboratory - University of Ni e-Sophia Antipolis / CNRS, Fran ehttp://www.i3s.uni e.fr/∼verelJuly, 8 2012S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveOutline of this tutorial1 Introdu tion : ontexte, goals, related theoreti al works, de�nition2 Fitness lands apes of multimodal, and neutral problems,a ase study3 Lo al optima network4 Fitness lands apes for ontinous and multobje tive problems

S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveOptimization frameworkInputsSear h spa e : Set of all feasible solutions,SObje tive fun tion : Quality riteriumf : S → IRGoalFind the best solution a ording to the riteriums⋆ = argmax f


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveOptimization frameworkInputsSear h spa e : Set of all feasible solutions,SObje tive fun tion : Quality riteriumf : S → IRGoalFind the best solution a ording to the riteriums⋆ = argmax fBut, sometime, the set of all best solutions, good approximation ofthe best solution, good 'robust' solution...S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveContexteBla k box S enarioWe have only {(s0, f (s0)), (s1, f (s1)), ...} given by an "ora le"No information is either not available or needed on the de�nition ofobje tive fun tionObje tive fun tion given by a omputation, or a simulationObje tive fun tion an be irregular, non di�erentiable, non ontinous, et .S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveContexteBla k box S enarioWe have only {(s0, f (s0)), (s1, f (s1)), ...} given by an "ora le"No information is either not available or needed on the de�nition ofobje tive fun tionObje tive fun tion given by a omputation, or a simulationObje tive fun tion an be irregular, non di�erentiable, non ontinous, et .(Very) large sear h spa e for dis rete ase ( ombinatorialoptimization), i.e. NP- omplete problemsS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveSear h algorithmsPrin iple Enumeration of the sear h spa eA lot of ways to enumerate the sear h spa eUsing random sampling : Monte Carlo te hni sLo al sear h te hni s :S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveSear h algorithmsPrin iple Enumeration of the sear h spa eA lot of ways to enumerate the sear h spa eUsing random sampling : Monte Carlo te hni sLo al sear h te hni s :If obje tive fun tion f has no propertie : random sear hIf not... S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveFitness lands apes : MotivationsWhy using �tness lands apes ?To analyse the stru ture of the sear h spa eTo study problem (sear h) di� ulty in ombinatorialoptimisation :information on runtime for a given problem and a lass of LSTo design e�e tive sear h algorithmsL. Barnett, U. Sussex, DPhil Diss. 2003"the more we know of the statisti al properties of a lass of �tnesslands apes, the better equipped we will be for the design ofe�e tive sear h algorithms for su h lands apes"S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveFitness lands apes in biology

Biologi al s ien e :Wright 1930 [36℄Biologi al evolution :a metaphori al uphillstruggle a ross a "�tnesslands ape"mountain peaks representhigh "�tness", or ability tosurvive,valleys represent low �tness.evolution pro eeds :population of organismsperforms an "adaptive walk"S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveFitness lands apes in biology In biology :Modelisation of spe iesevolutionUsed to model dynami alsystems :statisti al physi ,mole ular evolution,e ology, et S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveFitness lands apes in biology2 sides for Fitness Lands apes :Powerful metaphor : most profound on ept in evolutionarydynami sgive pi tures of evolutionary pro essbe areful of misleading pi tures : "smooth lands ape withoutnoise"Quantitative on ept : predi t the evolutionary pathsQuasispe ies equation : mean �eld analysis with di�erentialequationsSto hasti pro ess : markov hainNetwork analysis S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveIn ombinatorial optimizationFitness lands ape (S,N , f ) :S : set of admissible solutions,N : S → 2S : neighborhoodfun tion,f : S → IR : �tness fun tion.

2S is the set of sets : N (s) ⊂ SS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveFitness lands apes in Evolutionary Computation2 sides for Fitness Lands apes :Powerful metaphor : most profound on eptgive pi tures of the sear h dynami :"if the �tness lands apes have big valleys, I an use thisalgorithm"be areful of misleading pi tures : set of smooth mountainsQuantitative on ept : predi t the evolutionary dynami Quasispe ies equation : mean �eld analysis with di�erentialequationsSto hasti pro ess : markov hainNetwork analysis S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveWhat is a neighborhood ?Neighborhood fun tion :N : S → 2SSet of "neighbor" solutions asso iatedto ea h solution

N (x) = {y ∈ S | IP(y = op(x)) > 0}S. Verel Fitness lands apes and graphs


N (x) = {y ∈ S | IP(y = op(x)) > 0}orN (x) = {y ∈ S | IP(y = op(x)) > ǫ}



N (x) = {y ∈ S | IP(y = op(x)) > 0}orN (x) = {y ∈ S | IP(y = op(x)) > ǫ}orN (x) = {y ∈ S | d(y , x) ≤ 1}S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveExample of neighborhood : bit stringsSear h spa e : S = {0, 1}NAlgorithm : simple GA,hill- limbing, or simulatedannealing, et .N (01101) = {01100,01111,01001,00101,11101,}

Important !De�nition of neighborhoood must bebased on the lo al sear h operatorused in the algorithmNeighborhood ⇔ OperatorN (x) =

{y ∈ S | dHamming(y , x) = 1}S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveExample of neighborhood : permutations

Traveling Salesman Problem :�nd the shortest tour whi h ross one time every town

Sear h spa e :S = {σ | σ permutations }Algorithm : simple EAoperator : 2-opt

N (x) ={y ∈ S | IP(y = op2opt (x)) > 0}S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveExample of neighborhood

Traveling Salesman Problem :�nd the shortest tour whi h ross one time every town

Sear h spa e :S = {σ | σ permutations }Algorithm : simple EAoperators : 2-opt and 3-opt

N (x) ={y ∈ S | IP(y = op2opt (x)) >0 or IP(y = op3opt (x)) > 0}S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveExample of neighborhood : memeti algorithmsAlgorithm : memeti algorithm, EA + operator hill- limbingN (x) = {y ∈ S | y = opHC (x)}


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveExample of neighborhood : memeti algorithmsAlgorithm : memeti algorithm, EA + operator hill- limbingN (x) = {y ∈ S | y = opHC (x)}Algorithm : memeti algorithm, EA + operator hill- limbingand bit-�ip mutation2 possibilities :Study 2 lands apes :one for HC operator, one for bit-�ip mutationStudy 1 lands ape :

N (x) = {y ∈ S | y = opHC (x) or IP(y = opbit−�ip(x)) > ǫ}S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveExample of neighborhood : memeti algorithmsAlgorithm : memeti algorithm, EA + operator hill- limbingN (x) = {y ∈ S | y = opHC (x)}Algorithm : memeti algorithm, EA + operator hill- limbingand bit-�ip mutation2 possibilities :Study 2 lands apes :one for HC operator, one for bit-�ip mutationStudy 1 lands ape :

N (x) = {y ∈ S | y = opHC (x) or IP(y = opbit−�ip(x)) > ǫ}It depends on what you want to knowS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveGoal of the �tness lands apes study"Geometry" (features) of �tness lands ape⇒ dynami s of a lo al sear h algorithmGeometry is linked to the problem di� ulty :If there are a lot of lo al optima, the probability to �nd theglobal optimum is lower.If the �tness lands ape is �at, dis overing better solutions israre.What is the best sear h dire tion in the lands ape ?Study of the �tness lands ape featuresallows to studythe performan e of sear h algorithmsS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveGoal of the �tness lands apes study1 To ompare the di� ulty of two sear h spa es :One problem with 2 (or more) possible odings : (S1,N1, f1)and (S2,N2, f2)di�erent oding, mutation operator, �tness fun tion, et .Whi h one is easier to solve ?


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveGoal of the �tness lands apes study1 To ompare the di� ulty of two sear h spa es :One problem with 2 (or more) possible odings : (S1,N1, f1)and (S2,N2, f2)di�erent oding, mutation operator, �tness fun tion, et .Whi h one is easier to solve ?2 To sele t the algorithm :analysis of features of the lands apeWhi h algorithm an I use or design ?S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveGoal of the �tness lands apes study1 To ompare the di� ulty of two sear h spa es :One problem with 2 (or more) possible odings : (S1,N1, f1)and (S2,N2, f2)di�erent oding, mutation operator, �tness fun tion, et .Whi h one is easier to solve ?2 To sele t the algorithm :analysis of features of the lands apeWhi h algorithm an I use or design ?3 To tune the parameters :o�-line analysis of stru ture of �tness lands apeWhi h is the best mutation operator ? the size of thepopulation ? et .S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveGoal of the �tness lands apes study1 To ompare the di� ulty of two sear h spa es :One problem with 2 (or more) possible odings : (S1,N1, f1)and (S2,N2, f2)di�erent oding, mutation operator, �tness fun tion, et .Whi h one is easier to solve ?2 To sele t the algorithm :analysis of features of the lands apeWhi h algorithm an I use or design ?3 To tune the parameters :o�-line analysis of stru ture of �tness lands apeWhi h is the best mutation operator ? the size of thepopulation ? et .4 To ontrol the parameters during the run :on-line analysis of stru ture of �tness lands apeWhi h is the optimal mutation rate a ording to theestimation of stru ture ?S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveFitness lands apes point of viewFL = (Sol., Neighbors, Fitness)s_t

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Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveFitness lands apes point of viewFL = (Sol., Neighbors, Fitness)s_t

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Put prob. from your heuristi :s_t+1s_t

0

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No parti ular heuristi , heuristi sbased on the same neighborhoodrelationSample the neighborhood tohave information on lo alfeatures of the sear h spa eFrom lo al information :dedu e some global featureslike general shape of sear hspa e, "di� ulty", et .S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveGoal of the �tness lands apes studyStudy of the geometry of the lands apeallows tostudy the di� ulty, and design a good optimisation algorithmFitness lands ape is a graph (S,N , f ) where the nodes have avalue (�tness) : an be "pi tured" as a "real" lands apeTwo main geometries have been studied :multimodal and ruggednessneutral S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesMultimodal Fitness lands apesLo al optima s∗ (maximization) :no neighbor solution with higher �tness value∀s ∈ N (s∗), f (s) < f (s∗)

Search space

Fitness


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesMultimodal Fitness lands apesLo al optima s∗ (maximization) :no neighbor solution with higher �tness value∀s ∈ N (s∗), f (s) ≤ f (s∗)

Search space

Fitness


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesMultimodal Fitness lands apesAdaptive walk : (s0, s1, . . .) where si+1 ∈ N (si ) and f (si ) < f (si+1)Hill-Climbing (HC) algorithmChoose initial solution s ∈ Srepeat hoose s ′ ∈ N (s) su h that f (s ′) = maxx∈N (s) f (x)if f (s) < f (s ′) thens ← s ′end ifuntil s is a Lo al optimumBasin of attra tion of s∗ :{s ∈ S | HillClimbing(s) = s∗}.S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesMultimodal Fitness lands apes

Search space

Fitness

Optimisation di� ulty :number and size of attra tivebasins (Garnier et al [10℄)The idea :if the size of attra tive basinof global optima is relatively"small"the problem is di� ult tooptimizeThe measure :Length of adaptive walks(distribution, avg, et .)S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesWalking on �tness lands apesSearch space

Fitness

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Step�tness vs. step of a random walk(example of max-SAT problem)Random walk : (s1, s2, . . .)su h that si+1 ∈ N (si) andequiprobability on N (si )Fitness seems to be very" haoti "Analysis the �tness duringthe random walk as a signal


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesRugged/smooth �tness lands apes 350

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Auto orrelation of time series of�tnesses (f (s1), f (s2), . . .) alonga random walk (s1, s2, . . .) [35℄ :ρ(n) =

E [(f (si )− f )(f (si+n)− f )]var(f (si ))auto orrelation length τ = 1ρ(1)small τ : rugged lands apelong τ : smooth lands apeS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesResults on rugged �tness lands apes (Stadler 96 [26℄)Problem parameter ρ(1)symmetri TSP n number of towns 1− 4nanti-symmetri TSP n number of towns 1− 4n−1Graph Coloring Problem n number of nodes 1− 2α(α−1)n

α number of olorsNK lands apes N number of proteins 1− K+1NK number of epistasis linksRuggedness de reases with the size of thoses problems :small variation has less e�e t on the �tness valuesWeinberger, Stadler, Whitley, Sutton : elementary lands apesS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesFitness Distan e Correlation (FDC) (Jones 95 [15℄)Correlation between distan e to global optimum and �tness 0.35

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DistanceClassi� ation based on experimental studies :ρ < −0.15, easy optimizationρ > 0.15, hard optimization−0.15 < ρ < 0.15, unde ided zoneS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral Fitness Lands apesNeutral theory (Kimura ≈ 1960 [17℄)Theory of mutation and random driftA onsiderable number of mutations have no e�e ts on �tnessvaluesgenotypes space

Fitness plateausneutral degreeneutral networks[S huster 1994 [25℄,RNA folding℄S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral Fitness Lands apesCombinatorial optimizationRedundant problem (symetries, ...) (Goldberg 87 [12℄)Problem �not well� de�ned or dynami environment (Torres 04[14℄)genotypes space

Fitness Appli ative problems :Robot ontrolerCir uit designgeneti programmingProtein Foldinglearning problemsS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutrality and di� ultyIn our knowledge, there is no de�nitive answerabout neutrality / problem hardnessCertainly, it is dependent on the nature of neutrality of the�tness lands ape⇒ Sharp des riptionof the geometry of neutral �tness lands apes is needed


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutrality and di� ultyWe know for ertain that :No information is better than Bad information :Hard trap fun tions are more di� ult thanneedle-in-a-haysta k fun tionsGood information is better than No informationS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutrality and di� ultyWe know for ertain that :No information is better than Bad information :Hard trap fun tions are more di� ult thanneedle-in-a-haysta k fun tionsGood information is better than No informationWhen there is No information :you should have a good method to �nd it !S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesIn the followingDes ription of neutral �tness lands apes :Neutral sets :set of solutions with the same �tnessNeutral networks :add neighborhood information


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral sets : Density Of StatesSearch space

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FitnessDensity of states (D.O.S.)Introdu e in physi s(Rosé 1996 [24℄)Optimization(Belaidouni, Hao 00 [4℄)S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral sets : Density Of States 0

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FitnessDensity of states (D.O.S.)Informations given :Performan e of randomsear hTail of the distribution is anindi ator of di� ulty :the faster the de ay, theharder the problemBut do not are about theneighborhood relationS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral sets : Fitness CloudFitness f(s)

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(S,F , IP) : probability spa eop : S → S sto hasti operator of the lo al sear hX (s) = f (s)Y (s) = f (op(s))Fitness Cloud of opConditional probability densityfun tion of Y given XS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesFitness loud : Measure of evolvability

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EvolvabilityAbility to evolve : �tnessin the neighborhood ompared to the �tness ofthe solutionProbability of �ndingbetter solutionsAverage �tness ofbetter neighborsolutionsAverage and standarddeviation of �tnessesS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesFitness loud : Comparaison of di� ulty

Fitness f(s)

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Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesFitness loud : Comparaison of di� ulty

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Operator 1 > Operator 2Be ause Average 1 more orrelated to �tnessLinked to auto orrelationAverage is often a line :See works on ElementaryLands apes (D. Wihtleyand others)See Negative SlopeCoe� ient (NSC)S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesFitness loudPredi tion of �tness (CEC 2003)

Fitness f(s)

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f0 f1 f2 f3

Approximation (onlyapproximation) of the�tness value after fewsteps of lo al operatorIndi ation on the qualityof the operatorS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral �tness lands apesNeutral sets (done) :set of solutions with the same �tness⇒ No stru tureFitness loud (done) :Bivariate density (f (s), f (op(s)))⇒ Neighborhood relation between neutral setsNeutral networks (to be done) :⇒ Neighborhood stru ture into the neutral sets : Graph


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral networks (S huster 1994 [25℄)

genotypes space

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Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesDe�nitions (simple version)Neutral neighborhoodof s is the set of neighbors whi h have the same �tness f (s)

Nneut(s) = {s ′ ∈ N (s) | f (s) = f (s ′)}Neutral degree of sNumber of neutral neighbors : nDeg(s) = ♯(Nneut (s)− {s}).S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesDe�nitionsTest of neutralityisNeutral : S × S → {true, false}For example, isNeutral(s1, s2) is true if :f (s1) = f (s2).|f (s1)− f (s2)| ≤ 1/M with M is the sear h population size.|f (s1)− f (s2)| is under the evaluation error.Neutral neighborhoodof s is the set of neighbors whi h have the same �tness f (s)

Nneut(s) = {s ′ ∈ N (s) | isNeutral(s, s ′)}Neutral degree of sNumber of neutral neighbors : nDeg(s) = ♯(Nneut (s)− {s}).S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesDe�nitionsNeutral walkWneut = (s0, s1, . . . , sm)for all i ∈ [0,m − 1], si+1 ∈ N (si )for all (i , j) ∈ [0,m]2 , isNeutral(si , sj) is true.Neutral Networkgraph G = (N,E )N ⊂ S : for all s and s ′ from V , there is a neutral walkbelonging to V from s to s ′ ,(s1, s2) ∈ E if they are neutral neighbors : s2 ∈ Nneut(s1)A �tness lands ape is neutralif there are many solutions with high neutral degree.S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral Networks (NN) : Inside Metri sClassi al graph metri s :Size of NN :number of nodes of NN,Neutral degree distribution :measure of the quantity of "neutrality"Auto orrelation of neutral degree (Bastolla 03 [3℄) :during neutral random walk omparaison with random graph,measure of the orrelation stru ture of NNS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral Networks : Inside Metri s

3

2

6

6

4

2

Size : 15 solutionsDistribution of sizeoverall lands apesNeutral degreedistributionNeutral Degree

Fre

quen

cy0 1 2 3 4 5 6 7

01

23

45


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral Networks : Inside Metri s

3

2

6

6

4

2Size : 15 solutionsDistribution of sizeoverall lands apesNeutral degreedistributionAuto orrelation ofneutral degree :random walk on NNauto orrelation ofdegreesS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral Networks : Outside Metri sNeutral Network

Fitness

Neutral random walk

S0S3S2

S1

Number of portals(exits toward bettersolutions), spreadingof portalsAuto orrelation ofevolvability [33℄ :auto orrelation of thesequen e(evol(s0), evol(s1), . . .).S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral Networks : Outside Metri s

0.2

0.3

0.0

0.0

0.01

0.1

Auto orrelation ofevolvability :Evolvabilityevol = avg �tness inthe neighborhoodAuto orrelation of(evol(s0), evol(s1), . . .).Informations :if high orrelation⇒ "easy"(you an use thisinformation)if low orrelation⇒ "di� ult"S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesFrom analysis to design : a ase studyPhD work of Marie-Eléonore Marmion (De ember 2011)An example :From �tness lands apes analysis to design of e� ient lo al sear h


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesA ase study : The Permutation Flowshop S hedulingProblem (FSP)N jobs, M ma hinesPro essing time of ea h job an be di�erent on ea h ma hineEa h job an be pro essed on at most one ma hineEa h ma hine an pro ess at most one job at a timeJob order is the same on every ma hine :Representation = Permutation⇒ Makespan minimizationS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeighborhood relationInsertion operatorNeighborhood size : (N − 1)2Something strange in the state-of-art lo al sear h (Iterated Greedy)So a �tness analysis...S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutralityQuestion 1Is there some neutrality in the problem ?Neutral degree (ratio) of lo al optimaSample : set of lo al optimaN 20 50 100 2005 65 (18%) 792 (33%) 3920 (40%)10 7 ( 2%) 96 ( 4%) 784 ( 8%) 6732 (17%)20 4 ( 1%) 24 ( 1%) 98 ( 1%) 396 ( 1%)S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutralityQuestion 1Is there some neutrality in the problem ?Neutral degree (ratio) of lo al optimaSample : set of lo al optimaN 20 50 100 2005 65 (18%) 792 (33%) 3920 (40%)10 7 ( 2%) 96 ( 4%) 784 ( 8%) 6732 (17%)20 4 ( 1%) 24 ( 1%) 98 ( 1%) 396 ( 1%)Yes ! There is some neutrality.Continue the analysis to �nd information...S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral NetworksQuestion 2The neutral networks have some stru ture ?Auto orrelation of neutral degrees 0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250

ρ(1)

Number of Jobs

5 Machines10 Machines20 Machines


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutral NetworksQuestion 2The neutral networks have some stru ture ?Auto orrelation of neutral degrees 0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250

ρ(1)

Number of Jobs

5 Machines10 Machines20 Machines

Yes ! Neutral Networks are not random graphs.Continue the analysis to �nd information...S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesPortals, exitsQuestion 3Can we rea h a portal ?Typology of neutral networksT1 T2 T3

fitness

?No T1 for N = 50, 100, 200< 20% T1 and T2 for N = 20, < 3% T2 for N = 50, 100, 200> 97% T3 S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesPortals, exitsQuestion 3Can we rea h a portal ?Typology of neutral networksT1 T2 T3

fitness

?No T1 for N = 50, 100, 200< 20% T1 and T2 for N = 20, < 3% T2 for N = 50, 100, 200> 97% T3 Yes ! There is some portals.Continue the analysis to �nd information...S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutralityQuestion 4Is it easy to �nd a portal ?Average number of steps before a portalDuring random neutral walksN 20 50 100 200M = 5 17 33 3410 10 14 17 3020 6 6 6 6S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesNeutralityQuestion 4Is it easy to �nd a portal ?Average number of steps before a portalDuring random neutral walksN 20 50 100 200M = 5 17 33 3410 10 14 17 3020 6 6 6 6Yes ! The portals are losed to lo al optima.S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesDesign e� ient Lo al sear hNILS : Neutral Iterated Lo al Sear hLo al Sear h : First-improvement Hill-Climbing (until lo aloptimum)Perturbation : to es ape from plateauNeutral moves (until maximum number of steps)Ki k move (from the IG lo al sear h)⇒ Very ompetitive vs. the state-of-art IG, and we understand why.


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesSummary of metri sNeutral degrees distribution :"How neutral is the �tness lands ape ?"Auto orrelation of neutral degrees : network �stru ture�High

0.20.0 0.35 0.6 1.0

Middle strongLowPortals, exits :Es aping of plateau to �nd better solutionsAuto orrelation of evolvability :information on the links between NNS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesBasi Methodology of �tness lands apes analysisDensity of States : pure random sear h, initialization ?Length of adaptive walks : multimodality ?Auto orrelation of �tness : ruggedness ?Neutral Degree Distribution : neutrality ?Fitness Cloud : Quality of the operator, evolvability ?Fitness Distan e Correlation from best knownNeutral walks and evolvability : neutral information ?S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesBasi Methodology of �tness lands apes analysisDensity of States : pure random sear h, initialization ?Length of adaptive walks : multimodality ?Auto orrelation of �tness : ruggedness ?Neutral Degree Distribution : neutrality ?Fitness Cloud : Quality of the operator, evolvability ?Fitness Distan e Correlation from best knownNeutral walks and evolvability : neutral information ?... be reative from your algorithm and problem point of view... be areful on the omputed measures : one measure is notenough, and must be very well understandS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Multimodal and rugged �tness lands apesNeutral �tness lands apesSofware to perform �tness lands ape analysisFramework ParadisEO 1.3http://paradiseo.gforge.inria.fr/newWebsite/index.php?n=Do .Tutorialsand tutorials :http://paradiseo.gforge.inria.fr/newWebsite/index.php?n=Do .TutorialsmoAuto orrelationSampling<Neighbor> sampling(randomInit,neighborhood,fullEval,in rementalEval,nbStep);sampling();sampling.fileExport(str_out); S. Verel Fitness lands apes and graphs

http://paradiseo.gforge.inria.fr/newWebsite/index.php?n=Doc.Tutorials

http://paradiseo.gforge.inria.fr/newWebsite/index.php?n=Doc.Tutorials

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesMotivation and general idea : Levels of des riptionFDC, autocorrelation, etc.High level

Medium level

Low level

Local Optima Network

Fitness landscape

One metric

local optima, basins of attraction

SolutionsFitness lands apes : based on an huge number of solutionsOne metri : based on one real number, or urve to at h allthe omplexityLo al optima Network : based on lo al optimaS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesOverview and MotivationBring the tools of omplex networks analysis to the study thestru ture of ombinatorial �tness lands apesGoals : Understand problem di� ulty, design e�e tive heuristi sear h algorithmsMethodology : Extra t a network that represents the lands ape(Inspiration from energy lands apes (Doye, 2002 )1)Verti es : lo al optimaEdges : a notion of adja en y between basinsCondu t a network analysisRelate (exploit ?) network features to sear h algorithm design1J. P. K. Doye, The network topology of a potential energy lands ape : astati s ale-free network., Phys. Rev. Lett., 88 :238701, 2002.S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesSmall − world networks (Watts and Strogatz, 1998)Neither ordered nor ompletely randomNodes highly lustered yet path length is smallNetwork topologi al measures :C : lustering oe� ient, measure of lo al densityl : shortest path length global measure of separationS ale − free networks (Barabasi and Albert, 1999)The distribution of the number of neighbours (the degreedistribution) is right − skewed with a heavy tailMost of the nodes have less-than-average degree, whilst asmall fra tion of hubs have a large number of onne tionsDes ribed mathemati ally by a power-lawS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesEnergy surfa e and inherent networks (Doye, 2002)a Model of 2D energy surfa eb Contour plot, partition ofthe on�guration spa e intobasins of attra tionsurrounding minima lands ape as a networkInherent network :Nodes : energy minimaEdges : two nodes are onne ted if the energy barrierseparating them is su� iently low (transition state)S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesBasins of attra tion in ombinatorial optimisationExample of small NK lands ape with N = 6 and K = 2Bit strings of length N = 626 = 64 solutionsone point = one solution


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesBasins of attra tion in ombinatorial optimisationExample of small NK lands ape with N = 6 and K = 2Bit strings of length N = 6Neighborhood size = 6Line between points =solutions are neighborsHamming distan es betweensolutions are preserved(ex ept for at the border ofthe ube)S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesBasins of attra tion in ombinatorial optimisationExample of small NK lands ape with N = 6 and K = 2Color represent �tness value• high �tness• low �tness


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesBasins of attra tion in ombinatorial optimisationExample of small NK lands ape with N = 6 and K = 2Color represent �tness value• high �tness• low �tness→ point towards thesolution with highest �tnessin the neighborhoodExer ise :Why not make a Hill-Climbingwalk on it ?S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesBasins of attra tion in ombinatorial optimisationExample of small NK lands ape with N = 6 and K = 2Ea h olor orresponds toone basin of attra tionBasins of attra tion areinterlinked and overlappedBasins have no "interior"


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesBasins of attra tion in ombinatorial optimisationExample of small NK lands ape with N = 6 and K = 2

Basins of attra tion are interlinked and overlapped !Most neighbours of a given solution are outside its basinS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesLo al optima network

fit=0.7046

0.185

0.65

0.29

0.270.4

0.055 0.05

0.33

0.76

fit=0.7657 fit=0.7133 Nodes : lo al optimaEdges : transitionprobabilitiesS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesBasin of attra tionHill-Climbing (HC) algorithmChoose initial solution s ∈ Srepeat hoose s ′ ∈ N (s) su h that f (s ′) = maxx∈N (s) f (x)if f (s) < f (s ′) thens ← s ′end ifuntil s is a Lo al optimumBasin of attra tion of s∗ :{s ∈ S | HillClimbing(s) = s∗}.S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapeslo al optima networkLo al optima networkNodes : set of lo al optima S∗Edges : notion of onne tivity between basins of attra tioneij between i and j if there is at least a pair of neighbours siand sj ∈ N (si ) su h that si ∈ bi and sj ∈ bj (GECCO 2008[21℄)weights wij is atta hed to the edges, a ount for transitionprobabilities between basins (ALIFE 2008 [34℄, Phys. Rev. E2008 [30℄, CEC 2010)Es ape edges : eij between i and j ifBasins j whi h an be rea hed from the lo al optima i(Arti� ial Evolution 2011 [31℄)S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesBasin edges : Weights of edgesFrom ea h s and s ′ , p(s → s ′) = IP(s ′ = op(s))For example, S = {0, 1}N and bit-�ip operatorif s ′

∈ N (s) , p(s → s ′

) = 1Nif s ′

6∈ N (s) , p(s → s ′

) = 0



∈ N (s) , p(s → s ′

) = 1Nif s ′

6∈ N (s) , p(s → s ′

) = 0Probability that a on�guration s ∈ S has a neighbor in abasin bj p(s → bj) =∑s′∈bj p(s → s ′)



∈ N (s) , p(s → s ′

) = 1Nif s ′

6∈ N (s) , p(s → s ′

) = 0Probability that a on�guration s ∈ S has a neighbor in abasin bj p(s → bj) =∑s′∈bj p(s → s ′)wij : Total probability of going from basin bi to basin bj is theaverage over all s ∈ bi of the transition prob. to s ′ ∈ bj :p(bi → bj ) =1

♯bi ∑s∈bi p(s → bj)⇒ lo al optima network : weighted oriented graphS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesEs ape edgesEs ape edgesGiven distan e fun tion d , and positive integer D > 0.eij between LOi and LOj :if it exists a solution s su h that : d(s, LOi ) ≤ D andh(s) = LOj .Weight wij of eij :♯{s ∈ S | d(s, LOi) ≤ D and h(s) = LOj}normalized by ♯{s ∈ S | d(s, LOi ) ≤ D}.

⇒ lo al optima network : weighted oriented graphS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesBasin edges vs. Es ape edgesi

j

pi(s)pj(s')

p(s->s')

Basin edges i

j

Es ape edgesS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesNK �tness lands apes : ruggedness and epistasisNK -lands apes : Model of problemsN size of the bit-stringsK from 0 to N − 1, NK lands apes an be tuned from smooth torugged (easy to di� ult respe tively) :K = 0 no orrelations, f is an additive fun tion, and there is asingle maximumK = N − 1 lands ape ompletely random, the expe tednumber of lo al optima is 2NN+1Intermediate values of K interpolate between these twoextreme ases and have a variable degree of epistasis (i.e. geneintera tion) S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesMethodsExtra ted and analysed networksN ∈ {14, 16, 18},K ∈ {2, 4, . . . ,N − 2,N − 1}30 random instan es for ea h aseMeasures :Statisti s on basins sizes and �tness of optimaNetwork features : lustering oe� ient, shortest path to theglobal optimum, weight distribution, disparity, boundary ofbasins


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesRemember : Multimodal Fitness lands apesSearch space

Fitness

Optimisation di� ulty :number and size of attra tivebasins (Garnier et al [10℄)The idea :if the size of attra tive basinof global optima is relatively"small"the problem is di� ult tooptimizeS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesGlobal optimum basin size versus K 1e-05

0.0001

0.001

0.01

0.1

1

2 4 6 8 10 12 14 16 18

Nor

mal

ized

siz

e of

the

glob

al o

ptim

a’s

basi

n

K

N=16N=18

Size of the basin orresponding tothe global maximum for ea h KTrend : the basin shrinksvery qui kly with in reasingK.for higher K, more di� ultfor a sear h algorithm tolo ate the basin of attra tionof the global optimum


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesAnalysis of basins : basin size 0.1

1

10

100

1000

0 2000 4000 6000 8000 10000 12000

cum

ulat

ive

dist

ribut

ion

size of basin

exp.regr. line

Cumulative distribution of basinssizes for N = 18 and K = 4Trend : small number oflarge basin, large number ofsmall basinLog-normal umulativedistribution :not uniform !Slope of orrelation in reaseswith KWhen K large : basin sizesare nearly equalsthe distribution be omesmore uniformS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesAnalysis of basins : basin size 1e-20

1e-15

1e-10

1e-05

1

100000

0 100 200 300 400 500 600 700 800 900 1000

cum

ulat

ive

dist

ribut

ion

size of basin

K=2K=4K=6K=8

K=10K=12K=14K=16K=17

Trend : small number oflarge basin, large number ofsmall basinlog-normal umulativedistributionslope of orrelation in reaseswith Kwhen K large : basin sizesare nearly equalsS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesAnalysis of basins : �tness vs. basin size 1

10

100

1000

10000

0.5 0.55 0.6 0.65 0.7 0.75 0.8

basi

n of

attr

actio

n si

ze

fitness of local optima

exp.regr. line

Correlation �tness of lo al optimavs. their orresponding basinssizesTrend : lear positive orrelation between the�tness values of maxima andtheir basins' sizesThe highest, the largestOn average, the globaloptimum easier to �nd thanone other lo al optimumBut more di� ult to �nd, asthe number of lo al optimain reases exponentially within reasing KS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesGeneral network statisti sWeighted lustering oe� ientlo al density of the network w (i) =1si(ki − 1) ∑j ,h wij + wih2 aijajhahiwhere si =

∑j 6=i wij , anm = 1 if wnm > 0, anm = 0 if wnm = 0 andki =∑j 6=i aij .Disparitydishomogeneity of nodes with a given degreeY2(i) =

∑j 6=i (wijsi )2S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesGeneral network statisti s N = 16K ♯ nodes ♯ edges Cw Y d2 3315 516358 0.960.0245 0.3260.0579 56144 17833 91292930 0.920.0171 0.1370.0111 12686 46029 417914690 0.790.0154 0.0840.0028 17038 89033 933844394 0.650.0102 0.0620.0011 194210 1, 47034 1621394592 0.530.0070 0.0500.0006 206112 2, 25432 2279122670 0.440.0031 0.0430.0003 207114 3, 26429 2907322056 0.380.0022 0.0400.0003 203115 3, 86833 3212032061 0.350.0022 0.0390.0004 2001Clustering Coe� ient : For high K, transition between agiven pair of neighboring basins is less likely to o urDisparity : For high K the transitions to other basins tend tobe ome equally likely, an indi ation of the randomness of thelands ape S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesWeights distribution : transition probability between basins 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0001 0.001 0.01 0.1 1

P(w

ij=W

)

W

K=2K=4

K=10K=14K=17

distribution of the networkweights wij for outgoing edgeswith j 6= i in log-x s ale, N = 18Weights are smallFor high K the de ay isfasterLow K has longer tailsOn average, the transitionprobabilities are higher forlow K (less lo al optima)


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesWeight distribution remain in the same basin 0

5e-05

0.0001

0.00015

0.0002

0.00025

0.01 0.1 1

P(w

ii =

W)

Wii

K=2K=6

K=14

Average weight wii a ording tothe parameter N and KQuestion :Is it easy to es ape a basin ?Weights to remains in thesame are large ompare towij with i 6= jwii are higher for low KEasier to leave the basin forhigh K : high "natural"explorationBut : number of lo aloptima in reases fast with KS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesInterior and border size 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

2 4 6 8 10 12 14 16 18

aver

age

of th

e m

ean

size

K

N=14N=16N=18

Average of the mean size ofbasins interiorsQuestion :Do basins look like a "montain"with interior and border ?solution is in the interior if allneighbors are in the same basin


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesInterior and border size 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

2 4 6 8 10 12 14 16 18

aver

age

of th

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ean

size

K

N=14N=16N=18

Average of the mean size ofbasins interiorsQuestion :Do basins look like a "montain"with interior and border ?solution is in the interior if allneighbors are in the same basinAnswerInterior is very smallNearly all solution are in theborderS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesShortest path length between lo al optima 0

50

100

150

200

250

300

2 4 6 8 10 12 14 16 18

aver

age

path

leng

th

K

N=14N=16N=18Average distan e (shortest path)between nodes

Question :Are the basins "far" from ea hother ?In rease with N (♯ of nodesin reases exponentially)For a given N, in rease withK up to K = 10, thenstagnatesS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesShortest path length to global optima 0

50

100

150

200

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2 4 6 8 10 12 14 16 18

aver

age

path

leng

th to

the

optim

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K

N=14N=16N=18Average path length to the globaloptimum from all the other basins

Question :Is the global optimum basin isfar ? More relevant foroptimisationIn rease steadily within reasing KS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesSummary on lo al optima networkMedium level of des ription : proposed hara terization of ombinatorial lands apes as networksa new model for lands ape analysisNew �ndings about basin's stru ture :sizes, �tness vs. size, et .Related some network features to sear h di� ulty


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tive Complex networksDe�nitionsLON of NK landsapesSummary on lo al optima networkRelated some network features to sear h di� ultyWeights edges of NK-lands apes LONE�e tive frequen ies of TS and ILS from one basin to anotherWednesday, July 11th, 14 :32 "Lo al Optima Networks and thePerforman es of ILS", F. Daolio et al.S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveContinuous problemsFitness lands ape � FiL (X , N , f )X = IRnN de�ned by gaussian mutationf : IRn → IRP. Caamano, A. Prieto, J.A. Be erra, F. Bellas and R.J. Duro, Real ValuedMultimodal Fitness Lands ape Chara terization for Evolution, Le ture Notes inComputer S ien es, vol. 6443, pp. 567-574, Springer-Verlag, 2010


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveContinuous �tness lands apesMain di� ultyWhi h step size to de�ne the neighborhood ? 10−3 ?Di�erent step size, di�erent lands ape ?Use the standard de�nition of LO ?∃ǫ > 0, ∀x , |x − x⋆| ≤ ǫ⇒ f (x) ≤ f (x⋆)Toward measure like ||f (x)− f (y)||/||x − y || ?But is it better than derivate fun tion ?


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveContinuous �tness lands apesMain di� ultyWhi h step size to de�ne the neighborhood ? 10−3 ?Di�erent step size, di�erent lands ape ?Use the standard de�nition of LO ?∃ǫ > 0, ∀x , |x − x⋆| ≤ ǫ⇒ f (x) ≤ f (x⋆)Toward measure like ||f (x)− f (y)||/||x − y || ?But is it better than derivate fun tion ?Personal fealingMain di�erent between ombinatorial and ontinuous FiLNo sear h "dire tion" in ombinatorial problemProbability to improve the urrent solutionShape of the attra tion basinsS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveMultiobje tive optimizationWhat is the stru ture of a mutiobje tive sear h spa e ?Have you seen the multiobje tive tutorial by Dimo Bro kho� ?S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveMultiobje tive optimizationMultiobje tive optimization problemX : set of feasible solutions in the de ision spa eM > 2 obje tive fun tions f = (f1, f2, . . . , fM) (to maximize)Z = f (X ) ⊆ IRM : set of feasible out ome ve tors in theobje tive spa e

Decision space

x2

x1 Objective space

f

f1

2


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveDe�nitionPareto dominan e relationA solution x ∈ X dominates a solution x ′ ∈ X (x ′ ≺ x) i�∀i ∈ {1, 2, . . . ,M}, fi(x) > fi (x ′)∃j ∈ {1, 2, . . . ,M} su h that fj(x) > fj(x ′)

Decision space

x2

x1 Objective space

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f1

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non-dominated vector

non-dominated solution

vector

dominated


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tivePareto set, Pareto front

Decision space

x2

x1 Objective space

f

f1

2 Pareto front

Pareto optimalset

Paretooptimalsolution


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveSet-based multiobje tive �tness lands apeDe�nitionA set-based multiobje tive �tness lands ape is de�ned as atriplet (Σ, N, I) su h that :Σ ⊂ 2X is a set of feasible solution-sets(where X is the set of feasible solutions)N : Σ→ 2Σ is a neighborhood relation between solution-setsI : Σ→ IR is a unary quality indi ator,i.e. a �tness fun tion measuring the quality of solution-setsState-of-the-art tools from single-obje tive FiL an be reusedS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveSet-based sear h spa eIllustrative examplesPopulation-based approa hesΣ = {σ ∈ 2X : |σ| = µ}, where µ is the population sizeBounded ar hive-based approa hesΣ = {σ ∈ 2X : |σ| 6 µ}, where µ is the max ar hive sizeDominan e-based approa hes (with mutually n-d solutions)Σ = {σ ∈ 2X : ∀s, s ′ ∈ σ, s 6≺ s ′}Bounded ar hive- + Dominan e-based approa hesΣ = {σ ∈ 2X : |σ| 6 µ and ∀s, s ′ ∈ σ, s 6≺ s ′}No restri tionΣ = 2X. . . S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveSet-based neighborhood relationIllustrative examplesSet-level repla ement neighborhoodσ′ = σ ∪ {s ′} \ {s} su h that s ∈ σ, s ′ ∈ N (s)→ |N(σ)| 6 |σ| ·

∑s∈σ|N (s)|Set-level insertion neighborhood

σ′ = σ ∪ {s ′} su h that s ∈ σ, s ′ ∈ N (s)→ |N(σ)| 6

∑s∈σ|N (s)|Set-level deletion neighborhood

σ′ = σ \ {s} su h that s ∈ σ→ |N(σ)| = |σ|. . . (re ombination) S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveSet-based �tness fun tionIllustrative example : hypervolume (IH)Compliant with the (weak) Pareto dominan e relation

→ A ≺ B ⇒ IH(A) 6 IH(B)A single parameter : the referen e pointMinimal solution-set maximizing IH→ subset of the Pareto optimal setarg maxσ∈Σ IH(σ)


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveMain Fitness Lands ape measures an be usedRuggedness, auto orrelation→ Based on random walk samplingIs the �tness of neighboring solutions random?Auto orrelation length τ = 1

ρ, with ρ orrelation (f (s), f (s ′)), s ′ ∈ N (s)small τ : rugged lands ape large τ : smooth lands apeLo al optima

→ Based on adaptive walk samplingWhat is the number of lo al optima, and the size of basins ?(s0, s1, . . . , sL) where si+1 ∈ N (si ), f (si ) < f (si+1)small L : high multi-modality large L : low multi-modality(many lo al optima, small basin size) (few lo al optima, large basin size)S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveWhat is the stru ture of the sear h spa e ?Fa tsVery large sear h spa eVery large neighborhoodHigh omputational ost to ompute the �tnessS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveWhat is the stru ture of the sear h spa e ?Fa tsVery large sear h spa eVery large neighborhoodHigh omputational ost to ompute the �tnessOpen questions( ombinatorial) multiobje tive problems :Es aping from lo al loptima ?Exploring the neighborhood is an optimization problem ?S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveSummary on �tness lands apesFitness lands ape is a representation ofsear h spa enotion of neighborhood�tness of solutions


Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveSummary on �tness lands apesFitness lands ape is a representation ofsear h spa enotion of neighborhood�tness of solutionsGoal :lo al des ription : �tness between neighbor solutionsRuggedness, lo al optima, �tness loud, neutral networks, lo aloptima networks...and to dedu e global features :Di� ulty !To de ide (design, tune or ontrol) a good hoi e of therepresentation, operator and �tness fun tionS. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveOpen questionsHow to ontrol the parameters and/or operators of thealgorithm with the lo al des ription of �tness lands ape ?Can �tness lands ape des ribe the dynami s of a population ofsolutions ?Links between neutrality and �tness di� ulty ?Whi h intermediate des ription shows relevant properties ofthe optimization problem a ording to the lo al sear hheuristi ?Integration of the FL tools into the open framework paradisEOhttp://paradiseo.gforge.inria.frS. Verel Fitness lands apes and graphs

http://paradiseo.gforge.inria.fr

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveL. Barnett.Ruggedness and neutrality - the NKp family of �tnesslands apes.In C. Adami, R. K. Belew, H. Kitano, and C. Taylor, editors,ALIFE VI, Pro eedings of the Sixth International Conferen eon Arti� ial Life, pages 18�27. ALIFE, The MIT Press, 1998.Lionel Barnett.Net rawling - optimal evolutionary sear h with neutralnetworks.In Pro eedings of the 2001 Congress on EvolutionaryComputation CEC2001, pages 30�37, COEX, World TradeCenter, 159 Samseong-dong, Gangnam-gu, Seoul, Korea, 27-302001. IEEE Press.U. Bastolla, M. Porto, H. E. Roman, and M. Vendrus olo.Statis al properties of neutral evolution.Journal Mole ular Evolution, 57(S) :103�119, August 2003.S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveMeriema Belaidouni and Jin-Kao Hao.An analysis of the on�guration spa e of the maximal onstraint satisfa tion problem.In PPSN VI : Pro eedings of the 6th International Conferen eon Parallel Problem Solving from Nature, pages 49�58,London, UK, 2000. Springer-Verlag.P. Collard, M. Clergue, and M. Defoin Platel.Syntheti neutrality for arti� ial evolution.In Arti� ial Evolution : Fourth European Conferen e AE'99,pages 254�265. Springer-Verlag, 2000.Sele ted papers in Le ture Notes in Computer S ien es 1829.J. C. Culberson.Mutation- rossover isomorphisms and the onstru tion ofdis rimination fun tion.Evolutionary Computation, 2 :279�311, 1994.S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveJ. P. K. Doye.The network topology of a potential energy lands ape : a stati s ale-free network.Phys. Rev. Lett., 88 :238701, 2002.J. P. K. Doye and C. P. Massen.Chara terizing the network topology of the energy lands apesof atomi lusters.J. Chem. Phys., 122 :084105, 2005.Ri ardo Gar ia-Pelayo and Peter F. Stadler.Correlation length, isotropy, and meta-stable states.Physi a D, 107 :240�254, 1997.Santa Fe Institute Preprint 96-05-034.Josselin Garnier and Leila Kallel.E� ien y of lo al sear h with multiple lo al optima.SIAM Journal on Dis rete Mathemati s, 15(1) :122�141, 2002.S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveP. Git ho� and G. Wagner.Re ombination indu ed hypergraphs : A new approa h tomutation-re ombination isomorphism, 1996.David E. Goldberg and Philip Segrest.Finite markov hain analysis of geneti algorithms.In ICGA, pages 1�8, 1987.M. Huynen.Exploring phenotype spa e through neutral evolution.Journal Mole ular Evolution, 43 :165�169, 1996.E. Izquierdo-Torres.The role of nearly neutral mutations in the evolution ofdynami al neural networks.In J. Polla k and al, editors, Ninth International Conferen e ofthe Simulation and Synthesis of Living Systems (Alife 9), pages322�327. MIT Press, 2004.S. Verel Fitness lands apes and graphs

Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tiveT. Jones.Evolutionary Algorithms, Fitness Lands apes and Sear h.PhD thesis, University of New Mexi o, Albuquerque, 1995.S. A. Kau�man.The Origins of Order.Oxford University Press, New York, 1993.M. Kimura.The Neutral Theory of Mole ular Evolution.Cambridge University Press, Cambridge, UK, 1983.J. Lobo, J. H. Miller, and W. Fontana.Neutrality in te hnology lands ape, 2004.M. Newman and R. Engelhardt.E�e t of neutral sele tion on the evolution of mole ularspe ies. S. Verel Fitness lands apes and graphs

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Multimodal, rugged and neutral �tness lands apesLo al Optima NetworksContinous and multobje tivetea team.E. D. Weinberger.Correlated and un orrelatated �tness lands apes and how totell the di�eren e.In Biologi al Cyberneti s, pages 63 :325�336, 1990.S. Wright.The roles of mutation, inbreeding, rossbreeding, and sele tionin evolution.In Pro eedings of the Sixth International Congress of Geneti s1, pages 356�366, 1932.S. Verel Fitness lands apes and graphs

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