Global convergence analysis of the bat algorithm using a markovian framework and dynamical system theory

Article

Chen, S., Peng, G., He, X. and Yang, X. 2018. Global convergence analysis of the bat algorithm using a markovian framework and dynamical system theory. Expert Systems with Applications. 114, pp. 173-182. https://doi.org/10.1016/j.eswa.2018.07.036
TypeArticle
TitleGlobal convergence analysis of the bat algorithm using a markovian framework and dynamical system theory
AuthorsChen, S., Peng, G., He, X. and Yang, X.
Abstract

The bat algorithm (BA) has been shown to be effective to solve a wider range of optimization problems. However, there is not much theoretical analysis concerning its convergence and stability. In order to prove the convergence of the bat algorithm, we have built a Markov model for the algorithm and proved that the state sequence of the bat population forms a finite homogeneous Markov chain, satisfying the global convergence criteria. Then, we prove that the bat algorithm can have global convergence. In addition, in order to enhance the convergence performance of the algorithm and to identify the possible effect of parameter settings on convergence, we have designed an updated model in terms of a dynamic matrix. Subsequently, we have used the stability theory of discrete-time dynamical systems to obtain the stable parameter ranges for the algorithm. Furthermore, we use some benchmark functions to demonstrate that BA can indeed achieve global optimality efficiently for these functions.

KeywordsBat algorithm; Global convergence; Markov chain theory; Dynamical system theory; Parameters selection; Optimization; Swarm intelligence
PublisherElsevier
JournalExpert Systems with Applications
ISSN0957-4174
Electronic1873-6793
Publication dates
Online18 Jul 2018
Print30 Dec 2018
Publication process dates
Deposited08 Aug 2018
Accepted16 Jul 2018
Submitted06 Mar 2018
Output statusPublished
Accepted author manuscript
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Copyright Statement

© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/

Digital Object Identifier (DOI)https://doi.org/10.1016/j.eswa.2018.07.036
Web of Science identifierWOS:000446949300013
LanguageEnglish
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Analysis of randomisation methods in swarm intelligence
Fister, I. (Jr.), Yang, X., Brest, J., Fister, D. and Fister, I. 2015. Analysis of randomisation methods in swarm intelligence. International Journal of Bio-Inspired Computation. 7 (1), pp. 36-49. https://doi.org/10.1504/IJBIC.2015.067989
Analysis of quality-of-service aware orthogonal frequency division multiple access system considering energy efficiency
Ting, T., Yang, X., Lee, S. and Chien, S. 2014. Analysis of quality-of-service aware orthogonal frequency division multiple access system considering energy efficiency. IET Communications. 8 (11), pp. 1947-1954. https://doi.org/10.1049/iet-com.2013.1161
A novel improved accelerated particle swarm optimization algorithm for global numerical optimization
Wang, G., Hossein Gandomi, A., Yang, X. and Hossein Alavi, A. 2014. A novel improved accelerated particle swarm optimization algorithm for global numerical optimization. Engineering Computations. 31 (7), pp. 1198-1220. https://doi.org/10.1108/EC-10-2012-0232
Oil supply between OPEC and non-OPEC based on game theory
Chang, Y., Yi, J., Yan, W., Yang, X., Zhang, S., Gao, Y. and Wang, X. 2014. Oil supply between OPEC and non-OPEC based on game theory. International Journal of Systems Science. 45 (10), pp. 2127-2132. https://doi.org/10.1080/00207721.2012.762562
Advances of swarm intelligent systems in gene expression data classification
Talatahari, E., Talatahari, S., Gandomi, A. and Yang, X. 2014. Advances of swarm intelligent systems in gene expression data classification. Journal of Multiple-Valued Logic and Soft Computing. 22 (3), pp. 307-315.
Bat algorithm is better than intermittent search strategy
Yang, X., Deb, S. and Fong, S. 2014. Bat algorithm is better than intermittent search strategy. Journal of Multiple-Valued Logic and Soft Computing. 22 (3), pp. 223-237.
Nature-inspired optimization algorithms
Yang, X. 2014. Nature-inspired optimization algorithms. Elsevier.
Non-dominated sorting cuckoo search for multiobjective optimization
He, X., Li, N. and Yang, X. 2014. Non-dominated sorting cuckoo search for multiobjective optimization. 2014 IEEE Symposium on Swarm Intelligence (SIS). Orlando, Florida 09 - 12 Dec 2014 IEEE. pp. 1-7 https://doi.org/10.1109/SIS.2014.7011772
Computational optimization, modelling and simulation: past, present and future
Yang, X., Koziel, S. and Leifsson, L. 2014. Computational optimization, modelling and simulation: past, present and future. 5th workshop on Computational Optimization, Modelling and Simulations (COMS 2014) at the ICCS 2014. Cairns, Australia 10 - 12 Jun 2014 Elsevier. pp. 754-758 https://doi.org/10.1016/j.procs.2014.05.067
Discrete cuckoo search algorithm for job shop scheduling problem
Ouaarab, A., Ahiod, B., Yang, X. and Abbad, M. 2014. Discrete cuckoo search algorithm for job shop scheduling problem. 2014 IEEE International Symposium on Intelligent Control (ISIC). France 08 - 10 Oct 2014 IEEE. pp. 1872-1876 https://doi.org/10.1109/ISIC.2014.6967636
Nature-inspired framework for hyperspectral band selection
Nakamura, R., Garcia Fonseca, L., Dos Santos, J., Da S. Torres, R., Yang, X. and Papa, J. 2014. Nature-inspired framework for hyperspectral band selection. IEEE Transactions on Geoscience and Remote Sensing. 52 (4), pp. 2126-2137. https://doi.org/10.1109/TGRS.2013.2258351
A discrete firefly algorithm for the multi-objective hybrid flowshop scheduling problems
Marichelvam, M., Prabaharan, T. and Yang, X. 2014. A discrete firefly algorithm for the multi-objective hybrid flowshop scheduling problems. IEEE Transactions on Evolutionary Computation. 18 (2), pp. 301-305. https://doi.org/10.1109/TEVC.2013.2240304
Binary bat algorithm
Mirjalili, S., Mirjalili, S. and Yang, X. 2014. Binary bat algorithm. Neural Computing and Applications. 25 (3-4), pp. 663-681. https://doi.org/10.1007/s00521-013-1525-5
Bat algorithm based on simulated annealing and Gaussian perturbations
He, X., Ding, W. and Yang, X. 2014. Bat algorithm based on simulated annealing and Gaussian perturbations. Neural Computing and Applications. 25 (2), pp. 459-468. https://doi.org/10.1007/s00521-013-1518-4
Discrete cuckoo search algorithm for the travelling salesman problem
Ouaarab, A., Ahiod, B. and Yang, X. 2014. Discrete cuckoo search algorithm for the travelling salesman problem. Neural Computing and Applications. 24 (7-8), pp. 1659-1669. https://doi.org/10.1007/s00521-013-1402-2
Cuckoo search: recent advances and applications
Yang, X. and Deb, S. 2014. Cuckoo search: recent advances and applications. Neural Computing and Applications. 24 (1), pp. 169-174. https://doi.org/10.1007/s00521-013-1367-1
Chaotic bat algorithm
Gandomi, A. and Yang, X. 2014. Chaotic bat algorithm. Journal of Computational Science. 5 (2), pp. 224-232. https://doi.org/10.1016/j.jocs.2013.10.002
Mathematical modelling and parameter optimization of pulsating heat pipes
Yang, X., Karamanoglu, M., Luan, T. and Koziel, S. 2014. Mathematical modelling and parameter optimization of pulsating heat pipes. Journal of Computational Science. 5 (2), pp. 119-125. https://doi.org/10.1016/j.jocs.2013.12.003
An empirical study of test effort estimation based on bat algorithm
Srivastava, P., Bidwai, A., Khan, A., Rathore, K., Sharma, R. and Yang, X. 2014. An empirical study of test effort estimation based on bat algorithm. International Journal of Bio-Inspired Computation. 6 (1), pp. 57-70. https://doi.org/10.1504/IJBIC.2014.059966
Bio-inspired computation: success and challenges of IJBIC
Yang, X. and Cui, Z. 2014. Bio-inspired computation: success and challenges of IJBIC. International Journal of Bio-Inspired Computation. 6 (1), pp. 1-6. https://doi.org/10.1504/IJBIC.2014.059969
A wrapper approach for feature selection based on Bat Algorithm and Optimum-Path Forest
Rodrigues, D., Pereira, L., Nakamura, R., Costa, K., Yang, X., Souza, A. and Papa, J. 2014. A wrapper approach for feature selection based on Bat Algorithm and Optimum-Path Forest. Expert Systems with Applications. 41 (5), pp. 2250-2258. https://doi.org/10.1016/j.eswa.2013.09.023
Improved cuckoo search algorithm for hybrid flow shop scheduling problems to minimize makespan
Marichelvam, M., Prabaharan, T. and Yang, X. 2014. Improved cuckoo search algorithm for hybrid flow shop scheduling problems to minimize makespan. Applied Soft Computing. 19, pp. 93-101. https://doi.org/10.1016/j.asoc.2014.02.005
A bio-inspired algorithm for identification of critical components in the transportation networks
Zhang, X., Adamatzky, A., Yang, H., Mahadaven, S., Yang, X., Wang, Q. and Deng, Y. 2014. A bio-inspired algorithm for identification of critical components in the transportation networks. Applied Mathematics and Computation. 248, pp. 18-27. https://doi.org/10.1016/j.amc.2014.09.055
A firefly-inspired method for protein structure prediction in lattice models
Maher, B., Albrecht, A., Loomes, M., Yang, X. and Steinhofel, K. 2014. A firefly-inspired method for protein structure prediction in lattice models. Biomolecules. 4 (1), pp. 56-75. https://doi.org/10.3390/biom4010056
True global optimality of the pressure vessel design problem: a benchmark for bio-inspired optimisation algorithms
Yang, X., Huyck, C., Karamanoglu, M. and Khan, N. 2013. True global optimality of the pressure vessel design problem: a benchmark for bio-inspired optimisation algorithms. International Journal of Bio-Inspired Computation. 5 (6), pp. 329-335. https://doi.org/10.1504/IJBIC.2013.058910
Applications and analysis of bio-inspired eagle strategy for engineering optimization
Yang, X., Karamanoglu, M., Ting, T. and Zhao, Y. 2014. Applications and analysis of bio-inspired eagle strategy for engineering optimization. Neural Computing and Applications. 25 (2), pp. 411-420. https://doi.org/10.1007/s00521-013-1508-6
Flower pollination algorithm: a novel approach for multiobjective optimization
Yang, X., Karamanoglu, M. and He, X. 2014. Flower pollination algorithm: a novel approach for multiobjective optimization. Engineering Optimization. 46 (9), pp. 1222-1237. https://doi.org/10.1080/0305215X.2013.832237
Random walks, Lévy flights, Markov chains and metaheuristic optimization
Yang, X., Ting, T. and Karamanoglu, M. 2013. Random walks, Lévy flights, Markov chains and metaheuristic optimization. in: Future information communication technology and applications: ICFICE 2013 Springer Netherlands.
Are motorways rational from slime mould's point of view?
Adamatzky, A., Akl, S., Alonso-Sanz, R., Van Dessel, W., Ibrahim, Z., Ilachinski, A., Jones, J., Kayem, A., Martínez, G., De Oliveira, P., Prokopenko, M., Schubert, T., Sloot, P., Strano, E. and Yang, X. 2013. Are motorways rational from slime mould's point of view? International Journal of Parallel, Emergent and Distributed Systems. 28 (3), pp. 230-248. https://doi.org/10.1080/17445760.2012.685884
Advances in simulation-driven optimization and modelling
Koziel, S., Leifsson, L. and Yang, X. 2012. Advances in simulation-driven optimization and modelling. Journal of Computational Methods in Science and Engineering. 12 (1-2), pp. 1-4. https://doi.org/10.3233/JCM-2012-0400
Metaheuristic algorithms for self-organizing systems: a tutorial
Yang, X. 2012. Metaheuristic algorithms for self-organizing systems: a tutorial. in: Self-Adaptive and Self-Organizing Systems (SASO), 2012 IEEE Sixth International Conference on IEEE Conference Publications. pp. 249 -250
Integrating nature-inspired optimization algorithms to K-means clustering
Tang, R., Fong, S., Yang, X. and Deb, S. 2012. Integrating nature-inspired optimization algorithms to K-means clustering. in: Seventh International Conference on Digital Information Management (ICDIM), IEEE Conference Publications. pp. 116-123
Wolf search algorithm with ephemeral memory
Rui, T., Fong, S., Yang, X. and Deb, S. 2012. Wolf search algorithm with ephemeral memory. in: Seventh International Conference on Digital Information Management (ICDIM), 2012 IEEE Conference Publications. pp. 165 -172
Metaheuristic applications in structures and infrastructures
Gandom, A., Yang, X., Talatahari, S. and Alavi, A. 2013. Metaheuristic applications in structures and infrastructures. Elsevier.
Multi-objective flower algorithm for optimization
Yang, X., Karamanoglu, M. and He, X. 2013. Multi-objective flower algorithm for optimization. 2013 International Conference on Computational Science . Barcelona, Spain. 05 - 07 Jun 2013 Elsevier. pp. 861- 868 https://doi.org/10.1016/j.procs.2013.05.251
Optimal test sequence generation using firefly algorithm
Srivatsava, P., Mallikarjun, B. and Yang, X. 2013. Optimal test sequence generation using firefly algorithm. Swarm and Evolutionary Computation. 8, pp. 44-53. https://doi.org/10.1016/j.swevo.2012.08.003
Multiobjective firefly algorithm for continuous optimization
Yang, X. 2013. Multiobjective firefly algorithm for continuous optimization. Engineering with Computers. 29 (2), pp. 175-184. https://doi.org/10.1007/s00366-012-0254-1
Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems
Gandomi, A., Yang, X. and Alavi, A. 2013. Cuckoo search algorithm: a metaheuristic approach to solve structural optimization problems. Engineering with Computers. 29 (1), pp. 17-35. https://doi.org/10.1007/s00366-011-0241-y
Multiobjective cuckoo search for design optimization
Yang, X. and Deb, S. 2013. Multiobjective cuckoo search for design optimization. Computers and Operations Research. 40 (6), pp. 1616-1624. https://doi.org/10.1016/j.cor.2011.09.026
Flower pollination algorithm for global optimization
Yang, X. 2012. Flower pollination algorithm for global optimization. in: Unconventional Computation and Natural Computation: 11th International Conference, UCNC 2012, Orléan, France, September 3-7, 2012. Proceedings Berlin Springer.
Parameter estimation from laser flash experiment data
Wright, L., Yang, X., Matthews, C., Chapman, L. and Roberts, S. 2011. Parameter estimation from laser flash experiment data. in: Computational optimization and applications in engineering and industry studies in computational intelligence Berlin Springer.
Benchmark problems in structural optimization
Gandomi, A. and Yang, X. 2011. Benchmark problems in structural optimization. in: Koziel, S. and Yang, X. (ed.) Computational optimization, methods and algorithms Springer.
Computational optimization: an overview
Yang, X. and Koziel, S. 2011. Computational optimization: an overview. in: Computational Optimization, Methods and Algorithms Springer.
Engineering optimisation by cuckoo search
Yang, X. and Deb, S. 2010. Engineering optimisation by cuckoo search. International Journal of Mathematical Modelling and Numerical Optimisation. 1 (4), pp. 330 -343. https://doi.org/10.1504/IJMMNO.2010.03543
Engineering optimization: an introduction with metaheuristic applications
Yang, X. 2010. Engineering optimization: an introduction with metaheuristic applications. New Jersey John Wiley & Sons.
Metaheuristics in water, geotechnical and transport engineering
Yang, X., Gandomi, A., Talatahari, S. and Alavi, A. 2012. Metaheuristics in water, geotechnical and transport engineering. Elsevier.
Bat algorithm for topology optimization in microelectronic applications
Yang, X., Karamanoglu, M. and Fong, S. 2012. Bat algorithm for topology optimization in microelectronic applications. 1st International Conference on Future Generation Communication Technology. London, UK 12 - 14 Dec 2012 IEEE. pp. 150-155 https://doi.org/10.1109/FGCT.2012.6476566
Rare events forecasting using a residual-feedback GMDH neural network
Fong, S., Nannan, Z., Wong, R. and Yang, X. 2012. Rare events forecasting using a residual-feedback GMDH neural network. in: Seventh International Conference onDigital Information Management (ICDIM), 2012 IEEE Conference Publications. pp. 464-473
Cuckoo search for business optimization applications
Yang, X., Deb, S., Karamanoglu, M. and He, X. 2012. Cuckoo search for business optimization applications. National Conference on Computing and Communication Systems (NCCCS). Durgapur, West Bengal, India 21 - 22 Nov 2012 IEEE. https://doi.org/10.1109/NCCCS.2012.6412973
Bat algorithm for constrained optimization tasks
Gandomi, A., Yang, X., Alavi, A. and Talatahari, S. 2012. Bat algorithm for constrained optimization tasks. Neural Computing and Applications. 22 (6), pp. 1239-1255. https://doi.org/10.1007/s00521-012-1028-9
Efficiency analysis of swarm intelligence and randomization techniques
Yang, X. 2012. Efficiency analysis of swarm intelligence and randomization techniques. Journal of Computational and Theoretical Nanoscience. 9 (2), pp. 189-198. https://doi.org/10.1166/jctn.2012.2012
Modelling of a pulsating heat pipe and start up asymptotics
Yang, X. and Luan, T. 2012. Modelling of a pulsating heat pipe and start up asymptotics. Procedia Computer Science. 9, pp. 784-791. https://doi.org/10.1016/j.procs.2012.04.084
Computational optimization, modelling and simulation: smart algorithms and better models
Yang, X., Koziel, S. and Leifsson, L. 2012. Computational optimization, modelling and simulation: smart algorithms and better models. Procedia Computer Science. 9, pp. 852-856. https://doi.org/10.1016/j.procs.2012.04.091
Free lunch or no free lunch: that is not just a question?
Yang, X. 2012. Free lunch or no free lunch: that is not just a question? International Journal on Artificial Intelligence Tools. 21 (3). https://doi.org/10.1142/S0218213012400106
Coupled eagle strategy and differential evolution for unconstrained and constrained global optimization
Gandomi, A., Yang, X., Talatahari, S. and Deb, S. 2012. Coupled eagle strategy and differential evolution for unconstrained and constrained global optimization. Computers and Mathematics with Applications. 63 (1), pp. 191-200. https://doi.org/10.1016/j.camwa.2011.11.010
Bat algorithm: a novel approach for global engineering optimization
Yang, X. and Gandomi, A. 2012. Bat algorithm: a novel approach for global engineering optimization. Engineering Computations. 29 (5), pp. 464-483. https://doi.org/10.1108/02644401211235834
Evolutionary boundary constraint handling scheme
Gandomi, A. and Yang, X. 2012. Evolutionary boundary constraint handling scheme. Neural Computing and Applications. 21 (6), pp. 1449-1462. https://doi.org/10.1007/s00521-012-1069-0
Two-stage eagle strategy with differential evolution
Yang, X. and Deb, S. 2012. Two-stage eagle strategy with differential evolution. International Journal of Bio-Inspired Computation. 4 (1), pp. 1-5. https://doi.org/10.1504/IJBIC.2012.044932
Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect
Yang, X., Hosseini, S. and Gandomi, A. 2012. Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect. Applied Soft Computing. 12 (3), pp. 1180-1186. https://doi.org/10.1016/j.asoc.2011.09.017
Accelerated particle swarm optimization and support vector machine for business optimization and applications
Yang, X., Deb, S. and Fong, S. 2011. Accelerated particle swarm optimization and support vector machine for business optimization and applications. Fong, S. (ed.) NDT 2011: International Conference on Networked Digital Technologies. Macau, China 11 - 13 Jul 2011 Springer. pp. 53-66 https://doi.org/10.1007/978-3-642-22185-9_6
Bat algorithm for multi-objective optimisation
Yang, X. 2011. Bat algorithm for multi-objective optimisation. International Journal of Bio-Inspired Computation. 3 (5), pp. 267-274. https://doi.org/10.1504/IJBIC.2011.042259
Mixed variable structural optimization using Firefly Algorithm
Gandomi, A., Yang, X. and Alavi, A. 2011. Mixed variable structural optimization using Firefly Algorithm. Computers and Structures. 89 (23-24), pp. 2325-2336. https://doi.org/10.1016/j.compstruc.2011.08.002
A new metaheuristic bat-inspired algorithm
Yang, X. 2010. A new metaheuristic bat-inspired algorithm. in: González, J., Pelta, D., Cruz, C., Terrazas, G. and Krasnogor, N. (ed.) Nature Inspired Cooperative Strategies for Optimization (NISCO 2010) Berlin Springer.
Eagle strategy using Lévy walk and firefly algorithm for stochastic optimization
Yang, X. and Deb, S. 2010. Eagle strategy using Lévy walk and firefly algorithm for stochastic optimization. in: González, J., Pelta, D, Cruz, C., Terrazas, G. and Krasnogor, N. (ed.) Nature Inspired Cooperative Strategies for Optimization (NICSO 2010) Springer. pp. 101-111
Optimization and data mining for fracture prediction in geosciences
Shi, G. and Yang, X. 2010. Optimization and data mining for fracture prediction in geosciences. Procedia Computer Science. 1 (1), pp. 1359-1366. https://doi.org/10.1016/j.procs.2010.04.151
Firefly algorithm, stochastic test functions and design optimisation
Yang, X. 2010. Firefly algorithm, stochastic test functions and design optimisation. International Journal of Bio-Inspired Computation. 2 (2), pp. 78-84. https://doi.org/10.1504/IJBIC.2010.032124
Oil and gas assessment of the Kuqa depression of Tarim Basin in western China by simple fluid flow models of primary and secondary migrations of hydrocarbons
Shi, G., Zhang, Q., Yang, X. and Mi, S. 2010. Oil and gas assessment of the Kuqa depression of Tarim Basin in western China by simple fluid flow models of primary and secondary migrations of hydrocarbons. Journal of Petroleum Science and Engineering. 75 (1-2), pp. 77-90. https://doi.org/10.1016/j.petrol.2010.10.009
Firefly algorithms for multimodal optimization
Yang, X. 2009. Firefly algorithms for multimodal optimization. in: Stochastic Algorithms: Foundations and Applications; 5th International Symposium, SAGA 2009, Sapporo, Japan, October 26-28, 2009. Proceedings Springer.
Cuckoo search via Lévy flights
Yang, X. and Deb, S. 2009. Cuckoo search via Lévy flights. in: World Congress on Nature & Biologically Inspired Computing (NaBIC 2009) IEEE Publications. pp. 210-214