Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Set-based Multiobje tive Fitness Lands apes: denition, properties. ThRaSh 2010 Sébastien Verel1,2

Clarisse Dhaenens1,3 Arnaud Liefooghe4

Laetitia Jourdan1,3

http://www.i3s.uni e.fr/~verel

1 DOLPHIN team - INRIA Lille-Nord Europe (Fran e) 2 I3S Laboratory - University of Ni e-Sophia Antipolis / CNRS (Fran e) 3 LIFL - University of Lille 1 / CNRS (Fran e) 4 University of Coimbra (Portugal)

Multiobje tive Fitness Lands ape Multidimensional moFiL Di ulty in multiobje tive optimization

Indi ator moFiL

Set based moFiL

Multiobje tive problem Multiobje tive problem

S : set of admissible solutions,

f2

f : S → IRp : multiobje tive fun tion to minimize

Dominan e (minimization) Fesible Solution Ideal Point

Pareto Solution

Utopique Point

Nadir Point

f1

x dominate y : x ≺ y i : ∀k = 1..p , fk (x ) ≤ fk (y ) ∃k ∈ 1..p , fk (x ) < fk (y )

Multiobje tive Fitness Lands ape Multidimensional moFiL Di ulty in multiobje tive optimization

Indi ator moFiL

Set based moFiL

Solving a multiobje tive problem Pareto Set (minimization) f2

PS = {s ∈ S | ∀z ∈ S, z 6≺ s } Pareto Front

PF = {f (s ) | s ∈ PS } Fesible Solution Ideal Point

Pareto Solution

Utopique Point

Nadir Point

f1

Solving multiobje tive problem Find PS or at least X ⊂ PS whi h is a "good" approximation of PS

Multiobje tive Fitness Lands ape Multidimensional moFiL Di ulty in multiobje tive optimization

Indi ator moFiL

Set based moFiL

Di ulty in multiobje tive optimization

What does "di ulty" mean ? For a given omputational time, X far from a "good" approximation of the Pareto Set (distan e in obje tive spa e to PS, la k of diversity of X, et .) Time, or number of evaluations, for solving the problem is long : lass of omplexity

Multiobje tive Fitness Lands ape Multidimensional moFiL Di ulty in multiobje tive optimization

Indi ator moFiL

Set based moFiL

Di ulty in multiobje tive optimization What's make "di ult" a multiobje tive problem ? The number of obje tive ( riteria) The size of the feasible solutions spa e The "di ulty" of the single obje tive problem (time

omplexity, ...) The orrelation between the obje tives The onvexity of the Pareto Front The onne tedness of the Pareto Set ... Fitness lands ape point of view an help...

Multiobje tive Fitness Lands ape Mono-obje tive Fitness Lands apes

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Fitness lands apes in single-obje tive optimization

Fitness lands ape (S, N , f ) S : set of admissible solutions, N : S → 2S : neighborhood fun tion,

f : S → IR : tness fun tion. ∀x ∈ S , N (x ) = {y ∈ S | P (y = op (x )) > 0} or N (x ) = {y ∈ S | d (y , x ) ≤ k }

Multiobje tive Fitness Lands ape Mono-obje tive Fitness Lands apes

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Goal of the tness lands apes study "Geometry" (features) of tness lands ape ⇒ dynami s of a lo al sear h algorithm Geometry is linked to the problem di ulty :

If there are a lot of lo al optima, the probability to nd the global optimum is lower. If the tness lands ape is at, dis overing better solutions is rare. What is the best sear h dire tion in the lands ape ?

Study of the tness lands ape features allows to study the performan e of sear h algorithms

Multiobje tive Fitness Lands ape Mono-obje tive Fitness Lands apes

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Goal of the tness lands apes study 1 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 ) dierent oding, mutation operator, tness fun tion, et . Whi h one is easier to solve ? 2 To hoose the algorithm : analysis of global geometry of the lands ape Whi h algorithm an I use ? 3 To tune the parameters : o-line analysis of stru ture of tness lands ape Whi h is the best mutation operator ? the size of the population ? et . 4 To ontrol the parameters during the run : on-line analysis of stru ture of tness lands ape Whi h is the optimal mutation operator a

ording to the estimation of stru ture ?

Multiobje tive Fitness Lands ape Mono-obje tive Fitness Lands apes

Multidimensional moFiL

Links with lo al sear h Solutions, eighbors, tness : 6

4 s_t

3 7 1

Indi ator moFiL

Set based moFiL

Multiobje tive Fitness Lands ape Mono-obje tive Fitness Lands apes

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Links with lo al sear h Solutions, eighbors, tness : 6

Fitness lands ape point of view : before putting a parti ular heuristi

3

4 s_t

7 1

Put prob. from your heuristi : 0.5 0 s_t 0.5 0

s_t+1

Sample the neighborhood to have information on lo al features of the sear h spa e From lo al information : dedu e some global features like general shape of sear h spa e, "di ulty", et .

Multiobje tive Fitness Lands ape Mono-obje tive Fitness Lands apes

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Multiobje tive tness lands apes (moFiL)

3 ideas to dene multiobje tive Fitness Lands apes (moFiL)

Change nothing from the mono-obje tive denition (the most studied from 2003) : Fitness fun tion is the multiobje tive fun tion

Change the tness fun tion by an indi ator su h as hypervolume : Ruggedness an be studied

Change the sear h spa e by the set of sets of solutions : "Real" sear h spa e viewed by e ient lo al sear hes

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Fitness fun tion is the multiobje tive fun tion

Multidimensional multiobje tive Fitness Lands ape (S, N , f ) S : set of admissible solutions, N : S → 2S : neighborhood dened by lo al operator or distan e on S ,

f : S → IRp : multiobje tive fun tion.

Main idea to analysis of multiobje tive tness lands apes : Dominan e relation

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Overview on previous tools (1) Pareto Set [AT04℄ [AT07℄ :

Number, distribution over lands ape

+ :Information on the nal goal − :Information only on the nal goal Partition the sear h spa e into dominan e levels [PCS04℄ [GD09℄ :

onne tion between level of dominan e

+ :Information on stru ture of sear h spa e − :Ways to rea h PS ? small sear h spa e

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Overview on previous tools (2) Conne tedness [EK97℄ [GKR06℄ [PS09℄ :

number of lusters, size of luster by the graph indu ed by Pareto set and neighborhood relation (edges if the distan e ≤ k)

+ :If onne ted, easy way to nd the PS − :Huge PS are intra table, nal goal, often not onne ted

Fitness distan e orrelation, random walk from Pareto lo . opt. [GD06a℄ [GD08℄ [GD06b℄ :

orrelation between distan es in obj. and sol. spa e.

+ :Show the orrelation of tness − :The referen e solution hange, no orrelation after few steps,

no s ale

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Remarks on multidimensional moFiL

Majority of works use the on ept of dominan e :

onne tedness, size, estimation of distribution Pareto set, partition by dominan e levels Limited results : No enought information for the design of metaheuristi s Few understanding of the omplexity of multiobje tive problems from the tness lands apes analysis

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Change the tness fun tion by an indi ator Indi ator multiobje tive Fitness Lands ape (S, N , i ) S : set of admissible solutions, N : S → 2S : neighborhood dened by lo al operator or distan e on S ,

i : 2S → IR : indi ator (su h as hypervolume, ǫ dominan e) on

set of solutions

Main idea to analysis of multiobje tive tness lands apes : Value of the indi ator (hypervolume) of the neighborhood : i (N (s ) ∪ {s }) = hv (N (s ) ∪ {s })

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Rugged/smooth tness lands apes 405 400

Auto orrelation of time series of tnesses (f (s1 ), f (s2 ), . . .) along a random walk (s1 , s2 , . . .) [Wei90℄ :

395

performance

390 385 380 375 370 365 360 355 350 0

2000

4000

6000

8000

10000

pas s 1

ρ(n) =

Max-SAT NK

0.9 0.8

E [(f (si ) − f¯)(f (si +n ) − f¯)] var (f (si ))

0.7

auto orrelation length τ =

rho(s)

0.6 0.5 0.4

1

ρ(1)

small τ : rugged lands ape

0.3 0.2

long τ : smooth lands ape

0.1 0 0

5

10

15

20

25 pas s

30

35

40

45

50

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

A model of multiboje tive problem

Multiobje tive

MNK -lands apes with obje tive orrelation

MNK lands apes [AT04℄ ∀m ∈ [1, M ], fNK ,m (s ) =

N 1 X

N

i =1

fi ,m (si , si1 , . . . , siK )

M obje tives whi h are NK − lands apes Same intera tion : same K , and same si Ea h tness omponent fi ,m by extension : ysii,,msi1 ,...,siK .

im are randomly and independly drawn from [0, 1). MNK : y... in ρMNK : (ysii1,si ,...,si , . . . , ysiM i ,si1 ,...,siK ) ∼ multivariate 1 K uniform law of dimension M :

obje tive orrelation between obje tive an be tuned pre isely

Multiobje tive Fitness Lands ape

Results on

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

MNK -lands apes with obje tive orrelation

5

M=2 M=3 M=5

4.5

Correlation length

4 3.5 3 2.5 2

Dimension has low inuen e on orrelation length

1.5 1 0.5 2

4

6 K

5

10

Obje tive orrelation hange slowly the orrelation length

K=2 K=4 K=6 K=8 K=10

4.5 4 Correlation length

8

The non-linearity (epistasis) has the main inuen e

3.5 3 2.5 2 1.5 1 0.5 -1

-0.5

0 rho

0.5

1

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Analysis Denition of the ruggedness in indi ator moFiL. Ruggedness is not orrelated to the number of Pareto Lo al Optima in moFiL.

0.8

0.8

0.7

0.7

0.6

0.6 Objective 2

Objective 2

Argument in favour to optimize the hypervolume rather than the dominan e

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2 0.2

0.3

0.4 0.5 0.6 Objective 1

0.7

K = 4, ρ = −0.9

0.8

0.2

0.3

0.4 0.5 0.6 Objective 1

0.7

K = 4, ρ = 0.9

0.8

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Remarks on indi ator moFiL

Pros : Estimation of non-linearity / ruggedness of the lands apes Explain "lo al" di ulty of multiobje tive problem Cons : Metaheuristi s manipulate set of solutions, not single solution

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Change the sear h spa e by the set of sets of solutions Set based multiobje tive Fitness Lands ape (Σq , Nq , i ) Σq : set of sets of feasible solutions of maximum size q
Σq = {σ ⊂ S : |σ| ≤ q }, so |Σq | = |S| q

Nq : insert, delete, or hange one solution in the set i : Σq → IR : indi ator (su h as hypervolume, ǫ dominan e) on set of solutions

Main idea to analysis of multiobje tive tness lands apes : All the standard tools of Fitness Lands apes analysis

Multiobje tive Fitness Lands ape

Neighborhood

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Nq : Insert, delete, or hange 1 solution

Neighborhood Nq : Σq → 2Σq : dened from neighborhood N : S → 2S Neighbors of σ ∈ Σq are given by 3 operators :

Insert one solution whi h is neighbor of one of solution of σ Delete one solution Change one solution by its neighbor Nq (σ) = Nqi (σ) ∪ Nqd (σ) ∪ Nq (σ)

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Nqi (σ) : Insert one solution in the set ′

σ neighbor of σ = {s1 , . . . , sk } ′

if k = q : σ = σ ′

if 0 < k < q : σ = {s1 , . . . , sk , sk +1 } with sk +1 ∈ N (si ) ′ if k = 0 : σ = {s } with s ∈ S

Sizes when 0 < k < q , ′

if sk +1 is not in σ , |σ | = |σ| + 1 P |Nqi (σ)| ≤ ki=1 |N (si )| |Nqi (σ)| ≤ k λ if neighborhood size is st.

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Nqd (σ) : Delete one solution from the set ′

σ neighbor of σ = {s1 , . . . , sk } ′

if k = 0 : σ = σ ′

if 0 < k : σ = {s1 , . . . , si −1 , si +1 , . . . , sk } with i ∈ 1..k

Sizes when 0 < k , ′

|σ | = |σ| − 1 |Nqd (σ)| = k

Set based moFiL

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Nq (σ) : Change one solution of the set ′

σ neighbor of σ = {s1 , . . . , sk } ′

if k = 0 : σ = σ ′

′

if 0 < k : σ = {s1 , . . . , si −1 , si , si +1 , . . . , sk } ′ with i ∈ 1..k and si ∈ N (si ) Sizes when 0 < k , ′

′

if si is not in σ , |σ | = |σ| P |Nq (σ)| ≤ ki=1 |N (si )| |Nqi (σ)| ≤ k λ if neighborhood size is st. P Total size of Nq (σ) : |Nq (σ)| ≤ k + 2 ki=1 |N (si )| |Nq (σ)| ≤ k (2λ + 1) if neighborhood size is st.

Set based moFiL

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Links with lo al sear h : example of e ient LS

"On set based-based multiobje tive optimization" Zitzler et al, 2009

RandomSetMutation Set based Algo. for MO

Generate initial set P of size q with random solutions repeat P ′ ← randomSetMutation(P ) P ′′ ← heuristi SetMutation(P ) ′′ if

P  P then P ← P ′′

else if

P ′  P then P ← P′

end if end if until

terminaison riterion is not true

randomly hoose s1 , . . . , s ∈ S with s 6= s randomly sele t p1 , . . . , pkk ∈ P with pii 6= pjj P′

←

P \ {s1 , . . . , sk } ∪ {p1 , . . . , pk }

return P ′

heuristi SetMutation

Generate s1 , . . . , sk ∈ S based on P (sele tion and′ mutation of solutions from P )

P

← P ∪ {s1 , . . . , sk } ′′ while |P | > q do

for all

a P ′′ iP

∈ do ′′ ′′ δa ← ( \ { }) − ( )

end for

a

iP

hoose p ∈ P with δp = mina∈P ′′ δa P ′′

←

end while

′′

P ′′ \ {p}

return P ′′

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Tools to analysis set-based multiobje tive tness lands apes Density of states :

What an we ex ept from random sear h ?

Adaptive walk, lo al optima :

What is the number of lo al optima, and size of basin ? Random walk, ruggedness :

Is the solution in the neighborhood are random ?

Fitness loud, evolvability :

What is the distribution of tness in the neighborhood ? Can the solutions go to better tness values ? Neutral walk, neutrality :

Is there a lot of solutions with the same tness ? Is there information in and around the neutral networks ?

Lo al Optima Network :

What are the links between basins of attra tion in the lands apes ?

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Con lusion and perspe tives Summary Previous works on multidimensional moFiL : Fitness is the multiobje tive fun tion based on dominan e, information on Pareto Set (goal of optimization) Indi ator moFiL : Fitness is the value of indi ator based on random walk, information on ruggedness Set based moFiL : sear h spa e is the set of sets Use the "real" sear h spa e and elementary operators on sets

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Con lusion and perspe tives Summary Previous works on multidimensional moFiL : Fitness is the multiobje tive fun tion based on dominan e, information on Pareto Set (goal of optimization) Indi ator moFiL : Fitness is the value of indi ator based on random walk, information on ruggedness Set based moFiL : sear h spa e is the set of sets Use the "real" sear h spa e and elementary operators on sets Perspe tives Use it ! To de ide the best en oding and operator for example in multiobje tive TVRP (see M-E Marmion)

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

Non ommer ial ommuni ation Spe ial Session Fitness lands apes and metaheuristi s at international onferen e META'2010, Djerba Island, Tunisia, O tober 28-30th, 2010. Extended abstra t 2 pages Post publi ation for sele ted papers in COR, ITOR, JMMA Deadline may 5, 2010 web page : http://www2.lifl.fr/META10/index.php?n=Main.InfoFIL

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

For Further Reading Hernán E. Aguirre and Kiyoshi Tanaka. Insights on Properties of Multiobje tive MNK-Lands apes. In 2004 Congress on Evolutionary Computation (CEC'2004), volume 1, pages 196203, Portland, Oregon, USA, June 2004. IEEE Servi e Center. Hernán E. Aguirre and Kiyoshi Tanaka. Working prin iples, behavior, and performan e of moeas on mnk-lands apes. European Journal of Operational Resear h, 181(3) :1670  1690, 2007. M. Ehrgott and K. Klamroth. Conne tedness of e ient solutions in multiple riteria

ombinatorial optimization. European Journal of Operational Resear h, 97 :15966, 1997.

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

For Further Reading Deon Garrett and Dipankar Dasgupta. Analyzing the performan e of hybrid evolutionary algorithms for the multiobje tive quadrati assignment problem. In Congress on Evolutionary Computation (CEC 2006), pages 17101717. IEEE Servi e Center, July 2006. Deon Garrett and Dipankar Dasgupta. Analyzing the performan e of hybrid evolutionary algorithms for the multiobje tive quadrati assignment problem. In Congress on Evolutionary Computation (CEC'2006), pages . IEEE Servi e Center, July 2006. Deon Garrett and Dipankar Dasgupta. Multiobje tive lands ape analysis and the generalized assignment problem. pages 110124, 2008.

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

For Further Reading Deon Garrett and Dipankar Dasgupta. Plateau onne tion stru ture and multiobje tive metaheuristi performan e. In Congress on Evolutionary Computation (CEC'2009), pages . IEEE Servi e Center, July 2009. Jo hen Gorski, Kathrin Klamroth, and Stefan Ruzika. Conne tedness of E ient Solutions in Multiple Obje tive Combinatorial Optimization. Te hni al report, Kaiserslauterer uniweiter, 2006. Luis Paquete, Mar o Chiarandini, and Thomas Stützle. Pareto Lo al Optimum Sets in the Biobje tive Traveling Salesman Problem : An Experimental Study. In Xavier Gandibleux, Mar Sevaux, Kenneth Sörensen, and Vin ent T'kindt, editors, Metaheuristi s for Multiobje tive Optimisation, pages 177199, Berlin, 2004. Springer. Le ture

Multiobje tive Fitness Lands ape

Multidimensional moFiL

Indi ator moFiL

Set based moFiL

For Further Reading Luís Paquete and Thomas Stützle. Clusters of Non-dominated Solutions in Multiobje tive Combinatorial Optimization : An Experimental Analysis. In Vin ent Bari hard, Matthias Ehrgott, Xavier Gandibleux, and Vin ent T'Kindt, editors, Multiobje tive Programming and

Goal Programming. Theoreti al Results and Pra ti al Appli ations, pages 6977. Springer, Le ture Notes in

E onomi s and Mathemati al Systems, Vol. 618, 2009. ISBN 978-3-540-85645-0. E. D. Weinberger. Correlated and un orrelatated tness lands apes and how to tell the dieren e. In Biologi al Cyberneti s, pages 63 :325336, 1990.