Abstract
A discrete particle swarm optimization (PSO) algorithm is a randomized search heuristic for discrete optimization problems. A fundamental question about randomized search heuristics is how long it takes, in expectation, until an optimal solution is found. We give an overview of recent developments related to this question for discrete PSO algorithms. In particular, we give a comparison of known upper and lower bounds of expected runtimes and briefly discuss the techniques used to obtain these bounds.
About the authors
Moritz Mühlenthaler is currently a postdoctoral researcher at the Discrete Optimization group at TU Dortmund University. He received his diploma and doctoral degrees from the University of Erlangen-Nuremberg, Germany, under the supervision of Prof. Rolf Wanka. His research interests include analysis of algorithms, in particular approximation algorithms and randomized search heuristics, as well as combinatorial reconfiguration.
Alexander Raß is a doctoral student at the University of Erlangen-Nuremberg, where he already received his Master of Science in Mathematics. His research interests are runtime analysis of algorithms working on discrete domains and convergence analysis of algorithms working on continuous domains. In both cases the focus is on Particle Swarm Optimization (PSO). Additionally he established an open-source project for PSO with very high and adaptive precision.
References
1. S. Baswana, S. Biswas, B. Doerr, T. Friedrich, P. P. Kurur, and F. Neumann. Computing single source shortest paths using single-objective fitness. In Proc. of the 10th ACM/SIGEVO Workshop on Foundations of Genetic Algorithms (FOGA), pages 59–66, 2009.10.1145/1527125.1527134Search in Google Scholar
2. R. Bellman. Dynamic programming treatment of the travelling salesman problem. J. ACM, 9(1):61–63, Jan. 1962.10.1145/321105.321111Search in Google Scholar
3. M. R. Bonyadi and Z. Michalewicz. Particle swarm optimization for single objective continuous space problems: A review. Evolutionary Computation, 25(1):1–54, 2017.10.1162/EVCO_r_00180Search in Google Scholar PubMed
4. M. Clerc. Discrete particle swarm optimization, illustrated by the Traveling Salesman Problem. In New Optimization Techniques in Engineering, chapter 8, pages 219–239, 2004.10.1007/978-3-540-39930-8_8Search in Google Scholar
5. T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introduction to Algorithms. MIT Press, McGraw-Hill, 1990.Search in Google Scholar
6. B. Doerr and C. Winzen. Ranking-based black-box complexity. Algorithmica, 68(3):571–609, 2014.10.1007/s00453-012-9684-9Search in Google Scholar
7. S. Droste, T. Jansen, and I. Wegener. Upper and lower bounds for randomized search heuristics in black-box optimization. Theory of Computing Systems, 39(4):525–544, 2006.10.1007/s00224-004-1177-zSearch in Google Scholar
8. E. F. G. Goldbarg, G. R. de Souza, and M. C. Goldbarg. Particle swarm for the traveling salesman problem. In European Conference on Evolutionary Computation in Combinatorial Optimization, pages 99–110, Springer, 2006.10.1007/11730095_9Search in Google Scholar
9. W. J. Gutjahr. Ant Colony Optimization: Recent Developments in Theoretical Analysis, pages 225–254, 2011.10.1142/9789814282673_0008Search in Google Scholar
10. C. H. Papadimitriou and K. Steiglitz. Combinatorial Optimization: Algorithms and Complexity. Englewood Cliffs, N. J.: Prentice Hall, 1982.Search in Google Scholar
11. M. Held and R. M. Karp. A dynamic programming approach to sequencing problems. Journal of the Society for Industrial and Applied Mathematics, 10(1):196–210, 1962.10.1145/800029.808532Search in Google Scholar
12. S. Helwig and R. Wanka. Theoretical analysis of initial particle swarm behavior. In Proc. 10th Int. Conf. on Parallel Problem Solving from Nature (PPSN), pages 889–898, 2008.10.1007/978-3-540-87700-4_88Search in Google Scholar
13. M. Hoffmann, M. Mühlenthaler, S. Helwig, and R. Wanka. Discrete particle swarm optimization for TSP: Theoretical results and experimental evaluations. In Proc. 2nd Int. Conf. on Adaptive and Intelligent Systems (ICAIS), pages 416–427, 2011.10.1007/978-3-642-23857-4_40Search in Google Scholar
14. J. Kennedy and R. C. Eberhart. Particle swarm optimization. In Proc. IEEE International Conference on Neural Networks, volume 4, pages 1942–1948, 1995.10.1109/ICNN.1995.488968Search in Google Scholar
15. J. Kennedy and R. C. Eberhart. A discrete binary version of the particle swarm algorithm. In Proc. IEEE Int. Conf. on Systems, Man, and Cybernetics, volume 5, pages 4104–4108, 1997.Search in Google Scholar
16. P. K. Lehre and C. Witt. Black-box search by unbiased variation. Algorithmica, 64(4):623–642, 2012.10.1145/1830483.1830747Search in Google Scholar
17. M. Mühlenthaler, A. Raß, M. Schmitt, A. Siegling, and R. Wanka. Runtime analysis of a discrete particle swarm optimization algorithm on Sorting and OneMax. In Proc. 14th ACM/SIGEVO Workshop on Foundations of Genetic Algorithms (FOGA), pages 13–24, 2017.10.1145/3040718.3040721Search in Google Scholar
18. M. Mühlenthaler, A. Raß, M. Schmitt, and R. Wanka. Exact Markov chain-based runtime analysis of a discrete particle swarm optimization algorithm on Sorting and OneMax. arXiv:1902.01810, 2019. Extended version of [17].Search in Google Scholar
19. A. Raß, J. Schreiner, and R. Wanka. Runtime analysis of discrete particle swarm optimization applied to shortest paths computation. In Proc. 19th Evolutionary Computation in Combinatorial Optimization (EvoCOP), Springer International Publishing, 2019.10.1007/978-3-030-16711-0_8Search in Google Scholar
20. A. Rezaee Jordehi and J. Jasni. Particle swarm optimisation for discrete optimisation problems: a review. Artificial Intelligence Review, 43(2):243–258, Feb. 2015.10.1007/s10462-012-9373-8Search in Google Scholar
21. J. Scharnow, K. Tinnefeld, and I. Wegener. The analysis of evolutionary algorithms on sorting and shortest paths problems. Journal of Mathematical Modelling and Algorithms, 3(4):349–366, 2004.10.1023/B:JMMA.0000049379.14872.f5Search in Google Scholar
22. M. Schmitt and R. Wanka. Particle swarm optimization almost surely finds local optima. Theor. Comput. Sci., 561(PA):57–72, 2015.10.1145/2463372.2463563Search in Google Scholar
23. X. H. Shi, Y. C. Liang, H. P. Leeb, C. Lu, and Q. X. Wang. Particle swarm optimization-based algorithms for TSP and generalized TSP. Information Processing Letters, (103):169–176, 2007.10.1016/j.ipl.2007.03.010Search in Google Scholar
24. D. Sudholt and C. Witt. Runtime analysis of a binary particle swarm optimizer. Theoretical Computer Science, 411(21):2084–2100, 2010.10.1016/j.tcs.2010.03.002Search in Google Scholar
25. K.-P. Wang, L. Huang, C.-G. Zhou, and W. Pang. Particle swarm optimization for traveling salesman problem. In Machine Learning and Cybernetics, 2003 International Conference on, volume 3, pages 1583–1585, IEEE, 2003.Search in Google Scholar
26. I. Wegener. Methods for the analysis of evolutionary algorithms on pseudo-boolean functions. In R. Sarker, M. Mohammadian, and X. Yao, editors, Evolutionary Optimization, chapter 14, pages 349–369, Springer, 2002.10.1007/0-306-48041-7_14Search in Google Scholar
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