Multimodal Optimization (Niching)

1. Multimodal Optimization (Niching) A/Prof. Xiaodong Li School of Computer Science and IT, RMIT University Melbourne, Australia Email: xiaodong.li@rmit.edu.au April 2015

2. Niching • The notion of niching is inspired by nature. In natural ecosystems, individual species must compete to survive by taking on different roles. Different species evolve to fill different “niches” (or subspaces) in the environment that can support different types of life. • Instead of evolving a single population of individuals indifferently, natural ecosystems evolve different species (or subpopulations) to fill different niches. The terms species and niche are sometimes interchangeable.

3. Speciation and niching

4. Speciation and niching Biological species concept:a species is a group of actually or potentially interbreeding individuals who are reproductively isolated from other such groups. The definition of a species is still debatable. Most researchers believe either the morphological species concept (ie., members of a species look alike and can be distinguished from other species by their appearance), or the biological species concept (a species is a group of actually or potentially interbreeding individuals who are reproductively isolated from other such groups). Both definitions have their weaknesses.

5. Niching in Evolutionary Algorithms • Niching methods were introduced to EAs to allow maintenance of a population of diverse individuals so that multiple optima within a single population can be located. • As Mahoud described “A niching method must be able to form and maintain multiple, diverse, final solutions, whether these solutions are of identical fitness or of varying fitness. A niching method must be able to maintain these solutions for an exponential to infinite time period, with respect to population size.”

6. Why niching? • Many real-world problems are “multimodal” by nature, that is, multiple satisfactory solutions exist. For an optimization problem with multiple global and local optima, it might be desirable to locate all global optima and/or some local optima that are also considered as being satisfactory. • A niching method can be incorporated into a standard EA to promote and maintain formation of multiple stable subpopulations within a single population, with an aim to locate multiple optimal or suboptimal solutions.

7. Multimodal problems

8. Example 1 – Truss topologies Multiple optimal truss topologies (i.e., optimal solutions) found by using a niching method.

9. Example 2 – sports car chassis Between coupe and open-top, various different combinations of load cases. Each combination of load cases lead to different topologies. Interpretation of the results can become quite hard.

10. Classic niching methods • Fitness sharing and Crowding methods (early 70s and 80s). • In subsequent years, many niching methods have been proposed. Some representative examples include deterministic crowding, derating, restricted tournament selection, parallelization, clustering, and speciation. • More recently, NichePSO, Speciation-based PSO, Ring topology PSO.

11. Multimodal functions

12. Fitness Sharing

13. Fitness sharing The most widely used sharing function is the following: However, there is no easy task to set the niche radius parameter.

14. Crowding • De Jong’s crowding was initially designed only to preserve population diversity. • In crowding, an offspring is compared to a small random sample taken from the current population, and the most similar individual in the sample is replaced. • A parameter crowding factor (CF) is commonly used to determine the size of the sample.

15. Deterministic crowding • Mahfoud found that De Jong’s crowding method was unable to maintain more than two peaks of a five peaks fitness landscape due to stochastic replacement errors. • Mahfoud then made several modifications to crowding to reduce replacement errors, restore selection pressure, and also eliminate the crowding factor parameter. The resulting algorithm, deterministic crowding, was able to locate and maintain multiple peaks.

16. s1 s2 p s3 2rs x Speciation-based PSO f An example of how to determine the species seeds from the population at each iteration. s1, s2, and s3 are chosen as the species seeds. Note that p follows s2.

17. Performance measures Peak Ratio (PR) and Success Rate (SR):

18. How to determine solutions found? Here we assume the number of global peaks, and the distance between two closest global peaks are known in advance. For further information, see the technical report on CEC’13 competition benchmark on multimodal optimization.

19. Multimodal functions

20. Simulation runs Refer to Li (2004) for details.

21. Niching parameters Difficulty in choosing the niching parameters such as the species radius r . For example, for Shubert 2D, there is no value of r that can distinguish the global optima without individuals becoming overly trapped in local optima. Some recent works in handling this problem (Bird & Li, 2006a; Bird & Li, 2006b).

22. Ring topology based PSO

23. Ring topology based PSO A PSO algorithm using the ring topology can operate as a niching algorithm by using individual particles’ local memories to form a stable network retaining the best positions found so far, while these particles explore the search space more broadly. See (Li , 2010).

24. Further readings • Li, X. (2010), “Niching without Niching Parameters: Particle Swarm Optimization Using a Ring Topology”, IEEE Transactions on Evolutionary Computation, Volume 14, No.1, February 2010, pp.150-169. • Li, X., Engelbrecht, A. and Epitropakis,M.G. (2013), “Benchmark Functions for CEC’2013 Special Session and Competition on NichingMethods for Multimodal Function Optimization”. Technical Report, Evolutionary Computation and Machine Learning Group, RMIT University, Australia, 2013.