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Optimized PSO algorithm based on the simplicial algorithm of fixed point theory

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Abstract

Particle swarm optimization algorithm (PSO) has been optimized from various aspects since it was proposed. Optimization of PSO can be realized by optimizing its iterative process or the initial parameters and heuristic methods have been combined with the initial PSO algorithm to improve its performance. In this paper, we introduce the Simplicial Algorithm (SA) of fixed point theory into the optimization of PSO and proposed a FP-PSO (Fixed-point PSO) improved algorithm. In FP-PSO algorithm, the optimization of target function is converted into the problem of solving a fixed point equation set, and the solution set obtained by Simplicial Algorithm (SA) of fixed point theory is used as the initial population of PSO algorithm, then the remaining parameters can be obtained accordingly with classical PSO algorithm. Since the fixed point method has sound mathematical properties, the initial population obtained with FP-PSO include nearly all the approximate local extremes which maintain the diversity of population and can optimize the flight direction of particles, and shows their advantages on setting other initial parameters. We make an experimental study with five commonly used testing functions from UCI (University of California Irvine) which include two single-peak functions and three multi-peak functions. The results indicate that the convergence accuracy, stability, and robustness of FP-PSO algorithm are significantly superior to existing improve strategies which also optimize PSO algorithm by optimizing initial population, especially when dealing with complex situations. In addition, we nest the FP-PSO algorithm with four classical improved PSO algorithms that improve PSO by optimizing iterative processes, and carry out contrast experiments on three multi-peak functions under different conditions (rotating or non-rotating). The experimental results show that the performance of the improved algorithm using nested strategy are also significantly enhanced compared with these original algorithms.

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Notes

  1. The detailed process of K1 simplicial subdivision has been exhibited in the Appendix.

References

  1. Cavazzini G, et al (2017) A multi-fluid PSO-based algorithm for the search of the best performance of sub-critical organic rankine cycles. Energy 129:42–58

    Article  Google Scholar 

  2. Ishaque K, et al (2012) An improved particle swarm optimization (PSO)–based MPPT for PV with reduced steady-state oscillation. IEEE Trans Power Electron 27(8):3627–3638

    Article  Google Scholar 

  3. Kalaivaani PT et al (2019) A novel intelligent approach to simulate the blast-induced flyrock based on RFNN combined with PSO. Engin Comput, (2)

  4. Biller B, et al (2018) Simultaneous, multi-wavelength variability characterization of the free-floating planetary mass object PSO J318.5–22. Astron J 155(2):95–

    Article  Google Scholar 

  5. Tripathi A, Pathak I, Vidyarthi DP (2018) Energy efficient VM placement for effective resource utilization using modified binary PSO. Comput J 61(6):832–846

    Article  Google Scholar 

  6. Qin S, et al (2018) Dynamic model updating for bridge structures using the Kriging model and PSO algorithm ensemble with higher vibration modes. Sensors 18(6):1879

    Article  Google Scholar 

  7. Godio A, Santilano A (2018) On the optimization of electromagnetic geophysical data: application of the PSO algorithm. J Appl Geophys 148:163–174

    Article  Google Scholar 

  8. Carrillo-Santos CA, et al (2018) Wavenet identification of dynamical systems by a modified PSO algorithm. Eng Appl Artif Intel 73:1–9

    Article  Google Scholar 

  9. Shinoda M, Miyata Y (2018) PSO-based stability analysis of unreinforced and reinforced soil slopes using non-circular slip surface. Acta Geotechnica, (1), 1–13

  10. Lv Z, et al (2018) A multiple surrogates based PSO algorithm. Artif Intell Rev, (4), 1–22

  11. Yan J et al (2018) Visual analysis of collective anomalies using faceted high-order correlation graphs. IEEE Trans Visual Comput Graph PP(99):1–1

    Google Scholar 

  12. Wang KJ, et al (2014) A hybrid classifier combining SMOTE with PSO to estimate 5-year survivability of breast cancer patients. Appl Soft Comput J 20(7):15–24

    Article  Google Scholar 

  13. Sun J, Jing L, Xu W (2006) QPSO-based QoS multicast routing algorithm

  14. Li S, et al (2007) A new QPSO based BP neural network for face detection

  15. Shilaskar S, Ghatol A, Chatur P (2016) Medical decision support system for extremely imbalanced datasets. Inform Sci, 384

  16. Nandini DU, Leni ES (2018) Efficient shadow detection by using PSO segmentation and region-based boundary detection technique. J Supercomput, (6), 1–12

  17. Ding W, et al (2018) A layered-coevolution-based attribute-boosted reduction using adaptive quantum-behavior PSO and its consistent segmentation for neonates brain tissue. IEEE Trans Fuzzy Syst 26(3):1177–1191

    Article  Google Scholar 

  18. Wei Z, et al (2018) A load prediction model for cloud computing using PSO-based weighted wavelet support vector machine. Appl Intell, 1–12

  19. Kiran MS (2017) Particle swarm optimization with a new update mechanism. Appl Soft Comput 60:S1568494617304702

    Article  Google Scholar 

  20. Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1 (1):67–82

    Article  Google Scholar 

  21. Ghamisi P, Benediktsson JA (2014) Feature selection based on hybridization of genetic algorithm and particle swarm optimization. IEEE Geosci Remote Sens Lett 12(2):309–313

    Article  Google Scholar 

  22. Mozafar MR, Moradi MH, Amini MH (2017) A simultaneous approach for optimal allocation of renewable energy sources and charging stations based on improved GA-PSO Algorithm. Sustain Cities Soc, S2210670716306035

  23. Benvidi A, et al (2017) Spectrophotometric determination of synthetic colorants using PSO–GA-ANN. Food Chem 220:377–384

    Article  Google Scholar 

  24. Peterson LE (2016) Maximum likelihood logistic regression using metaheuristics. IEEE, 509–514

  25. Lin G, Zhao K, Qin W (2016) Takagi-sugeno fuzzy model identification using coevolution particle swarm optimization with multi-strategy. Appl Intell 45(1):187–197

    Article  Google Scholar 

  26. Liu ZH, et al (2013) Coevolutionary particle swarm optimization using AIS and its application in multiparameter estimation of PMSM. IEEE Trans Cybern 43(6):1921–1935

    Article  Google Scholar 

  27. Jármai K, Snyman JA, Farkas J (2006) Minimum cost design of a welded orthogonally stiffened cylindrical shell. Comput Struct 84(12):787–797

    Article  Google Scholar 

  28. Soodi HA, Vural AM (2018) STATCOM estimation using back- propagation, PSO, shuffled frog leap algorithm, and genetic algorithm based neural networks. Comput Intell Neurosc 2018(1):1–17

    Article  Google Scholar 

  29. Cormier N, et al (2012) Automated critical point identification for PIV data using multimodal particle swarm optimization. Int J Numer Methods Fluids 70(7):923–938

    Article  MATH  Google Scholar 

  30. Li G, et al (2017) An intelligent wheel position searching algorithm for cutting tool grooves with diverse machining precision requirements. Int J Mach Tools Manuf 122:149–160

    Article  Google Scholar 

  31. Khatami A, et al (2017) A new PSO-based approach to fire flame detection using K-Medoids clustering. Expert Syst Appl 68(C):69–80

    Article  Google Scholar 

  32. Liu R, et al (2018) Fast stability scanning for future grid scenario analysis. IEEE Trans Power Syst PP (99):1–1

    Google Scholar 

  33. Tomikawa Y, et al (2014) Construction of double-eight and double-trefoil polymer topologies with core-clickable kyklo-Telechelic precursors. Macromolecules 47(23):8214–8223

    Article  Google Scholar 

  34. Ludwig SA (2013) Particle swarm optimization approach with parameter-wise hill-climbing heuristic for task allocation of workflow applications on the cloud. In: IEEE International conference on tools with artificial intelligence

  35. Shi YH, Eberhart RC (1998) A modified particle swarm optimizer. In: The 1998 IEEE International conference on evolutionary computation proceedings. IEEE World congress on computational

  36. Nickabadi A, Ebadzadeh MM, Safabakhsh R (2011) A novel particle swarm optimization algorithm with adaptive inertia weight. Appl Soft Comput J 11(4):3658–3670

    Article  Google Scholar 

  37. Harrison KR, Engelbrecht AP, Ombuki-Berman BM (2016) Inertia weight control strategies for particle swarm optimization. Swarm Intell 10(4):1–39

    Article  Google Scholar 

  38. Liu J, Mei Y, Li X (2016) An analysis of the inertia weight parameter for binary particle swarm optimization. IEEE Trans Evol Comput 20(5):666–681

    Article  Google Scholar 

  39. Garriga JL, Soroush M (2010) Model predictive control tuning methods: a review. Indust Eng Chem Res 49(8):3505–3515

    Article  Google Scholar 

  40. Gupta IK, Choubey A, Choubey S (2017) Particle swarm optimization with selective multiple inertia weights. In: International conference on computing

  41. Wang X, et al (2014) Optimization of liquid desiccant regenerator with multiobject particle swarm optimization algorithm. Indust Eng Chem Res 53(49):19293–19303

    Article  Google Scholar 

  42. Yao J, et al (2010) Application of particle swarm optimization with stochastic inertia weight and adaptive mutation in target localization. In: International conference on computer application & system modeling

  43. Li L, et al (2009) The novel non-linear strategy of inertia weight in particle swarm optimization. In: International conference on bio-inspired computing

  44. Amoshahy MJ, Shamsi M, Sedaaghi MH (2016) A novel flexible inertia weight particle swarm optimization algorithm. Plos One 11(8):e0161558

    Article  Google Scholar 

  45. Du C, et al (2019) Research on active disturbance rejection control with parameter autotune mechanism for induction motors based on adaptive particle swarm optimization algorithm with dynamic inertia weight. IEEE Trans Power Electron 34(3):2841–2855

    Article  Google Scholar 

  46. Adeli A, Broumandnia A (2018) Image steganalysis using improved particle swarm optimization based feature selection. Appl Intell 48(6):1609–1622

    Article  Google Scholar 

  47. Druzeta S, Ivic S (2017) Examination of benefits of personal fitness improvement dependent inertia for particle swarm optimization. Soft Comput 21(12):3387–3400

    Article  Google Scholar 

  48. Ghatak SR, Sannigrahi S, Acharjee P (2018) Comparative performance analysis of DG and DSTATCOM using improved PSO based on success rate for deregulated environment. IEEE Syst J 12(3):2791–2802

    Article  Google Scholar 

  49. Zhang Y, et al (2016) A feature extraction method of the particle swarm optimization algorithm based on adaptive inertia weight and chaos optimization for Brillouin scattering spectra. Opt Commun 376:56–66

    Article  Google Scholar 

  50. Long Z, et al (2017) Hysteresis compensation of the Prandtl-Ishlinskii model for piezoelectric actuators using modified particle swarm optimization with chaotic map. Rev Sci Instrum 88(7):2205–2213

    Article  Google Scholar 

  51. Zhuang W, Zhang S, Wang T (2018) A cooperative particle swarm optimization with constriction factor based on simulated annealing. Computing 100(8):861–880

    Article  MathSciNet  MATH  Google Scholar 

  52. Mao KF, Bao GQ, Chi XU (2010) Particle swarm optimization algorithm based on non-symmetric learning factor adjusting. Comput Eng 36(19):182–184

    Google Scholar 

  53. Liu ZH et al (2018) Global identification of electrical and mechanical parameters in PMSM drive based on dynamic self-learning PSO. IEEE Trans Power Electron PP(99):1–1

    Google Scholar 

  54. Clerc M (2002) The swarm and the queen: towards a deterministic and adaptive particle swarm optimization. In: Congress on evolutionary computation

  55. Chen DB, Zhao CX (2009) Particle swarm optimization with adaptive population size and its application. Appl Soft Comput J 9(1):39–48

    Article  Google Scholar 

  56. Yan Y, et al (2018) Modified PSO algorithms with “request and reset” for leak source localization using multiple robots. Neurocomputing

  57. Yassin IM, et al (2010) Comparison between NARX parameter estimation methods with binary particle swarm optimization-based structure selection method. In: Evolutionary computation

  58. Yassin IM, et al (2012) Effect of swarm size parameter on binary particle swarm optimization-based NARX structure selection. In: Industrial electronics & applications

  59. Park JS (2013) A general iterative algorithm for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Fixed Point Theory

  60. Xiong L, Liu X (2012) Extreme points and support points of a class of analytic functions with fixed coefficient. Journal of Hubei University

  61. Amann H (1976) Fixed point equations and nonlinear eigenvalue problems in ordered banach spaces. SIAM Rev 18(4):620–709

    Article  MathSciNet  MATH  Google Scholar 

  62. Kuno T (2018) A modified simplicial algorithm for convex maximization based on an extension of (ω)-subdivision. J Glob Optim 71:1–15

    Article  MathSciNet  Google Scholar 

  63. Endres SC, Sandrock C, Focke WW (2018) A simplicial homology algorithm for Lipschitz optimisation. J Global Optim 1:1–37

    MathSciNet  MATH  Google Scholar 

  64. Beigvand SD, Abdi H, La Scala M (2017) Hybrid gravitational search algorithm-particle swarm optimization with time varying acceleration coefficients for large scale CHPED problem. Energy 126:841–853

    Article  Google Scholar 

  65. Park JB, et al (2010) An improved particle swarm optimization for nonconvex economic dispatch problems. IEEE Trans Power Syst 25(1):156–166

    Article  Google Scholar 

  66. Zhang Y, Wang S, Ji G (2015) A comprehensive survey on particle swarm optimization algorithm and its applications. Math Probl Eng, 2015

  67. Gandomi AH, et al (2013) Chaos-enhanced accelerated particle swarm optimization. Commun Nonlinear Sci Numer Simul 18(2):327–340

    Article  MathSciNet  MATH  Google Scholar 

  68. Ding W, et al (2018) A layered-coevolution-based attribute-boosted reduction using adaptive quantum behavior PSO and its consistent segmentation for neonates brain tissue. IEEE Trans Fuzzy Syst 26(3):1177–1191

    Article  Google Scholar 

  69. Li Z, et al (2019) Path planning method for AUV docking based on adaptive quantum-behaved particle swarm optimization. IEEE Access 7:78665–78674

    Article  Google Scholar 

  70. Salomon R (1996) Re-evaluating genetic algorithm performance under coordinate rotation of benchmark functions. A survey of some theoretical and practical aspects of genetic algorithms. Biosystems 39(3):263

    Article  Google Scholar 

  71. Khare A, Rangnekar S (2013) A review of particle swarm optimization and its applications in solar photovoltaic system. Appl Soft Comput J 13(5):2997–3006

    Article  Google Scholar 

  72. Shafiullah M, et al (2019) A modified optimal PMU placement problem formulation considering channel limits under various contingencies. Measurement 135:875–885

    Article  Google Scholar 

  73. Reddy SS, Wajid S (2019) Swarm intelligent-based congestion management using optimal rescheduling of generators. Int J Bio-Inspired Comput 13(3):159–168

    Article  Google Scholar 

  74. Reddy SS, Panigrahi BK (2017) Application of swarm intelligent techniques with mixed variables to solve optimal power flow problems. Int J Bio-Inspired Comput 10(4):283–292

    Article  Google Scholar 

  75. Du WB, et al (2015) Adequate is better: particle swarm optimization with limited-information. Appl Math Comput 268:832–838

    MathSciNet  MATH  Google Scholar 

  76. Du W-B, et al (2017) Heterogeneous strategy particle swarm optimization. IEEE Trans Circ Systems II Express Briefs 64(4):467–471

    Article  Google Scholar 

  77. Wang J-J, Liu G-Y (2019) Saturated control design of a quadrotor with heterogeneous comprehensive learning particle swarm optimization. Swarm Evol Comput 46:84–96

    Article  Google Scholar 

  78. Lin A, et al (2019) Adaptive comprehensive learning particle swarm optimization with cooperative archive. Appl Soft Comput 77:533–546

    Article  Google Scholar 

Download references

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant No.71531008, No.71521001, No.71490720 and No.71801076.

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Authors and Affiliations

  1. School of Management, Hefei University of Technology, Hefei, 230009, China

    Minglun Ren, Xiaodi Huang, Xiaoxi Zhu & Liangjia Shao

  2. Key Laboratory of Process Optimization and Intelligent Decision-making of Ministry of Education, Hefei, 230009, China

    Minglun Ren

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Appendix: the process of K1 simplicial subdivision

Appendix: the process of K1 simplicial subdivision

As shown in Fig. 4, a cube in the three-dimensional coordinate system, n = 3, N = (1, 2, 3), π = 3!. There six kinds of substitution: π1 = (1, 2, 3), π2 = (1, 3, 2), π3 = (2, 1, 3), π4 = (2, 3, 1), π5 = (3, 1, 2), π6 = (3, 2, 1). Assumed that the cube vertex coordinates are A(0, 0, 0), B(0, 1, 0), C(1, 1, 0), D(1, 0, 0), E(1, 0, 1), F(0, 0, 1), G(0, 1, 1), H(1, 1, 1).

Fig. 4
figure 4

A cube in the three-dimensional coordinate

Full size image

Carry out K1 simplicial subdivision though integer point y0 = A(0, 0, 0):

If \({\pi _{1}} =(1,2,3), y^{0} =A (0,0,0), {y^{1}} =y^{0} + u^{{\pi }_{1} (1)} =y^{0} +{u_{1} =} (1,0,0), {y^{2}} ={y^{1}} + u^{{\pi }_{1} (2)} =y^{0} +{u_{2} =} (1,1,0), {y^{3}} ={y^{2}} + u^{{\pi }_{1} (3)} ={y^{2}} +{u_{3} =} (1,1,1)\), then the simplex is < A, D, C, H >.

If \({\pi _{2}} =(1,3,2), y^{0} =A (0,0,0), {y^{1}} =y^{0} + u^{{\pi }_{2} (1)} =y^{0} +{u_{1} =} (1,0,0), {y^{2}} ={y^{1}} + u^{{\pi }_{2} (2)} =y^{0} +{u_{3} =} (1,0,1), {y^{3}} ={y^{2}} + u^{{\pi }_{2} (3)} ={y^{2}} +{u_{2} =} (1,1,1)\), then the simplex is < A, D, E, H >.

If \({\pi _{3}} =(2,1,3), y^{0} =A (0,0,0), {y^{1}} =y^{0} + u^{{\pi }_{3} (1)} =y^{0} +{u_{2} =} (0,1,0), {y^{2}} ={y^{1}} + u^{{\pi }_{3} (2)} =y^{0} +{u_{1} =} (1,1,0), {y^{3}} ={y^{2}} + u^{{\pi }_{3} (3)} ={y^{2}} +{u_{3} =} (1,1,1)\), then the simplex is < A, B, C, H >.

If \({\pi _{4}} =(2,3,1), y^{0} =A (0,0,0), {y^{1}} =y^{0} + u^{{\pi }_{4} (1)} =y^{0} +{u_{2} =} (0,1,0), {y^{2}} ={y^{1}} + u^{{\pi }_{4} (2)} =y^{0} +{u_{3} =} (0,1,1), {y^{3}} ={y^{2}} + u^{{\pi }_{4} (3)} ={y^{2}} +{u_{1} =} (1,1,1)\), then the simplex is < A, B, G, H >.

If \({\pi _{5}} =(3,1,2), y^{0} =A (0,0,0), {y^{1}} =y^{0} + u^{{\pi }_{5} (1)} =y^{0} +{u_{3} =} (0,0,1), {y^{2}} ={y^{1}} + u^{{\pi }_{5} (2)} =y^{0} +{u_{1} =} (1,0,1), {y^{3}} ={y^{2}} + u^{{\pi }_{5} (3)} ={y^{2}} +{u_{2} =} (1,1,1)\), then the simplex is < A, F, E, H >.

If \({\pi _{6}} =(3,2,1), y^{0} =A (0,0,0), {y^{1}} =y^{0} + u^{{\pi }_{6} (1)} =y^{0} +{u_{3} =} (0,0,1), {y^{2}} ={y^{1}} + u^{{\pi }_{6} (2)} =y^{0} +{u_{2} =} (0,1,1), {y^{3}} ={y^{2}} + u^{{\pi }_{6} (3)} ={y^{2}} +{u_{3} =} (1,1,1)\), then the simplex is < A, F, G, H >.

All the simplexes from point A(0, 0, 0) :< A, D, C, H >,< A, D, E, H >,< A, B, C, H >,< A, B, G, H >,< A, F, E, H >,< A, F, G, H > consist of a K1 simplicial subdivision.

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Ren, M., Huang, X., Zhu, X. et al. Optimized PSO algorithm based on the simplicial algorithm of fixed point theory. Appl Intell 50, 2009–2024 (2020). https://doi.org/10.1007/s10489-020-01630-6

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