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



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).


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  • 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 $\boldsymbol{100}$ $\boldsymbol{57}$ $\boldsymbol{21}$