Approximation Model Guided Selection for Evolutionary Multiobjective ...
Approximation Model Guided Selection for Evolutionary Multiobjective Optimization
Aimin Zhou1, Qingfu Zhang2, and Guixu Zhang1 1Each
China Normal University, Shanghai, China 2University
of Essex, Colchester, UK 03/2013
Outline
§ Background § Approximation Model Guided Selection § Experimental Results § Discussions & Conclusions 2
Approximation Model Guided Selection, EMO 2013
Outline
§ Background § Approximation Model Guided Selection § Experimental Results § Discussions & Conclusions 3
Approximation Model Guided Selection, EMO 2013
Multiobjective Optimization Problem o Definition
Pareto set (PS)
min F (x) = ( f1 (x), f 2 (x),…, f m (x)) s.t x ∈ D where D : decision (variable) space. fi : D → R, objective function F : D → R m , objective vector function
f2
F
o Pareto domination F (D )
o Optimum • Pareto set (PS) • Pareto front (PF) 4
Pareto front (PF) Approximation Model Guided Selection, EMO 2013
f1
Target of MOEA o Find an approximation set,
Pareto set (P)
which is • as diverse as possible • as close to the PF (PS) as possible
F
f2
Lower dimensional problems! F (D )
Pareto front (PF) 5
Approximation Model Guided Selection, EMO 2013
f1
A General MOEA Framework o Reproduction
Population
• Generate new trial solutions
o Selection Reproduction
Selection
• Select fittest ones into the next generation
New Solutions
6
Approximation Model Guided Selection, EMO 2013
Dominance based Selection o Step 1: rank population • dominance rank • dominance count • dominance strength
o Step 2: estimate density • niche and fitness sharing • crowding distance • K-nearest neighbor • gridding
Define a complete order over individuals! 7
Approximation Model Guided Selection, EMO 2013
Indicator based Selection o Convert an MOP into an SOP • Obj. = performance metric
Define a complete order over populations!
8
Approximation Model Guided Selection, EMO 2013
Model Guided Selection o Step 1: model the PF o Step 2: choose reference points o Step 3: select promising solutions Much work needs to be done along this direction! o
H. J. F. Moen, et al., Many-objective Optimization Using Taxi-Cab Surface Evolutionary Algorithm, EMO, 2013.
o
H. Jain, and K. Deb, An improved Adaptive Appraoch for Elitist Nondominated Sorting Genetic Algorithm for Many-Objective Optimization, EMO, 2013.
9
Approximation Model Guided Selection, EMO 2013
Outline
§ Background § Approximation Model Guided Selection § Experimental Results § Discussions & Conclusions 10
Approximation Model Guided Selection, EMO 2013
AMS Framework
Model PF
Define sub-problem
Select
o
A. Zhou, Estimation of Distribution Algorithms for Continuous Multiobjective Optimization, Ph.D Thesis, University of Essex, 2009. (Chapter 5.3)
o
A. Zhou, Q. Zhang, Y. Jin, and B. Sendhoff, Combination of EDA and DE for Continuous Biobjective Optimization, CEC 2008.
o
Q. Zhang and H. Li, MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition, IEEE Trans. on Evolutionary Computation, 2007.
11
Approximation Model Guided Selection, EMO 2013
Zero-order Approximation(AMS0) o A single point to approximate the PF o Model (ideal point)
o Distance to utopian PF (sub-problem)
Simple, but does not consider the shape of PF!
12
Approximation Model Guided Selection, EMO 2013
First-order Approximation(AMS1) o A simplex to approximate the PF o Model (vertices of simplex)
o Distance to simplex
Simple, and consider the shape of PF in a sense! 13
Approximation Model Guided Selection, EMO 2013
Outline
§ Background § Approximation Model Guided Selection § Experimental Results § Discussions & Conclusions 14
Approximation Model Guided Selection, EMO 2013
Settings o Offspring reproduction operator • A probability model based reproduction operator (RM-MEDA)
o Comparison strategy • NDS: non-dominated sorting scheme (NSGA-II)
o Test instances • 8 instances with different properties
o Performance metric • IGD: inverted general distance
15
Approximation Model Guided Selection, EMO 2013
Convex PF
NDS
16
AMS0
AMS1
Approximation Model Guided Selection, EMO 2013
Concave PF
NDS
17
AMS0
AMS1
Approximation Model Guided Selection, EMO 2013
Tri-objective MOP
NDS
18
AMS0
AMS1
Approximation Model Guided Selection, EMO 2013
Statistical Results
NDS:1 19
AMS0:4
AMS1:4
Approximation Model Guided Selection, EMO 2013
Outline
§ Background § Approximation Model Guided Selection § Experimental Results § Discussions & Conclusions 20
Approximation Model Guided Selection, EMO 2013
o AMS works better than NDS in most of the test instances o AMS0 is more stable than AMS1 o AMS1 works better than AMS0 if a good simplex model can be found stability of AMS
stability of AMS
order of model 21
number of objectives
Approximation Model Guided Selection, EMO 2013
o How to build a high-quality model? • model should be cheap • model should be stable
o What’s the performance on complicated problems? • non-concave (non-convex) • with disconnected PF
o What’s the performance on many-objective problems? • interesting sub-problems (reference points, targets points)
22
Approximation Model Guided Selection, EMO 2013
Thanks!
The source code is available from [email protected]
23
Approximation Model Guided Selection, EMO 2013
Aimin Zhou1, Qingfu Zhang2, and Guixu Zhang1 1Each
China Normal University, Shanghai, China 2University
of Essex, Colchester, UK 03/2013
Outline
§ Background § Approximation Model Guided Selection § Experimental Results § Discussions & Conclusions 2
Approximation Model Guided Selection, EMO 2013
Outline
§ Background § Approximation Model Guided Selection § Experimental Results § Discussions & Conclusions 3
Approximation Model Guided Selection, EMO 2013
Multiobjective Optimization Problem o Definition
Pareto set (PS)
min F (x) = ( f1 (x), f 2 (x),…, f m (x)) s.t x ∈ D where D : decision (variable) space. fi : D → R, objective function F : D → R m , objective vector function
f2
F
o Pareto domination F (D )
o Optimum • Pareto set (PS) • Pareto front (PF) 4
Pareto front (PF) Approximation Model Guided Selection, EMO 2013
f1
Target of MOEA o Find an approximation set,
Pareto set (P)
which is • as diverse as possible • as close to the PF (PS) as possible
F
f2
Lower dimensional problems! F (D )
Pareto front (PF) 5
Approximation Model Guided Selection, EMO 2013
f1
A General MOEA Framework o Reproduction
Population
• Generate new trial solutions
o Selection Reproduction
Selection
• Select fittest ones into the next generation
New Solutions
6
Approximation Model Guided Selection, EMO 2013
Dominance based Selection o Step 1: rank population • dominance rank • dominance count • dominance strength
o Step 2: estimate density • niche and fitness sharing • crowding distance • K-nearest neighbor • gridding
Define a complete order over individuals! 7
Approximation Model Guided Selection, EMO 2013
Indicator based Selection o Convert an MOP into an SOP • Obj. = performance metric
Define a complete order over populations!
8
Approximation Model Guided Selection, EMO 2013
Model Guided Selection o Step 1: model the PF o Step 2: choose reference points o Step 3: select promising solutions Much work needs to be done along this direction! o
H. J. F. Moen, et al., Many-objective Optimization Using Taxi-Cab Surface Evolutionary Algorithm, EMO, 2013.
o
H. Jain, and K. Deb, An improved Adaptive Appraoch for Elitist Nondominated Sorting Genetic Algorithm for Many-Objective Optimization, EMO, 2013.
9
Approximation Model Guided Selection, EMO 2013
Outline
§ Background § Approximation Model Guided Selection § Experimental Results § Discussions & Conclusions 10
Approximation Model Guided Selection, EMO 2013
AMS Framework
Model PF
Define sub-problem
Select
o
A. Zhou, Estimation of Distribution Algorithms for Continuous Multiobjective Optimization, Ph.D Thesis, University of Essex, 2009. (Chapter 5.3)
o
A. Zhou, Q. Zhang, Y. Jin, and B. Sendhoff, Combination of EDA and DE for Continuous Biobjective Optimization, CEC 2008.
o
Q. Zhang and H. Li, MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition, IEEE Trans. on Evolutionary Computation, 2007.
11
Approximation Model Guided Selection, EMO 2013
Zero-order Approximation(AMS0) o A single point to approximate the PF o Model (ideal point)
o Distance to utopian PF (sub-problem)
Simple, but does not consider the shape of PF!
12
Approximation Model Guided Selection, EMO 2013
First-order Approximation(AMS1) o A simplex to approximate the PF o Model (vertices of simplex)
o Distance to simplex
Simple, and consider the shape of PF in a sense! 13
Approximation Model Guided Selection, EMO 2013
Outline
§ Background § Approximation Model Guided Selection § Experimental Results § Discussions & Conclusions 14
Approximation Model Guided Selection, EMO 2013
Settings o Offspring reproduction operator • A probability model based reproduction operator (RM-MEDA)
o Comparison strategy • NDS: non-dominated sorting scheme (NSGA-II)
o Test instances • 8 instances with different properties
o Performance metric • IGD: inverted general distance
15
Approximation Model Guided Selection, EMO 2013
Convex PF
NDS
16
AMS0
AMS1
Approximation Model Guided Selection, EMO 2013
Concave PF
NDS
17
AMS0
AMS1
Approximation Model Guided Selection, EMO 2013
Tri-objective MOP
NDS
18
AMS0
AMS1
Approximation Model Guided Selection, EMO 2013
Statistical Results
NDS:1 19
AMS0:4
AMS1:4
Approximation Model Guided Selection, EMO 2013
Outline
§ Background § Approximation Model Guided Selection § Experimental Results § Discussions & Conclusions 20
Approximation Model Guided Selection, EMO 2013
o AMS works better than NDS in most of the test instances o AMS0 is more stable than AMS1 o AMS1 works better than AMS0 if a good simplex model can be found stability of AMS
stability of AMS
order of model 21
number of objectives
Approximation Model Guided Selection, EMO 2013
o How to build a high-quality model? • model should be cheap • model should be stable
o What’s the performance on complicated problems? • non-concave (non-convex) • with disconnected PF
o What’s the performance on many-objective problems? • interesting sub-problems (reference points, targets points)
22
Approximation Model Guided Selection, EMO 2013
Thanks!
The source code is available from [email protected]
23
Approximation Model Guided Selection, EMO 2013
