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SCIENCE CHINA Information Sciences, Volume 62, Issue 7: 070203(2019) https://doi.org/10.1007/s11432-018-9546-4

A hybrid quantum-based PIO algorithm for global numerical optimization

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  • ReceivedJun 21, 2018
  • AcceptedJul 3, 2018
  • PublishedApr 25, 2019

Abstract

A novel hybrid quantum-based pigeon-inspired optimization (PIO) algorithm for global numerical optimization is proposed to perceive deceptiveness and preserve diversity.In the proposed algorithm,the current best solution is regarded as a linear superposition of two probabilistic states, namely positive and deceptive.Through a quantum rotation gate, the positive probability is either enhanced or reset to balance exploration and exploitation.Simulation results reveal that the hybrid quantum-based PIO algorithm demonstrates an outstanding performance in global optimization owing to preserving diversity in the early evolution.As a result, the stability of the algorithm is enhanced so that the precision of optimization is improved statistically.The proposed algorithm is demonstrated to be effective for solving multimodal and non-convex problems in higher dimension with a smaller population size.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61403191, 11572149), Funding of Jiangsu Innovation Program for Graduate Education (Grant Nos. KYLX 0281, KYLX15 0318, NZ2015205), and Fundamental Research Funds for the Central Universities, Aerospace Science and Technology Innovation Fund (CASC).


References

[1] Duan H, Qiao P. Pigeon-inspired optimization: a new swarm intelligence optimizer for air robot path planning. Int Jnl Intel Comp Cyber, 2014, 7: 24-37 CrossRef Google Scholar

[2] Lei X, Ding Y, Wu F X. Detecting protein complexes from DPINs by density based clustering with Pigeon-Inspired Optimization Algorithm. Sci China Inf Sci, 2016, 59: 070103 CrossRef Google Scholar

[3] Qiu H X, Duan H B. Multi-objective pigeon-inspired optimization for brushless direct current motor parameter design. Sci China Technol Sci, 2015, 58: 1915-1923 CrossRef Google Scholar

[4] Deng Y M, Zhu W R, Duan H B. Hybrid membrane computing and pigeon-inspired optimization algorithm for brushless direct current motor parameter design. Sci China Technol Sci, 2016, 59: 1435-1441 CrossRef Google Scholar

[5] Zhao J, Zhou R. Pigeon-inspired optimization applied to constrained gliding trajectories. NOnlinear Dyn, 2015, 82: 1781-1795 CrossRef Google Scholar

[6] Sun Y, Xian N, Duan H. Linear-quadratic regulator controller design for quadrotor based on pigeon-inspired optimization. Aircraft Eng Aerospace Tech, 2016, 88: 761-770 CrossRef Google Scholar

[7] Dou R, Duan H. Pigeon inspired optimization approach to model prediction control for unmanned air vehicles. Aircraft Eng Aerospace Tech, 2016, 88: 108-116 CrossRef Google Scholar

[8] Zhang X M, Duan H B, Yang C. Pigeon-inspired optimization approach to multiple UAVs formation reconfiguration controller design. In: Proceeding of 2014 IEEE Chinese Guidance, Navigation and Control Conference (CGNCC), Yantai, 2014. 2707--2712. Google Scholar

[9] Wang Y, Wang D. Variable thrust directional control technique for plateau unmanned aerial vehicles. Sci China Inf Sci, 2016, 59: 033201 CrossRef Google Scholar

[10] Hao R, Luo D L, Duan H B. Multiple UAVs mission assignment based on modified pigeon-inspired optimization algorithm. In: Proceeding of 2014 IEEE Chinese Guidance, Navigation and Control Conference (CGNCC), Yantai, 2014. 2692--2697. Google Scholar

[11] Sun H, Duan H B. PID controller design based on prey-predator pigeon-inspired optimization algorithm. In: Proceedings of 2014 IEEE International Conference on Mechatronics and Automation, Tianjin, 2014. Google Scholar

[12] Duan H, Wang X. Echo State Networks With Orthogonal Pigeon-Inspired Optimization for Image Restoration.. IEEE Trans Neural Netw Learning Syst, 2016, 27: 2413-2425 CrossRef PubMed Google Scholar

[13] Tilahum S L. Prey predator algorithm: a new metaheuristic optimization approach. Dissertation for Ph.D. Degree. Penang: University Sains Malaysia, 2013. Google Scholar

[14] Zhang S, Duan H. Gaussian pigeon-inspired optimization approach to orbital spacecraft formation reconfiguration. Chin J Aeronautics, 2015, 28: 200-205 CrossRef Google Scholar

[15] Oftadeh R, Mahjoob M J, Shariatpanahi M. A novel meta-heuristic optimization algorithm inspired by group hunting of animals: Hunting search. Comput Math Appl, 2010, 60: 2087-2098 CrossRef Google Scholar

[16] Lu T C, Juang J C. A region-based quantum evolutionary algorithm (RQEA) for global numerical optimization. J Comput Appl Math, 2013, 239: 1-11 CrossRef Google Scholar

[17] Deng G, Wei M, Su Q, et al. An effective co-evolutionary quantum genetic algorithm for the no-wait flow shop scheduling problem. Adv Mech Eng, 2015, 7: 1--10. Google Scholar

[18] Deutsch D. Quantum theory, the Church-Turing principle and the universal quantum computer. Proc R Soc A-Math Phys Eng Sci, 1985, 400: 97-117 CrossRef ADS Google Scholar

[19] Zhang G, Jin W. Quantum evolutionary algorithm for multi-objective optimization problems. In: Proceedings of the 2003 IEEE International Symposium on Intelligent Control, Houston, 2003. Google Scholar

[20] Zhang R, Gao H. Improved quantum evolutionary algorithm for combinatorial optimization problem. In: Proceedings of the 6th International Conference on Machine Learning and Cybernetics, HongKong, 2007. 19--22. Google Scholar

[21] Tsoulos I G, Stavrakoudis A. Enhancing PSO methods for global optimization. Appl Math Comput, 2010, 216: 2988--3001. Google Scholar

[22] Sivanandam S N. Genetic algorithm implementation using matlab. In: Introduction to Genetic Algorithms. Berlin: Springer, 2008. 211--262. Google Scholar

[23] Motiian H, Soltanian-Zadeh H. Improved particle swarm optimization and applications to hidden Markov model and Ackley function. In: Proceedings of IEEE International Conference on Computational Intelligence for Measurement Systems and Applications (CIMSA), 2011. Google Scholar

[24] Lee J, Song S, Yang Y, et al. Multimodal function optimization based on the survival of the fitness kind of the evolution strategy. In: Proceeding of the 29th Annual International Conference of the IEEE EMBS, Lyon, 2007. Google Scholar

[25] Bouvry P, Arbab F, Seredynski F. Distributed evolutionary optimization, in Manifold: Rosenbrock's function case study. Inf Sci, 2000, 122: 141-159 CrossRef Google Scholar

[26] Pehlivanoglu Y V. Hybrid intelligent optimization methods for engineering problems. Dissertation for Ph.D. Degree. Norfolk: Old Dominion University, 2010. Google Scholar

  • Figure 1

    Procedure of the PIO algorithm.

  • Figure 2

    Modification of the evolutionary strategy of the QPIO algorithm.

  • Figure 3

    Procedure of the QPIO algorithm.

  • Figure 4

    (Color online) Illustration of the test functions in two-dimensional variables. (a) Ackley; (b) Rastrigin; protect łinebreak (c) Rosenbrock.

  • Figure 5

    Evolution of the population diversity in two-dimensional problem with $N=8$. (a) Dimensionless diversity; protect łinebreak(b) diversity in logarithmic unit.

  • Figure 6

    Statistical sensitivity of population size on the optimized value in two-dimensional problems. (a) Rastrigin function; (b) Rosenbrock function.

  • Figure 7

    Statistical sensitivity of dimension on the optimized value with $N=8$. (a) Rastrigin function; (b) Rosenbrock function.

  • Table 1   Control parameters of GA, PSO, PIO, and QPIO
    Control parameter Symbol GA PSO PIO QPIO
    Propulsion size $N$ $6$ $6$$6$ $6$
    Maximum number of iterations $T$ $40$ $40$ $40$ $40$
    Inertia factor/map factor $w/R$ $\text{exp}{(-0.2t)}$ $0.2$ $0.2$
    Learning factor $[c_1,c_2]$ $[2,2]$ $[0,2]$ $[0,2]$
    Constraint factor $f_C$ $0.618$ $0.618$ $0.618$
    Number of iterations for the map operator $T_{\text{m}}$ $20$ $20$
    Rotating angle $(^{\circ})$ $\Delta\theta$$-$11
  • Table 2   Properties of the test functions
    2*Function Optimal value, Optimal solution, Suboptimal value, Number of suboptimal solutions,
    $y_\text{opt}$ $\boldsymbol{x}_\text{opt}$ $y_\text{subopt}$ $N_\text{subopt}$
    Ackley $0$ $\boldsymbol{1}$ $2.6375$ $4$
    Rastrigin $0$ $\boldsymbol{1}$ $2/4$ $4/4$
    Rosenbrock $0$ $\boldsymbol{1}$
  • Table 3   Optimal results of GA, PSO, PIO, and QPIO for different functions ($~n=2$, $N=6$, $T=40$, $E=100$)
    Function Algorithm $\bar{y}_{\text{opt}}$ $\min({y}_{\text{opt}})$ $\max({y}_{\text{opt}})$ $\text{Var}({y}_{\text{opt}})$ Time (s)
    4*AckleyGA $1.69\times~10^{0}$ $1.24\times~10^{-4}$ $5.82$ $2.0567$ $0.045$
    PSO $3.69\times~10^{-1}$ ${2.27\times~10^{-5}}$ $2.59$ $0.6371$ $0.011$
    PIO $3.62\times~10^{-1}$ $6.14\times~10^{-5}$ $2.58$ $0.6560$ $0.012$
    QPIO $\boldsymbol{3.12\times~10^{-2}}$ $\boldsymbol{2.40\times~10^{-7}}$$\boldsymbol{0.50}$ $\boldsymbol{0.0051}$ $\boldsymbol{0.009}$
    4*RastriginGA $5.68$ $1.82\times~10^{-5}$ $25.09$ $32.6172$ $0.044$
    PSO $2.75$ ${8.77\times~10^{-7}}$ $9.95$ $5.9639$ $0.012$
    PIO $2.84$ $\boldsymbol{1.05\times~10^{-8}}$ $9.90$ $5.8967$ $0.013$
    QPIO $\boldsymbol{1.06}$ $3.78\times~10^{-7}$ $\boldsymbol{4.09}$$\boldsymbol{1.1147}$ $\boldsymbol{0.009}$
    4*RosenbrockGA $3.37\times~10^{0}$ $8.91\times~10^{-4}$ $58.63$ $74.7925$ $0.046$
    PSO $5.81\times~10^{-1}$ ${9.05\times~10^{-11}}$ $6.03$ $1.0709$ $0.011$
    PIO $4.76\times~10^{-1}$ $\boldsymbol{2.09\times~10^{-12}}$$7.04$ $0.8972$ $0.012$
    QPIO $\boldsymbol{0.90\times~10^{-1}}$$2.45\times~10^{-9}$ $\boldsymbol{0.94}$ $\boldsymbol{0.0287}$$\boldsymbol{0.009}$
  • Table 4   Global convergence of GA, PSO, PIO, and QPIO ($n=2$, $N=6$, $T=40$, $E=100$)
    2*Algorithm Global convergence percent, $p_\text{g}~(%)$
    Ackley Rastrigin Rosenbrock
    GA $37$ $20$ $1$
    PSO $86$ $22$ $9$
    PIO $85$ $26$ ${10}$
    QPIO $\boldsymbol{100}$ $\boldsymbol{57}$ $\boldsymbol{21}$

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