Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
Introduction to Multi-Objective Evolutionary
Algorithms
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
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
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
Multi-Objective Problems
• With multiple objectives
• Examples:
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna Design
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]
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
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)
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna Design
Multi-Objective ProblemsMOEA‘s Difference from Single-Objective EASelection: Non-Dominated Sorting <非劣排序>
Reference: Wikipedia[Multiobjective optimization] - http://en.wikipedia.org/wiki/Multiobjective_optimization
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
General Framework of Evolutionary Algorithms
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna Design
Multi-Objective ProblemsMOEA‘s Difference from Single-Objective EASelection: Non-Dominated Sorting <非劣排序>
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.
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
MOEA’s Difference from Single-Objective EA
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna Design
Multi-Objective ProblemsMOEA‘s Difference from Single-Objective EASelection: Non-Dominated Sorting <非劣排序>
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!
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
Selection: Non-Dominated Sorting
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna Design
Multi-Objective ProblemsMOEA‘s Difference from Single-Objective EASelection: Non-Dominated Sorting <非劣排序>
• 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
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
Knapsack with Multiple Values
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
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]
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
Parameters & Knapsack Data
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna DesignProblem DescriptionA Test Case
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
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
Knapsack - Result
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna DesignProblem DescriptionA Test Case
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
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
Knapsack - Result
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna DesignProblem DescriptionA Test Case
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
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
Demand of Micro Antennas
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna Design
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
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
Objectives and Contraints of X-Band 5
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna Design
• 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
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
MOEA
Designing with MOEA
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna Design
Encoding Evaluator
chromosome : vector<double>
Electromagnetic Simulator{NEC, HFSS, etc.}
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
Comparison between traditional and evolvable Antenna
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna Design
NASA X-Band 5 Antenna designed by EA
QHA Handmade Antenna
Song Gao Introduction to Multi-Objective Evolutionary Algorithms Sunday, July 04, 2010
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
Multi-Objective Evolutionary Algorithms(MOEA)MOEA solving Knapsack
MOEA Application: Automated Antenna Design