An Alternative Preference Relation to Deal with Many-Objective - - PowerPoint PPT Presentation

An Alternative Preference Relation to Deal with Many-Objective Optimization Problems

Antonio López-Jaimes

Japan Aerospace Exploration Agency Institute of Space and Astronautical Science

December 17th, 2012

Outline of the Presentation

Motivation The Chebyshev Achievement Function Approximating the Entire Pareto Front Experimental Results Conclusions

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Outline of the Presentation

Motivation The Chebyshev Achievement Function Approximating the Entire Pareto Front Experimental Results Conclusions

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Difficulties of Many-objective Problems (1/3)

Number of nondominated

vectors quickly increases with the number of

• bjectives: (2k − 2)/(2k).

z f2 f1

Number of dominance resistant

solutions (DRSs) increases with the

• bjectives:

Solutions with good values in most

• bjectives, but a poor value in at

least one objective.

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Motivation and Proposal (2/3)

Need for techniques with good scalability with respect to the

number of objectives.

Both in terms of convergence, and computational time. December 17th, 2012

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Motivation and Proposal (2/3)

Need for techniques with good scalability with respect to the

number of objectives.

Both in terms of convergence, and computational time. On the other hand, some researchers (Schütze, Lara, Coello

2011) have found that in some problems the difficulty does not significantly increases with the number of objectives.

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Motivation and Proposal (3/3)

Therefore...

• 1. We propose an alternative preference relation to deal with

Many-objective problems.

Good convergence without sacrificing distribution. Efficient and suitable for parallel implementation. Easy to use it just by replacing the Pareto dominance relation.

• 2. We also study the effect of DRSs in the widely used DTLZ test

problem suite and also in some WFG problems.

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Outline of the Presentation

Motivation The Chebyshev Achievement Function Approximating the Entire Pareto Front Experimental Results Conclusions

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Chebyshev Achievement Function

The new preference relation is based on the Chebyshev

achievement function.

An achievement function is a special scalarizing function,

s : Rk → R, parameterized by a reference point, zref.

max{zi − zref

i }

(z1 − zref

1 )2

(z2 − zref

2 )2

(z1 − zref

1 )2

(z2 − zref

2 )2

z1 z2 zref

Unlike Euclidean distance, December 17th, 2012

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Chebyshev Achievement Function

The new preference relation is based on the Chebyshev

achievement function.

An achievement function is a special scalarizing function,

s : Rk → R, parameterized by a reference point, zref.

z1 z2 zref max {zi − zref

i }

max {zi − zref

i }

Unlike Euclidean distance, the Chebyshev function takes

• nly the maximum

difference between a vector z and a reference point zref.

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Augmented Chebyshev Achievement Function

Definition (Wierzbicki 1980, Ehrgott 2005)

The augmented Chebyshev achievement function is defined by s(z | zref) = max

i=1,...,k{λi(zi − zref i )}

The reference point: aspiration levels for each objective. The weight vector, λ, normalize the objective values. December 17th, 2012

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Augmented Chebyshev Achievement Function

Definition (Wierzbicki 1980, Ehrgott 2005)

The augmented Chebyshev achievement function is defined by s(z | zref) = max

i=1,...,k{λi(zi − zref i )} + ρ k

• i=1

λi(zi − zref

i ),

The reference point: aspiration levels for each objective. The weight vector, λ, normalize the objective values. The second term helps to remove weakly Pareto optimal

solutions.

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Augmented Chebyshev Achievement Function

Definition (Wierzbicki 1980, Ehrgott 2005)

The augmented Chebyshev achievement function is defined by s(z | zref) = max

i=1,...,k{λi(zi − zref i )} + ρ k

• i=1

λi(zi − zref

i ),

The reference point: aspiration levels for each objective. The weight vector, λ, normalize the objective values. The second term helps to remove weakly Pareto optimal

solutions.

Any solution of the Pareto front can be generated by

s(z | zref).

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The Chebyshev Preference Relation

Combines Pareto dominance + Chebyshev function. Defines a Region of Interest (RoI) around a reference point.

solutions with s(z | zref) smin + δ,

best achievement size of the RoI

smin δ z2 z1 zmin zref RoI

Solutions outside RoI are

compared with Chebyshev value.

Solutions in RoI are compared

using Pareto dominance.

Solutions in RoI dominate

solutions outside RoI.

Pareto C h e b y s h e v

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Application of the Chebyshev Relation

Unfeasible reference point

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

zref = ( , 0.3) z∞

* = (0.7057, 0.5055)

0.5

Optimal PF

Feasible space

NSGA−II

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Application of the Chebyshev Relation

Feasible reference point

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Outline of the Presentation

Motivation The Chebyshev Achievement Function Approximating the Entire Pareto Front Experimental Results Conclusions

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Approximating the Entire Pareto Front

To approximate the entire Pareto front we used as reference point the approximation of the ideal point maintained by the Chebyshev relation.

Use a stringent criterion for

solutions far from the Pareto front for guiding the solutions towards the ideal point,

and Pareto dominance for

solutions near the Pareto front for covering the entire Pareto front. smin δ = 0.9 zmin z⋆

Points always kept

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Replacing Pareto Dominance by Another Relation

In order to improve efficiency...

Instead of the Pareto dominance, another secondary preference

relation can be used instead.

Here we used a preference relation based on the binary

ǫ-indicator.

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Replacing Pareto Dominance by ǫ-indicator Relation

Epsilon Indicator, Iǫ

Iǫ(z2, z1) = min

ǫ∈R {z2 i ǫ + z1 i for i = 1, . . . , k}

“Extent” by which a solution dominates another one: ǫ = 0 z2 ≺ z1 dominates by needs to dominate z1 ǫ < 0 ǫ > 0

Using a fitness function, Fǫ(z), based on the ǫ-indicator we can

define a preference relation:

Epsilon relation, Rǫ

A solution z1 “dominates” z2 wrt the relation Rǫ if and only if: Fǫ(z1) > Fǫ(z2).

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Replacing Pareto Dominance by ǫ-indicator Relation

Two examples of fitness functions (P is the current population):

Maximim fitness function (Balling 2003)

Fmin

ǫ (z1) =

min

z2∈P\{z1} Iǫ(z2, z1)

Sum of Iǫ values (Zitzler and Künzli 2004)

Fsum

ǫ

(z1) =

• z2∈P\{z1}

− exp

• −Iǫ({z2}, {z1})
• ǫ = 0

z2 ≺ z1 dominates by ǫ < 0 needs to dominate z1 ǫ > 0

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Outline of the Presentation

Motivation The Chebyshev Achievement Function Approximating the Entire Pareto Front Experimental Results Analysis of Dominance Resistant Solutions Conclusions

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Algorithms and Parameter Settings (1/2)

NSGA2 used as baseline optimizer: Pareto dominance. Chebyshev relation with Isum

ǫ

.

Chebyshev relation with Imin

ǫ .

Test Problems:

Problem Features DTLZ1, DTLZ3 Multiple local Pareto fronts. DTLZ4 Nonuniform solution density. DTLZ7, WFG2 Disconnected PFopt. WFG6 Nonseparable MOP.

Objectives were varied from 3 to 14 objectives. Number of distance-related variables: 20 (5 for DTLZ1). Number of position-related variables: k − 1. December 17th, 2012

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Algorithms and Parameter Settings (2/2)

Performance Indicators:

• Generational Distance

Problem GD = (z/|P|) − 0.5 DTLZ1 GD = (z2/|P|) − 1 DTLZ2-4 GDg = g(x) − 1 DTLZ7 GDx in design space WFG2, WFG6

Inverted Generational distance. Epsilon Indicator. Other parameters:

Parameter Value Population size 200 Generations 200 Crossover rate 0.9 Mutation rate 1/n Crossover index 20 Mutation index 20

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Generational Distance

GD values for DTLZ1 and WFG6 using from 3 to 14 objectives.

3 4 6 8 10 12 14 50 100 150 200 250 300 350 400 450

• Num. of objectives

Generational Distance (average)

DTLZ1 NSGA2 NSGA2−Isum

ε

NSGA2−Imin

ε 3 4 6 8 10 12 14 0.01 0.02 0.03 0.04 0.05 3 4 6 8 10 12 14 0.65 0.7 0.75 0.8 0.85 0.9 0.95

• Num. of objectives

Generational distance (average)

WFG6 NSGA2 NSGA2−Isum

ε

NSGA2−Imin

ε

Remarkable improvement Marginal improvement

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Generational Distance

GD values for DTLZ1 and WFG6 using from 3 to 14 objectives.

3 4 6 8 10 12 14 50 100 150 200 250 300 350 400 450

• Num. of objectives

Generational Distance (average)

DTLZ1 NSGA2 NSGA2−Isum

ε

NSGA2−Imin

ε 3 4 6 8 10 12 14 0.01 0.02 0.03 0.04 0.05 3 4 6 8 10 12 14 0.65 0.7 0.75 0.8 0.85 0.9 0.95

• Num. of objectives

Generational distance (average)

WFG6 NSGA2 NSGA2−Isum

ε

NSGA2−Imin

ε

Remarkable improvement Marginal improvement Does Chebyshev approach performs bad

• r NSGA2 performs well?

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Inverted Generational Distance

MOP MOEA

3 6 10 14

DTLZ2 NSGA2

mean 0.0151 0.0239 0.0469 0.0443 std 0.0010 0.0030 0.0113 0.0104

NSGA2-Isum

ǫ

mean 0.0079 0.0132 0.0183 0.0331 std 0.0025 0.0047 0.0091 0.0185

NSGA2-Imin

ǫ

mean 0.0087 0.0121 0.0165 0.0220 std 0.0027 0.0028 0.0070 0.0099

DTLZ7 NSGA2

mean 0.0896 0.3300 12.7540 38.0094 std 0.2740 0.0830 4.8488 9.5044

NSGA2-Isum

ǫ

mean 0.0138 0.2226 4.9511 9.1685 std 0.0029 0.0083 0.3701 0.3093

NSGA2-Imin

ǫ

mean 0.0150 0.2229 4.9978 9.2333 std 0.0061 0.0073 0.2918 0.1852

WFG6 NSGA2

mean 0.0075 0.0120 0.0187 0.0271 std 0.0034 0.0049 0.0052 0.0071

NSGA2-Isum

ǫ

mean 0.0073 0.0091 0.0141 0.0232 std 0.0024 0.0020 0.0038 0.0240

NSGA2-Imin

ǫ

mean 0.0075 0.0087 0.0132 0.0209 std 0.0015 0.0025 0.0050 0.0085

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Outline of the Presentation

Motivation The Chebyshev Achievement Function Approximating the Entire Pareto Front Experimental Results Analysis of Dominance Resistant Solutions Conclusions

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Analysis of Dominance Resistant Solutions

DRSs in

DTLZ3

using

NSGA2

(objs. values are divided by 20) 20 000 random solutions for DTLZ2

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Maximum Tradeoff to Detect DRSs

Classification of DRSs

Maximum tradeoff: T max(z) =

max(zi) min(zi)+1

for all i = 1, ..., k.

DRSs have a very large

T max. Small value in some

• bjective and large value in
• ther objective.

Not all solutions far from

the Pareto front obtain a large T max. z1

Pareto front

z2 z3

large tradeoff small tradeoff small tradeoff

T max = 1 1 1

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Maximum Tradeoff Results

Maximum tradeoff distribution in DTLZ1 DRSs: T max >> 1 NSGA2 NSGA2-Isum

ǫ

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Maximum Tradeoff Results

Maximum tradeoff distribution in WFG6 DRSs: T max >> 1

0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 16 18 20 % of solutions in PFapprox Maximum tradeoff

NSGA2, WFG6 3 objectives 6 objectives 12 objectives

0.4 0.5 0.6 0.7 0.8 0.9 1 2 4 6 8 10 12 14 16 18 % of solutions in PFapprox Maximum tradeoff

NSGA2−Isum

ε

, WFG6 3 objectives 6 objectives 12 objectives

NSGA2 NSGA2-Isum

ǫ

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Outline of the Presentation

Motivation The Chebyshev Achievement Function Approximating the Entire Pareto Front Experimental Results Conclusions

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Conclusions

The Chebyshev relation + ǫ-indicator is also useful to deal

with Many-Objective problems.

NSGA-II’s convergence was improved without sacrificing

distribution.

Comparing solutions using, either the achievement function or

the ǫ-indicator value, help to eliminate DRSs.

The main source of difficulty of DTLZ problems is the presence

• f dominance resistant solutions.

The difficulty of some WFG problems is not considerably

increased with the number of objectives. Future Work:

Compare the Chebyshev relation against optimization

techniques that have shown good scalability in Many-objective

• problems. E.g., MOEA/D or ǫ-MOEA.

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Space Trajectory Design Problem

Design Parameters: Thrust schedule to raise

perigee and apogee.

Initial mass of the spacecraft. Propagation date. Objectives:

• 1. Min. time to reach the Moon.

2,3. Min. time in radiation belts.

• 4. Min. operation time of the Engine.
• 5. Min. time under Earth’s shadow.
• 6. Max. initial mass of the spacecraft.

Constraint:

• 1. Flight time 1.5 years.

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Chebyshev Relation for Preference Incorporation

Specific desirable objective values:

• 1. Terminal condition: Reach the Moon orbit by 1.5 years after

launch date.

• 2. Duration in the radiation belt (h2e4 ) 1500h.
• 3. Longest eclipse time (maxEclipse ) 1h.
• 4. Initial mass of the spacecraft = 400kg.

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Reference Point and Parameters

maxEclipse, h2e4, mass have a stringent value, zref =         mass = 400 h2e4 = 1500 IES = 10000 TOF = 550 h5e3 = 500 maxEclipse = 1         . Parameter Value

• Pop. size

180 Generations 120

• Func. evals.

21 600 Archive size 929 RoI size 0.075 Running time 7h:17m:1s

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Variation of mass for solutions with maxEclipse 1

There are design with maxEclipse 1 even with mass > 400kg.

380 385 390 395 400 405 410 415 0.85 0.9 0.95 1 Mass (kg) maxEclipse (hours)

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Variation of h2e4 for solutions with maxEclipse 1

1400 1420 1440 1460 1480 1500 1520 0.85 0.9 0.95 1 h<2e4 (hours) maxEclipse (hours)

Mass (kg)

380 385 390 395 400 405 410 385kg ± 0.1

For designs with same mass (e.g., ≈ 385kg), Time in radiation belt slightly increases when maxEclipse is improved.

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Variation of IES for solutions with maxEclipse 1

7800 8000 8200 8400 8600 8800 9000 9200 0.85 0.9 0.95 1 IES (hours) maxEclipse (hours)

Mass (kg)

380 385 390 395 400 405 410

For designs with same mass, Engine Usage increases when maxEclipse is improved.

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Designs with 400 ± 0.25kg, maxEclipse 1h, h2e4 1500h Objective values

mass h<2e4 IES TOF h<5e3 mxEclipse (kg) (hours) (hours) (days) (hours) (hours) 400.2 1447.3 8853.2 401.93 219.51 0.967 400.2 1452.1 8808.1 402.02 219.73 0.967 399.9 1472.1 8693.4 410.35 223.44 0.980 399.9 1481.7 8770.3 404.72 224.27 0.996

Parameter values

startDate ChngPrd f_apoUP f_periUP (day/hour) (day) (deg) (deg) 33/0.74 124.09 179.89 0.7812 33/0.74 99.517 179.4 0.7812 359/1.77 272.26 171.01 0.3387 350/1.93 213.98 178.27 1.0393

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