Combination of Evolutionary Algorithms with Other Techniques

1. Combination of Evolutionary Algorithms with Other Techniques Qingfu Zhang Department of Computing & Electronic Systems, University of Essex, UK

2. Q. Zhang & Y. W. Leung, Orthogonal GA for Routing, IEEE Trans on EC, 1999 • Q. Zhang, J. Sun & E. Tsang, EA with guided mutation, IEEE Trans on EC, 2005. • Q. Zhang & H. Li, MOEA/D, IEEE Trans on EC, 2007 • Q. Zhang, A. Zhou, and Y. Jin, RM-MEDA, IEEE Trans on EC, 2008. • H. Li and Q. Zhang, MOP with complicated PS, MOEA/D and NSGA-II, IEEE Trans on EC, 2008, accepted.

3. Motivations • Two heads are better one. • EAs are general ideas, and an EA alone is often not very efficient. • Problem-specific knowledge could be used EA + other techniques.

4. Contents • Orthogonal Crossover: Crossover based on experimental design • Guided Mutation: EDA+GA • RM-MEDA: Multiobjective Evolutionary Algorthim based on Regularity Property. • MOEA/D: Multiobjective Evolutionary Algorithm Based on Aggregation

5. Crossover Based on Experimental Design • Crossover • Find nrepresentative points in a area determined by m parent points. • “Representative” has been well studied in experimental design. Parent solutions

6. Orthogonal design: Example • The quality of a product depends on three factors A, B and C. Each factor has three levels (three possible values). • An experiment=A combination of factor-levels, e.g, (80, 20, 3). • In the case of k factors at n levels, we have nk possible experiments. • Very often, it is impossible to test all the experiments. • Experimental design tell us how to select a small but representative experiments for testing. • Orthogonal Array: tool for selecting representative experiments:

7. Orthogonal Crossover Based on L4(23) Example • Given 2 parent solutions: • (Randomly) divide two parents into three parts (as in 2-point Xover) • We have three factors: red, black, blue, each of them has two levels: $’s and £’s. • We can use L4(23) to generate 4 offspring. ($=1, £=2). X=($$$$$$$$$$$$$$$$$$$$$$$) Y=(£££££££££££££££££££££££) X=($$$$$$$$$$$$$$$$$$$$$$$) Y=(£££££££££££££££££££££££) ($$$$$$$$$$$$$$$$$$$$$$$) (££££££££$$$$$$$$$$$$$$$) (££££££££$$$$$$$$$££££££) (£££££££££££££££££$$$$$$)

8. Contents • Orthogonal Crossover: Crossover based on experimental design • Guided Mutation: EDA+GA • RM-MEDA: Multiobjective Evolutionary Algorthim based on Regularity Property. • MOEA/D: Multiobjective Evolutionary Algorithm Based on Aggregation

9. Guided Mutation • Motivations • EDA : build a prob. model to model “good solutions” based on info from the previous generations, and then sample new solutions from the model. • EDAs can use globally statistical information. But the location information of each individual is not directly used in generating new solutions. They may ignore isolated good solutions In modelling, two isolated points (red ones) may be ignored.

10. Sometimes, we couldn’t build a correct model. • EDAs have no mechanism to control the similarity (distance) between new solutions and a given solution. Proximate Optimality Principle (POP): good solutions are similar. • Blue points: parent sets. • We build a normal distribution model to model the distribution of the blue points. • Red points are sampled from the model. • Some red points are far from the blue ones. Guided mutation is an operator based on location information and global statistical information.

11. Basic idea Model ($$$$$$$$$$$$$$$$) £ $ ($£$$$£$$£$££$$$$)

12. Example • Input: • The prob. model • A point • Control parameter: • Output: • How to generate

13. Contents • Orthogonal Crossover: Crossover based on experimental design • Guided Mutation: EDA+GA • MOEA/D: Multiobjective Evolutionary Algorithm Based on Aggregation • RM-MEDA: Multiobjective Evolutionary Algorthim based on Regularity Property.

14. Pareto set (PS) Pareto front (PF) Task of MOEAs Solving a MOP requires interactions between decision makers and engineers. Very often, A decision maker wants: A representative set of Pareto optimal solutions (uniformly distributed along the PF or PS ) The task of most Multiobjective Evolutionary Algorithms (MOEAs)

15. N min g ( x, λ ) Background • Most current MOEAs are based on Pareto domination. • Very hard to make its solutions uniformly distributed along the PF, particularly if the population size is small. • Ideas in traditional math program methods: Decomposition (aggregation) N problemS. Not a N-obj opt problem! Finding a set of N uniformly distributed Pareto optimal solutions Solve these N problems one by one. The distribution of final solutions could be very uniform if g( ) and λ are properly chosen.

16. N min g ( x, λ ) Idea • These problems are related with each other. • If λiand λj are close, we can call neighbours. • neighbouring problems should have similar solutions. • N agents are used for solving these N problems • During the search, neighbouring agents can help each other.

17. Algorithm Framework • Agent i records , the best solution he has found so far for his problem. • At each generation, each agent i does the following: • Randomly select several neighbours and obtain their best solutions. • Apply genetic operators on these selected solutions and generate a new solution . • Apply single opt. local search on to optimise its obj and obtain . • Replace by if . • Let its neighbours replace their best solutions by if is better than their current best solutions (measured by their individual objectives).

18. More Details of MOEA/D • Initialisation: each agent can initialise randomly or by using problem-specific knowledge, e.g. • Randomly generate a point in the decision space and then use a single obj LS to improve it. • Decomposition Method: • Any methods should do. We have tried weight sum approach, Tchebycheff approach and Penalty based boundary intersection (PBI) approach (which we proposed by modifying the NBI method). • The setting of uniformly distributed in Other methods can also be considered, e.g., dynamically tuning. • Reference point (needed in PBI and Tchebycheff approaches) is estimated from the previous search.

19. Contents • Orthogonal Crossover: Crossover based on experimental design • Guided Mutation: EDA+GA • MOEA/D: Multiobjective Evolutionary Algorithm Based on Aggregation • RM-MEDA: Multiobjective Evolutionary Algorthim based on Regularity Property

20. Pareto set (PS) Pareto front (PF) Motivations • Regularity of continuous MOP: • This property has been ignored by MOEA researchers. • The PSs in most commonly-used test problems are too simple (a line or a plane). • There is very little research on the shape of PSs. Under certain conditions, the PS (PF) is a (m-1)-dimensional piecewise continuous manifold in decision (objective) space. Where m is the # of the objs. How can we deal with a continuous MOP if its PS is (m-1)-D piecewise continuous manifold?

21. Suppose we use the following commonly-used framework: Why commonly-used genetic operators do not work well for complicated PSs? When two parents are in the PS, their offspring may not be close to the PS. The PS is not an equilibrium. Population selection Reproduction operators Competition Replacement PS in decision Space. New Solutions Very little work Lots of work So we resort to EDA (Modelling and Sampling).

22. In the case of 2 objs The PS is a 1-D curve. If the algorithm works well. The principal (centroid) curve of the population could be an approximation to the PS. Population selection Modelling & Sampling Competition Replacement Population PS New Solutions Basic Idea

23. Population selection Modelling & Sampling Competition Replacement New Solutions This model is different from other models in EDAs. How to model C and ?

24. : point in the current population. : centriod: C simplification The number of clusters needs to be preset. Modelling: How to model C and ? We assume that: • The centroid curve C consists of several line segments. This assumption makes C computable. • How to model C • Divide the population into several clusters by local PCA. • Compute the central line of each cluster. • How to model • the deviation of the points in each cluster to its central line.

25. x1 x2 x Sampling • How to sample new solutions: • The number of new solutions sampled around : • Sampling around • Uniformly randomly pick a point x1 in • x2~N(0, ), • x=x1+x2.

26. We use the non-dominated sorting (Deb et al) in selection. Convergence: prefer solutions close to the PF. Diversity: prefer “isolated” solutions Population selection Modelling & Sampling Competition Replacement New Solutions Selection

27. Conclusion • Traditional methods (ideas) + Population based algorithms works Source codes of these methods can be downloaded from http://dces.essex.ac.uk/staff/qzhang/

28. Papers and codes can be downloaded from: http://cswww.essex.ac.uk/staff/qzhang/ Thanks!