RM-MEDA: A REGULARITY
MODEL-BASED
MULTIOBJECTIVE ESTIMATION
OF DISTRIBUTION
ALGORITHM
BISCuit EDA Seminar
2008. 07. 16.
Introduction
2

Multiobjective Optimization Problem
 Optimization
problem with multiple conflicting
objectives
 Most of real-world optimization problem belongs to
multiobjective optimization problem
 Multiple optimal solutions: trade-off solutions

Multiobjective Evolutionary Algorithm
 Can
handle multiple objective directly
 Based on population: find multiple trade-off solutions
simultaneously
(c)2004, SNU Biointelligence Lab., http://bi.snu.ac.kr
Multiobjective Optimization Problem
3

General formulation
Optimize
f i ( X),
i  1,, M
subject to
g j ( X)  0,
j  1,, J
hk ( X)  0, k  1,, K



Where, X=[x1,…xn]T
X is better than Y (X dominates Y)

X is better than or equal to Y in all objectives ( fi s)

X is strictly better than Y in at least one objective
f2
The solutions that are not dominated by any other
solution are Pareto-optimal solutions.
(c)2004, SNU Biointelligence Lab., http://bi.snu.ac.kr
f1
Contributions of the Paper

An estimation of distribution algorithm for continuous
multiobjective optimization based on the regularity
property.
 Karush-Kuhn-Tucker
condition: PS of a continuous
MOP defines a piecewise continuous (m-1)-D manifold
in the decision space.

Systematic experiments on test instances with linear
or nonlinear variable linkage.
RM-MEDA

Centroid
Model

Each centroid is uniformly
distributed over the
piecewise continuous (m1)-D manifold.
m=2: each manifold is a
line segment.
 m=3: each manifold is a
2-D rectangle.


Each individual is
sampled by adding noise
to centroid.
Individual
RM-MEDA

Modeling
Partition the population into K disjoint clusters
 For each cluster, estimate parameters



Shape of manifold, variance (noise, size of manifold), relative
probability for the cluster
Clustering: (m-1)-D local principle component analysis.
RM-MEDA

Algorithm
Experimental Setting



10 test functions
 Convex/concave
 Linear/nonlinear variable linkage
 Uniform/nonuniform distribution over Pareto front
 Multimodal/unimodal
 2-3 objectives
Compared with
 Generalized differential evolution
 NSGA-II : non-EDA style
 MIDEA : mixture of Gaussians
Performance metric: inverted generational distance
 Average of minimum distance from PS to obtained non-dominated set

Convergence and diversity
Experimental Results

Linear variable linkages

RM-MEDA, GDE >> NSGA-II, MIDEA
Experimental Results

Nonlinear variable linkage

RM-MEDA performs better
 Able
to model nonlinear variable linkages.
Experimental
Results

Many local Pareto fronts
Experimental Results

Sensitivity to the number of clusters
Experimental Results

Scalability on different numbers of decision
variables.
• The number of evaluations
required to achieve the given
performance level.
• GDE : dashed line
• RM-MEDA : solid line
Conclusion

Reproduction operators developed for scalar optimization
may not fit for multiobjective optimization problems (MOPs).


RM-MEDA do not directly use location information of
individual solutions.


One of the reasons for the failure of current MOEAs on MOPs
with variable linkages.
RM-MEDA may fail for MOPs with many local Pareto fronts.
Future research topics


Combination of location information and global statistical
information. Ex) Guided mutation.
Use of other machine learning techniques.