Multi-Objective Evolutionary Algorithms

Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010

Introduction to Multi-Objective Evolutionary

Algorithms


Table of Content

• Multi-Objective Evolutionary Algorithms(MOEA)

– Multi-Objective Problems

– MOEA’s Difference from Single-Objective EA

– Selection: Non-Dominated Sorting <非劣排序>

• MOEA solving Knapsack

– Problem Description

– A Test Case

• MOEA Application: Automated Antenna Design

Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack

MOEA Application: Automated Antenna Design


Multi-Objective Problems

• With multiple objectives

• Examples:



Multi-Objective ProblemsMOEA‘s Difference from Single-Objective EASelection: Non-Dominated Sorting <非劣排序>

)](,),(),([ 21 xxxMIN nx

)2,2(

3,5 22

x

xzxy

- Minimize y,z- 凤姐征婚：Maximize [学历，相貌，国际

视野，……]

- Knapsack with multiple values

- TSP<旅行商问题> Minimize [Time,

Distance, Cost, Risk]

- 选修课：Minimize [所花精力] Maximize

[最终得分]


Solution methods for Multi-Objective problems

• Constructing a single aggregate objective function (AOF) <聚合为单目标>

• Normal Boundary Intersection (NBI) method

• Normal Constraint (NC) method

• Successive Pareto Optimization (SPO) method

• Multiobjective Optimization using Evolutionary Algorithms (MOEA).

• PGEN (Pareto surface generation for convex multiobjective instances)

• IOSO (Indirect Optimization on the basis of Self-Organization)




Reference: Wikipedia[Multiobjective optimization] - http://en.wikipedia.org/wiki/Multiobjective_optimization

http://en.wikipedia.org/wiki/Multiobjective_optimization


General Framework of Evolutionary Algorithms




Initialize Population

Breeding Operation

Crossover, Mutation, etc.

Selection

Termination Conditions?

Termination(Output)

Evaluation

Evaluation

Randomly create a certain number of solutions(individuals)

Apply breeding operators on individuals in the population,

producing a group of new individuals.

Apply selection operator on individuals in the population, shrink the

population size to the initial size.

Calculate the objective value based on decision variables of a individual.


MOEA’s Difference from Single-Objective EA




Initialize Population

Breeding Operation

Crossover, Mutation, etc.

Selection

Termination Conditions?

Termination(Output)

Evaluation

Normal Sorting Non-Dominated Sorting <非劣排序>Producing a group of Pareto Fronts <Pareto 前沿>

Single Objective Multiple Objectivesdouble Evaluate(vector<double> dna)vector<double> Evaluate(vector<double> dna)

Optimization Result: one optimal solution a group of non-dominated solutions

内涵在这里！


Selection: Non-Dominated Sorting




• Non-dominated comparison

– 对于给定的两个非全等向量: A(x1, x2, …, xn), B(y1, y2, …, yn)

• 当且仅当∀i: xi ≥ yi，有A 大于B

• 当且仅当∀i: xi ≤ yi ，有B大于A

• 其余情况，即∃i, ∃j: i≠j AND xi > yi AND xj < yj则A与B不可比

(2, 3)

(3, 2)

国际视野

学历


Knapsack with Multiple Values


MOEA Application: Automated Antenna DesignProblem DescriptionA Test Case

• DefinitionGiven a set of items

– each with a set of costs [e.g. Weight, Volume, etc. ]

– and a set of values

determine the number of each item to include in a collection, so that the total costs are less than given limits and the total values are as large as possible.

给定一组物品

每种物品有确定的 {数量成本[重量，体积，神秘成本]价值[价值1，价值2]

}确定每种物品的选取数量，在满足总成本在给定的限定范围之内的前提下，极大化总价值[价值1，价值2]


Parameters & Knapsack Data



Count Weight(kg) Volume(m3) MysteryCost Value0 Value1

0 8 6 2 303 95

6 5 6 5 244 5

5 6 5 0 62 103

5 3 4 6 27 330

8 3 2 9 154 279

8 0 3 0 212 183

7 2 3 4 115 431

6 5 4 1 420 237

9 4 2 9 453 322

7 3 7 9 193 147

9 6 1 0 373 235

6 4 2 2 162 446

8 0 1 6 9 233

2 6 4 7 492 443

2 8 1 7 168 19

6 7 8 0 280 290

6 6 8 9 265 188

2 7 0 3 68 424

5 1 3 2 359 349

8 1 6 3 382 455

3 9 1 2 313 298

0 7 4 5 476 194

6 3 1 0 281 109

7 9 6 2 110 109

6 3 0 1 120 201

9 3 8 9 286 311

2 9 1 5 168 178

3 9 1 9 335 198

5 7 5 4 260 431

2 0 6 9 275 162

Properties

Objectives

Weight Volume MysteryCostLimit 446 187 377

Constraints

AlgorithmProblem

Population Size 300

Number of Crossover Parents 5

Crossover Probability 1

Mutation Probability 0.05

Copy Probability 0.05

Max Generation 800

Total Evaluation Used: 240300


Knapsack - Result



0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 104

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4x 10

4

2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3

x 104

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5x 10

4

2.7 2.8 2.9 3 3.1 3.2 3.3 3.4

x 104

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7x 10

4

3.2 3.25 3.3 3.35 3.4 3.45 3.5 3.55 3.6 3.65 3.7

x 104

3.5

3.55

3.6

3.65

3.7

3.75

3.8

3.85

3.9

3.95

4x 10

4

3 3.1 3.2 3.3 3.4 3.5 3.6

x 104

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9x 10

4

3.4 3.45 3.5 3.55 3.6 3.65 3.7 3.75

x 104

3.55

3.6

3.65

3.7

3.75

3.8

3.85

3.9

3.95

4

4.05x 10

4

Generation = 0 Generation = 100 Generation = 200

Generation = 400 Generation = 600 Generation = 800

Distribution of Objectives


Knapsack - Result



0.5 1 1.5 2 2.5 3 3.5 4

x 104

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

Generation 0

Generation 100

Generation 200

Generation 400

Generation 600

Generation 800

Objectives’ evolutionary progress in same scale

0 200 400 600 8000

1

2

3

4x 10

4

Generation

Mean V

alu

e1

0 200 400 600 8001.5

2

2.5

3

3.5

4x 10

4

Generation

Mean V

alu

e2

Evolutionary progress of average value of each objective


Demand of Micro Antennas



用于微型卫星

空间和能量都十分宝贵

对天线各种性能要求极高


Objectives and Contraints of X-Band 5



• VSWR(Objectives)• Minimize: T_VSWR(r0, r1, …, r4, x1, y1, z1, …, x4, y4, z4)• Minimize: R_VSWR(r0, r1, …, r4, x1, y1, z1, …, x4, y4, z4)

• Gain(Objectives)• Maximize: T_Gainθ,φ(r0, r1, …, r4, x1, y1, z1, …, x4, y4, z4)• Maximize: R_Gainθ,φ(r0, r1, …, r4, x1, y1, z1, …, x4, y4, z4)

• Constraints• T_Gainθ,φ≥ 0 • R_Gainθ,φ ≥ 0• T_VSWR < 1.5• R_VSWR < 1.5• Diameter < 15.24cm• Height < 15.24cm• Antenna Mass < 165g• Θ= 5.0i, i = 8,9,…,16• φ = 5.0j, j = 0,1,…,71


MOEA

Designing with MOEA



Encoding Evaluator

chromosome : vector<double>

Electromagnetic Simulator{NEC, HFSS, etc.}


Comparison between traditional and evolvable Antenna



NASA X-Band 5 Antenna designed by EA

QHA Handmade Antenna


Resources

• MOEA Competition @ CEC 2009– http://dces.essex.ac.uk/staff/qzhang/moeacompetition09.htm

• Knapsack problem– http://en.wikipedia.org/wiki/Knapsack_problem

• NASA – Evolvable Systems– http://www.nasa.gov/centers/ames/research/exploringtheuniverse/exploringtheuniverse-

evolvablesystems.html



http://dces.essex.ac.uk/staff/qzhang/moeacompetition09.htm

http://en.wikipedia.org/wiki/Knapsack_problem

http://www.nasa.gov/centers/ames/research/exploringtheuniverse/exploringtheuniverse-evolvablesystems.html



Technology

Multi-Objective Evolutionary Algorithms