RM-MEDA: A Regularity Model-Based Multiobjective Estimation of Distribution Algorithm BISCuit EDA Seminar 2008. 07. 16.

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., http://bi.snu.ac.kr

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., http://bi.snu.ac.kr

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 (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

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). 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.