Relation between Neighborhood Size and MOEA/D .../file/...Hisao Ishibuchi, Naoya Akedo, Yusuke...

Relation between Neighborhood Size and MOEA/D Performance

on Many-Objective Problems

Hisao Ishibuchi, Naoya Akedo, Yusuke Nojima Osaka Prefecture University, JAPAN

Content of This Presentation 1. Motivation We briefly explain our motivation. 2. MOEA/D and Its Modified Version We explain MOEA/D and its slightly modified version with two neighborhood structures. 3. Experimental Results We report some interesting observations on the behavior of MOEA/D with various settings of neighborhood size.

Motivation Performance of MOEA/D In a series of our studies on MOEA/D, its performance was almost always high. [1] Ishibuchi et al.: “Adaptation of Scalarizing Functions in

MOEA/D”. EMO 2009. [2] Ishibuchi et al.: “Evolutionary Many-Objective Optimization

by NSGA-II and MOEA/D with Large Populations”. SMC 2009. [3] Ishibuchi et al.: “Simultaneous Use of Different Scalarizing

Functions in MOEA/D”. GECCO 2010. [4] Ishibuchi et al. “Effects of the Existence of Highly Correlated

Objectives on the Behavior of MOEA/D”. EMO 2011. [5] Ishibuchi et al.: “A Study on the Specification of a Scalarizing

Function in MOEA/D for Many-Objective Knapsack Problems”. LION 2013.

However, in other authors’ studies, the performance of MOEA/D was sometimes surprisingly bad when it was used for performance comparison with their EMO algorithms. Question: Why was the performance of MOEA/D poor in their computational experiments (whereas its performance was high in our own studies)?

Motivation Performance of MOEA/D

Question: Why was the performance of MOEA/D poor in their computational experiments (whereas its performance was high in our own studies)? Possibilities: (0) Use of different test problems. (1) Use of inappropriate scalarizing functions: In MOEA/D, each solution is evaluated by a scalarizing

function with a different weight vector. (2) Use of inappropriate neighborhood specifications: - Parents are selected from neighbors (Local Mating). - Generated offspring are compared with neighbors

(Local Replacement).


Question: Why was the performance of MOEA/D poor in their computational experiments (whereas its performance was high in our own studies)? Possibilities: (0) Use of different test problems. (1) Use of inappropriate scalarizing functions: In MOEA/D. each solution is evaluated by a scalarizing

function with a different weight vector. ==> LION 2013 (2) Use of inappropriate neighborhood specifications: - Parents are selected from neighbors (Local Mating). - Generated offspring are compared with neighbors

(Local Replacement).


M i i f ( )

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e f 2(

x)Final GenerationPareto Front

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Results on Two-Objective Knapsack Problem (LION 2013) Performance of MOEA/D depends on Scalarizing Function

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PBI (θ = 1) NSGA-II


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Average Hypervolume on Two-Objective Problem (100 runs) Performance of MOEA/D depends on Scalarizing Function

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HV on Two-Objective: 1.00






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Best Specification for Two-Objective Problem PBI Function with θ = 5

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Average Hypervolume on Ten-Objective Problem

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HV on Ten-Objective: 1.00






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The best specification for the two-objective problem is the worst for the ten-objective problem

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Observation (when PBI (θ = 5) was used): MOEA/D is outperformed by NSGA-II in their applications to the 10-objective problem.

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NSGA-II







Observation from the weighted sum: MOEA/D is much better than NSGA-II in their applications to the 10-objective problem.

Question: Why was the performance of MOEA/D poor in their computational experiments (whereas its performance was high in our own studies). Possibilities: (1) Use of inappropriate scalarizing functions: In MOEA/D. each solution is evaluated by a scalarizing

function with a different weight vector. ==> LION 2013 (2) Use of inappropriate neighborhood specifications: - Parents are selected from neighbors (Local Mating). - Generated offspring are compared with neighbors

(Local Replacement). In EMO 2013, we examine the effect of neighborhood size.


Decomposition: A multi-objective problem is handled as a set of scalarizing function optimization problems with different weight vectors. Uniformly distributed weight vectors were used in the original study (2007).

Basic Idea of MOEA/D Q. Zhang and H. Li (IEEE TEVC 2007)

Weight vectors (2-objective case) Weight vectors (3-objective case)

Single-objective problem to optimize a scalarizing function with a different weight vector

Decomposition: A multi-objective problem is handled as a set of scalarizing function optimization problems with different weight vectors. Uniformly distributed weight vectors were used in the original study (2007).

Basic Idea of MOEA/D Q. Zhang and H. Li (IEEE TEVC 2007)

Weighted Sum

f(x) = f2(x)

f(x) = 0.25 f1(x) + 0.75 f2(x)c

f(x) = 0.5 f1(x) + 0.5 f2(x)

f(x) = 0.75 f1(x) + 0.25 f2(x)

f(x) = f1(x)

Characteristic Features of MOEA/D Local Mating in Parent Selection

(0.25, 0.25, 0.5)

(0.25, 0.5, 0.25)

(0, 0, 1)(0.25, 0, 0.75)

(0.5, 0, 0.5)(0.75, 0, 0.25)

(1, 0, 0)

(0, 1, 0)

(0.25, 0.75, 0)

(0.5, 0.5, 0)

(0.75, 0.25, 0)

(0, 0.75, 0.25)

(0, 0.5, 0.5)

(0, 0.25, 0.75)w1

w3

w2

(0.5, 0.25, 0.25)

To generate a new offspring for the current solution , a pair of parents are selected from its neighbors . Then crossover and mutation are used.

(0.25, 0.25, 0.5)

(0.25, 0.5, 0.25)

(0, 0, 1)(0.25, 0, 0.75)

(0.5, 0, 0.5)(0.75, 0, 0.25)

(1, 0, 0)

(0, 1, 0)

(0.25, 0.75, 0)

(0.5, 0.5, 0)

(0.75, 0.25, 0)

(0, 0.75, 0.25)

(0, 0.5, 0.5)

(0, 0.25, 0.75)w1

w3

w2

(0.5, 0.25, 0.25)

A newly generated offspring is compared with the current solution and its neighbors . (If better then replace).

Characteristic Features of MOEA/D Local Replacement in Solution Update

Our Model: Slightly Modified Version Local Mating and Local Replacement

(0.25, 0.25, 0.5)

(0.25, 0.5, 0.25)

(0, 0, 1)(0.25, 0, 0.75)

(0.5, 0, 0.5)(0.75, 0, 0.25)

(1, 0, 0)

(0, 1, 0)

(0.25, 0.75, 0)

(0.5, 0.5, 0)

(0.75, 0.25, 0)

(0, 0.75, 0.25)

(0, 0.5, 0.5)

(0, 0.25, 0.75)w1

w3

w2

(0.5, 0.25, 0.25)

In MOEA/D, the same neighborhood structure was used for both the local mating and the local replacement.

Our Implementation: We use different neighborhood structures for local mating and local replacement.

Mating


(0.25, 0.25, 0.5)

(0.25, 0.5, 0.25)

(0, 0, 1)(0.25, 0, 0.75)

(0.5, 0, 0.5)(0.75, 0, 0.25)

(1, 0, 0)

(0, 1, 0)

(0.25, 0.75, 0)

(0.5, 0.5, 0)

(0.75, 0.25, 0)

(0, 0.75, 0.25)

(0, 0.5, 0.5)

(0, 0.25, 0.75)w1

w3

w2

(0.5, 0.25, 0.25)



Mating Replacement

(0.25, 0.25, 0.5)

(0.25, 0.5, 0.25)

(0, 0, 1)(0.25, 0, 0.75)

(0.5, 0, 0.5)(0.75, 0, 0.25)

(1, 0, 0)

(0, 1, 0)

(0.25, 0.75, 0)

(0.5, 0.5, 0)

(0.75, 0.25, 0)

(0, 0.75, 0.25)

(0, 0.5, 0.5)

(0, 0.25, 0.75)w1

w3

w2

(0.5, 0.25, 0.25)



Mating Replacement

This modification is to examine the effects of local mating and local replacement separately.


Scalarizing Functions

1. Weighted Sum:

2. Weighted Tchebycheff:

∑=

⋅=m

iii

WS fg1

)(xλ

|)(|max *

...,,2,1xkiimi

TE fzg −⋅==

λ

We used the following scalarizing functions:

)(|)(max1.1* tfz ii Ω∈⋅= xx

Six- and Eight-Objective Knapsack Problems

Two- and Four-Objective Knapsack Problems

Scalarizing Functions Their Contour Lines

Maximize f1(x)

Max

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e f 2

(x)

Reference PointWeighted Sum

Contour line of each scalarizing function for the weighted vector λ = (0.5, 0.5)

Maximize f1(x)

Max

imiz

e f 2

(x)

Reference PointWeighted Tchebycheff

∑=

⋅=m

iii

WS fg1

)(xλ |)(|max *

...,,2,1xkiimi

TE fzg −⋅==

λ

*iz

Good Bad

Content of This Presentation 1. Motivation We briefly explain our motivation. 2. MOEA/D and Its Modified Version We explain MOEA/D and its slightly modified version with two neighborhood structures. 3. Experimental Results We report some interesting observations on the behavior of MOEA/D with various settings of neighborhood size. (1) Mating neighborhood: α % of the population size (2) Replacement neighborhood: β % of the population size α = 1, 2, 4, 6, 8, 10, 20, 40, 60, 80, 100 β = 1, 2, 4, 6, 8, 10, 20, 40, 60, 80, 100

Computational Experiments Original 2-500 Knapsack Problem Two-objective 500-item knapsack problem in Zitzler & Thiele (IEEE TECV 1999) with randomly generated two objectives.

Maximize ,)(1

∑=

=n

jjiji xpf x i = 1, 2

Subject to ,1

i

n

jjij cxw ≤∑

=

i = 1, 2

pij : Profit of item j according to knapsack i wij : Weight of item j according to knapsack i cij : Capacity of knapsack i

Value of pij and wij were randomly specified in [10, 100], and wij is 50% of the sum of wij .

Computational Experiments Settings of Computational Experiments

EMO Algorithms: MOEA/D Coding: Binary string of length 500 Crossover: Uniform crossover with the probability 0.8 Mutation: Bit-flip mutation with the probability 1/500 Constraint Handling: Greedy repair in Zitzler & Thiele (1999) Termination: 400,000 solution evaluations Neighborhood Size in MOEA/D: Mating Neighborhood: α % of the population size

α = 1, 2, 4, 6, 8, 10, 20, 40, 60, 80, 100 Replacement Neighborhood: β % of the population size

β = 1, 2, 4, 6, 8, 10, 20, 40, 60, 80, 100 Scalarizing Function in MOEA/D: Weighted Tchebycheff Population Size: 200

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2-500 Problem

Two-Objective 500-Item Problem Average Hypervolume over 100 runs

Good Results: Mating Neighborhood: 2%-10% of Pop size Replacement: Any size is OK. Pop size: 200 2 4 6 8 10

Two-Objective 500-Item Problem Results of a Single Run

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Mating: 1% Replacement: 1%




Knapsack Problems with 4-10 Objectives (4-500, 6-500, 8-500, 10-500)

Many-Objective Knapsack Problems (n-500 Problems) Randomly generated objectives were added to the original 2-500 problems (n = 4, 6, 8)

Maximize ,)(1

∑=

=n

jjiji xpf x i = 1, 2, ..., n

Subject to ,1

i

n

jjij cxw ≤∑

=

i = 1, 2

pij : Profit of item j according to knapsack i wij : Weight of item j according to knapsack i cij : Capacity of knapsack i

Values of pij were randomly specified in [10, 100].

Computational Experiments Settings of Computational Experiments

EMO Algorithms: MOEA/D Coding: Binary string of length 500 Crossover: Uniform crossover with the probability 0.8 Mutation: Bit-flip mutation with the probability 1/500 Constraint Handling: Greedy repair in Zitzler & Thiele (1999) Termination: 400,000 solution evaluations Mating Neighborhood: α % of the population size

α = 1, 2, 4, 6, 8, 10, 20, 40, 60, 80, 100 Replacement Neighborhood: β % of the populations size

β = 1, 2, 4, 6, 8, 10, 20, 40, 60, 80, 100 Scalarizing Function in MOEA/D:

Tchebycheff (4-500), Weighted Sum (6-500, 8-500) Population Size: 220 (4-500), 252 (6-500), 120 (8-500)

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4-500 Problem

Four-Objective 500-Item Problem Average Hypervolume over 100 runs

Good Results: Mating Neighborhood: 2%-10% of Pop size Replacement: 4%-100% of Pop size Pop size: 220 4 6 8 10

4

2

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6-500 Problem

Six-Objective 500-Item Problem Average Hypervolume over 100 runs

Good Results: Mating Neighborhood: 2%-6% of Pop size Replacement: 4%-100% of Pop size Pop size: 252

2 4 6 4

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8-500 Problem

Eight-Objective 500-Item Problem Population Size: 120

Good Results: Mating Neighborhood: 6%-8% of Pop size Replacement: 6%-100% of Pop size Pop size: 120

6 8

1.10 NSGA-II

6

Eight-Objective 500-Item Problem Projection onto the f1-f2 Subspace





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2-500 Problem

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4-500 Problem

Knapsack Problems with 2-8 Objectives Average Hypervolume

Pop size: 200

Pop size: 220

Pop size: 256

Pop size: 120

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20

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60

80

100

x1

x2

Other Test Problem Six-Objective Distance Minimization Problem

f1(x) = d(x, A) f2(x) = d(x, B) f3(x) = d(x, C) f4(x) = d(x, D) f5(x) = d(x, E) f6(x) = d(x, F)

Minimize f(x) = ( f1(x), f2(x), f3(x), f4(x), f5(x), f6(x))

A

D

F

E

B

C

x1

x2

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20

40

60

80

100

x1

x2

0 20 40 60 80 100

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60

80

100

x1

x2

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100

x1

x2

0 20 40 60 80 100

20

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80

100






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20

40

60

80

100

x1

x2

Six-Objective Problem with Four Equivalent Pareto Regions

f1: mind(x, A1), d(x, A2), d(x, A3), d(x, A4)

Minimize f(x) = ( f1(x), f2(x), f3(x), f4(x), f5(x), f6(x))

f2: mind(x, B1), d(x, B2), d(x, B3), d(x, B4) f3: mind(x, C1), d(x, C2), d(x, C3), d(x, C4) f4: mind(x, D1), d(x, D2), d(x, D3), d(x, D4) f5: mind(x, E1), d(x, E2), d(x, E3), d(x, E4) f6: mind(x, F1), d(x, F2), d(x, F3), d(x, F4)

A4

B4

C4

D4

F4

E4

A3

B3

C3

D3

F3

E3

A1

B1

C1

D1

F1

E1

A2

B2

C2

D2

F2

E2

x1

x2

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80

100x1

x2

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100






x1

x2

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20

40

60

80

100

x1

x2

0 20 40 60 80 100

20

40

60

80

100

Conclusions

We examined the relation between the performance of MOEA/D and the size of the two neighborhoods. For many-objective knapsack problems, we obtained the following observations:

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8-500 Problem

Conclusions

We examined the relation between the performance of MOEA/D and the size of the two neighborhoods. For many-objective knapsack problems, we obtained the following observations: (1) Good results were obtained from the combination of

a small (but not too small) mating neighborhood and a large replacement neighborhood.

Conclusions



(2) Careful specifications of the two neighborhoods are needed for many-objective problems.

Conclusions




(3) When the mating neighborhood was large, good results were not obtained.

Conclusions




(3) When the mating neighborhood was large, good results were not obtained.

(4) When the replacement neighborhood was large, good results were obtained.

Conclusions

Thank you very much !

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Weighted Tchebycheff and Weighted Sum

Weighted Sum

Weighted Sum

Tchebycheff

Tchebycheff

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Knapsack Problems with 2-8 Objectives Average Hypervolume

Pop size: 200

Pop size: 220

Pop size: 256

Pop size: 330

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Eight-Objective 500-Item Problem Population Size: 330

Good Results: Mating Neighborhood: 2-6% of Pop size Replacement: 4-100% of Pop size

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Relation between Neighborhood Size and MOEA/D .../file/...Hisao Ishibuchi, Naoya Akedo, Yusuke...

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