Abstract
A new hybrid chaos optimization algorithm (COA), namely, the accelerated particle swarm optimization combined with different chaotic maps (APSOC), is proposed for global optimization with continuous and discrete variables in this paper. Computational efficiency of APSOC and other three COAs (CPSO1–CPSO3) is compared for nonlinear benchmark functions. And the three influencing factors of chaotic maps on efficiency are considered, namely, the Lyapunov exponent (LE) which quantifies the search speed of chaotic sequence, the probability distribution function (PDF), and the dispersion degree of chaotic sequence which is defined as an index to measure the computational performance of evolutionary algorithm herein. To investigate the influence of CPSOs with different one-dimensional chaotic maps on efficiency of global optimization, three cases are examined, such as: different chaotic maps with close LE and different PDF; the same chaotic map with the same PDF and different LE; and the identical chaotic map with equal or close LE and different PDF. Optimization results demonstrate that the probability distribution, search speed, and dispersion degree of chaotic sequences affect remarkably the performance of CPSOs. Finally, statistic results and evolution curves of APSOC with Circle map are compared with those of other three COAs, and the optimal design of trusses with discrete variables are performed by APSOC. It is indicated that APSOC with Circle map is superior to other CPSOs and has greater exploration ability and faster convergence rate.
Similar content being viewed by others
References
Ott E (2002) Chaos in dynamical systems. Cambridge University Press, Cambridge
Yuan XF, Li ST, Wang YN, Sun W, Wu LH (2011) Parameter identification of electronic throttle using a hybrid optimization algorithm. Nonlinear Dyn 63(4):549–557
Li B, Jiang WS (1998) Optimizing complex function by chaos search. Cybern Syst 29(4):409–419
Yang DX, Li G, Cheng GD (2007) On the efficiency of chaos optimization algorithms for global optimization. Chaos Solitons Fract 34(4):1366–1375
Yang DX, Liu ZJ, Zhou JL (2014) Chaos optimization algorithms based on chaotic maps with different probability distribution and search speed for global optimization. Commun Nonlinear Sci Numer Simul 19(4):1229–1246
Yuan XF, He YQ, Liu LJ (2015) Parameter extraction of solar energy models using chaotic asexual reproduction optimization. Neural Comput Appl 26:1227–1237
Meng HJ, Zheng P, Wu RY, Hao XJ, Xie Z (2004) A hybrid particle swarm algorithm with embedded chaotic search. In: Proceeding of IEEE conference on cybernetics and intelligent systems, Singapore, vol 1 (3), pp 367–371
Liu B, Wang L, Tang F, Huang DX (2005) Improved particle swarm optimization combined with chaos. Chaos Solitons Fract 25(5):1261–1271
Xiang T, Liao XF, Wong KW (2007) An improved particle swarm optimization algorithm combined with piecewise linear chaotic map. Appl Math Comput 190(2):1637–1645
Alatas B (2011) Uniform big bang–chaotic big crunch optimization. Commun Nonlinear Sci Numer Simul 16(9):3696–3703
Talatahari S, Azar BF, Sheikholeslami R, Gandomi AH (2012) Imperialist competitive algorithm combined with chaos for global optimization. Commun Nonlinear Sci Numer Simul 17(3):1312–1319
Gandomi AH, Yang XS, Talatahari S, Alavi AH (2013) Firefly algorithm with chaos. Commun Nonlinear Sci Numer Simul 18(1):89–98
Ghasemi M, Ghavidel S, Aghaei J, Gitizadeh M, Falah H (2014) Application of chaos-based chaotic invasive weed optimization techniques for environmental OPF problems in the power system. Chaos Solitons Fract 69:271–284
Mokhtari H, Salmasnia A (2015) A Monte Carlo simulation based chaotic differential evolution algorithm for scheduling a stochastic parallel processor system. Expert Syst Appl 42(20):7132–7147
Kennedy J, Eberhart RC (1995) Particle swarm optimization. In: Proceedings of IEEE international conference on neural networks, vol 4, pp 1942–1948
Chan WD, Shih SP (2010) PID controller design of nonlinear systems using an improved particle swarm optimization approach. Commun Nonlinear Sci Numer Simul 15(11):3632–3639
Mahmoodabadi MJ, Bagheri A, Nariman-zadeh N, Jamali A (2012) A new optimization algorithm based on a combination of particle swarm optimization, convergence and divergence operators for single-objective and multi-objective problems. Eng Optim 44(10):1167–1186
Yu X, Zhang XQ (2014) Enhanced comprehensive learning particle swarm optimization. Appl Math Comput 242(1):265–276
Jiang CW, Etorre B (2005) A hybrid method of chaotic particle swarm optimization and linear interior for reactive power optimization. Math Comput Simul 68(1):57–65
Alatas B, Akin E (2009) Chaotically encoded particle swarm optimization algorithm and its applications. Chaos Solitons Fract 41(2):939–950
Gandomi AH, Yun GJ, Yang XS, Talatahari S (2013) Chaos-enhanced accelerated particle swarm optimization. Commun Nonlinear Sci Numer Simul 18(2):327–340
Kaveh A, Sheikholeslami R, Talatahari S, Keshvari-Ilkhichi M (2014) Chaotic swarming of particles: a new method for size optimization of truss structures. Adv Eng Softw 67:136–147
He YY, Zhou JZ, Xiang XQ, Chen H, Qin H (2009) Comparison of different chaotic maps in particle swarm optimization algorithm for long-term cascaded hydroelectric system scheduling. Chaos Solitons Fract 42(5):3169–3176
Wu Q, Law R, Wu E, Lin JX (2013) A hybrid-forecasting model reducing Gaussian noise based on the Gaussian support vector regression machine and chaotic particle swarm optimization. Inf Sci 238(20):96–110
Shirazi MJ, Vatankhah R, Boroushaki M, Salarieh H, Alasty A (2012) Application of particle swarm optimization in chaos synchronization in noisy environment in presence of unknown parameter uncertainty. Commun Nonlinear Sci Numer Simul 17(2):742–753
He YY, Yang SL, Xu QF (2013) Short-term cascaded hydroelectric system scheduling based on chaotic particle swarm optimization using improved logistic map. Commun Nonlinear Sci Numer Simul 18(7):1746–1756
Shi Y, Eberhart RC (1998) A modified particle swarm optimizer. In: Proceedings of the IEEE congress on evolutionary computation, Anchorage, Alaska, May 4–9, 1998, pp 69–73
Park JB, Jeong YW, Kim HH, Shin JR (2006) An improved particle swarm optimization for economic dispatch with valve-point effect. Int J Innov Energy Syst Power 1(1):1–7
Chuang LY, Yang CH, Li JC (2011) Chaotic maps based on binary particle swarm optimization for feature selection. Appl Soft Comput 11(1):239–248
Tatsumi K, Ibuki T, Tanino T (2015) Particle swarm optimization with stochastic selection of perturbation-based chaotic updating system. Appl Math Comput 269:904–929
Mariani VC, Duck ARK, Guerra FA, Coelho LS, Rao RV (2012) A chaotic quantum-behaved particle swarm approach applied to optimization of heat exchangers. Appl Therm Eng 42:119–128
Chen GG, Liu LL, Song PZ, Du YW (2014) Chaotic improved PSO-based multi-objective optimization for minimization of power losses and L index in power systems. Energy Convers Manag 86:548–560
Yang CH, Tsai SW, Chuang LY, Yang CH (2012) An improved particle swarm optimization with double-bottom chaotic maps for numerical optimization. Appl Math Comput 219(1):260–279
Coelho LS, Coelho AAR (2009) Model-free adaptive control optimization using a chaotic particle swarm approach. Chaos Solitons Fract 41(4):2001–2009
Coelho LS, Mariani VC (2009) A novel chaotic particle swarm optimization approach using Hénon map and implicit filtering local search for economic load dispatch. Chaos Solitons Fract 39(2):510–518
Pluhacek M, Senkerik R, Davendra D, Oplatkova ZK, Zelinka I (2013) On the behavior and performance of chaos driven PSO algorithm with inertia weight. Comput Math Appl 66(2):122–134
Pluhacek M, Senkerik R, Davendra D (2015) Chaos particle swarm optimization with ensemble of chaotic systems. Swarm Evol Comput 25:29–35
He YY, Xu QF, Yang SL, Liao L (2014) Reservoir flood control operation based on chaotic particle swarm optimization algorithm. Appl Math Model 38(17–18):4480–4492
Coelho LS (2008) A quantum particle swarm optimizer with chaotic mutation operator. Chaos Solitons Fract 37(5):1409–1418
Cai JJ, Ma XQ, Li LX, Peng HP (2007) Chaotic particle swarm optimization for economic dispatch considering the generator constraints. Energy Convers Manag 48(2):645–653
Hong WC (2009) Chaotic particle swarm optimization algorithm in a support vector regression electric load forecasting model. Energy Convers Manag 50(1):105–117
Wang Y, Zhou JZ, Lu YL, Qin H, Wang YQ (2011) Chaotic self-adaptive particle swarm optimization algorithm for dynamic economic dispatch problem with valve-point effects. Expert Syst Appl 38(11):14231–14237
Turgut OE (2016) Hybrid chaotic quantum behaved particle swarm optimization algorithm for thermal design of plate fin heat exchangers. Appl Math Model 40(1):50–69
Acharjee P, Mallick S, Thakur SS, Ghoshal SP (2011) Detection of maximum loadability limits and weak buses using chaotic PSO considering security constraints. Chaos Solitons Fract 44(8):600–612
Zhao ZS, Feng X, Lin YY, Wei F et al (2015) Evolved neural network ensemble by multiple heterogeneous swarm intelligence. Neurocomputing 149(A):29–38
Yang XS (2010) Engineering optimization: an introduction with metaheuristic applications. Wiley, London
Liu ZJ, Yang DX (2014) Computational performance of chaos-enhanced accelerated particle swarm optimization with different chaotic maps. In: Proceedings of the 4th international conference of dynamics, vibration and control, 2014, Aug. 23–25, Shanghai, China
Li HS, Au SK (2010) Design optimization using subset simulation algorithm. Struct Saf 32(6):384–392
John KV, Ramakrishnan CV (1987) Minimum weight design of trusses using an improved move limit method of sequential linear programming. Comput Struct 27(5):583–591
Wu SJ, Chow PT (1995) Steady-state genetic algorithms for discrete optimization of trusses. Comput Struct 56(6):979–991
Parsopoulos KE, Vrahatis MN (2002) Recent approaches to global optimization problems through particle swarm optimization. Nat Comput 1(2–3):235–306
Lee KS, Geem ZW, Lee SH, Bae KW (2005) The harmony search heuristic algorithm for discrete structural optimization. Eng Optim 37(7):663–684
Li LJ, Huang ZB, Liu F (2009) A heuristic particle swarm optimization method for truss structures with discrete variables. Comput Struct 87(7):435–443
Kaveh A, Talatahari S (2010) Optimum design of skeletal structures using imperialist competitive algorithm. Comput Struct 88(21–22):1220–1229
Zhang YC, Hou YP, Liu ST (2014) A new method of discrete optimization for cross-section selection of truss structures. Eng Optim 46(8):1052–1073
Bedeian AG, Mossholder KW (2000) On the use of the coefficient of variation as a measure of diversity. Organ Res Methods 3(3):285–297
Acknowledgments
The supports of the National Natural Science Foundation of China (Grant Nos. 51478086 and 11332004), and the Key Laboratory Foundation of Science and Technology Innovation in Shaanxi Province (No. 2013SZS02-K02, State Key Laboratory Base of Eco-hydraulic Engineering in Arid Area, Xi’an University of Technology) are much appreciated.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yang, D., Liu, Z. & Yi, P. Computational efficiency of accelerated particle swarm optimization combined with different chaotic maps for global optimization. Neural Comput & Applic 28 (Suppl 1), 1245–1264 (2017). https://doi.org/10.1007/s00521-016-2433-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00521-016-2433-2