1. RM-MEDA: A Regularity Model-Based Multiobjective Estimation of Distribution Algorithm
BISCuit EDA Seminar

2. Introduction 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.,

3. Multiobjective Optimization Problem
General formulation
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
The solutions that are not dominated by any other solution are Pareto-optimal solutions. f2
The multi-objective optimization problem is generally formalized like this (-> 수식)
It has M objective functions and J inequality constraints and K equality constraints.
To compare solutions in MOP, the domination relation is used. X dominates Y if and only if X satisfies these two conditions.
X is better than or equal to Y in all objective functions f and X is strictly better than Y in at least one ojective.
The blue point in the graph is dominated by the yellow point (파란 원하고 화살표로 이어진 것). We suppose f1 and f2 as minimization functions.
The blue point has greater function value than yellow point in both objectives. And the yellow points cannot dominate each other, but dominate points upper-right points. We call these yellow points as non-dominated set and the non-dominated set which are not dominated by any other solution in search space is called as Pareto-optimal solutions. f1
(c)2004, SNU Biointelligence Lab.,

4. 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.

5. RM-MEDA
Centroid
Model
Each centroid is uniformly distributed over the piecewise continuous (m- 1)-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

6. 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.

7. RM-MEDA
Algorithm

8. 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

9. Experimental Results Linear variable linkages
RM-MEDA, GDE >> NSGA-II, MIDEA

10. Experimental Results Nonlinear variable linkage
RM-MEDA performs better
Able to model nonlinear variable linkages.

11. Experimental Results
Many local Pareto fronts

12. Experimental Results
Sensitivity to the number of clusters

13. 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

14. Conclusion
Reproduction operators developed for scalar optimization may not fit for multiobjective optimization problems (MOPs).
One of the reasons for the failure of current MOEAs on MOPs with variable linkages.
RM-MEDA do not directly use location information of individual solutions.
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.