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Weighted vertices optimizer (WVO): A novel metaheuristic optimization algorithm

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  • This paper introduces a novel optimization algorithm that is based on the basic idea underlying the bisection root-finding method in mathematics. The bisection method is modified for use as an optimizer by weighting each agent or vertex, and the algorithm developed from this process is called the weighted vertices optimizer (WVO). For exploitation and exploration, both swarm intelligence and evolution strategy are used to improve the accuracy and speed of WVO, which is then compared with six other popular optimization algorithms. Results confirm the superiority of WVO in most of the test functions.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

    Citation:

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  • Figure 1.  a) the bisection method b) the raw concept of WVO algorithm using two vertices

    Figure 2.  the graphical description of proposed method for five vertices mode

    Figure 3.  the flowchart of WVO algorithm

    Figure 4.  2D plot of test functions

    Figure 5.  3D sketch of Shurb's function (F1)

    Figure 6.  the positions of WVO vertices in first iteration

    Figure 7.  A) the positions of vertices in second iteration B)the positions of vertices in 5th iteration C)the positions of vertices in 9th iteration D)the positions of vertices in 13th iteration

    Figure 8.  the cost value versus iteration

    Figure 9.  cost value of F2 function in each iteration

    Figure 10.  cost value of F3 function in each iteration

    Figure 11.  cost value of F4 function in each iteration

    Figure 12.  cost value of F5 function in each iteration

    Figure 13.  cost value of F6 function in each iteration

    Figure 14.  block diagram of AVR along with PID controller [17]

    Figure 15.  cost value of each method for AVR's PID

    Figure 16.  the step response of without PID controller and with optimized gains

    Figure 17.  the logarithmic plot of cost function versus iteration for F5 function and different N${}_{V}$

    Figure 18.  the logarithmic plot of cost function versus iteration for F6 function and different N${}_{V}$

    Table 1.  parameters of WVO

    C${}_{F}$ C${}_{B}$ C${}_{G}$ N${}_{V}$ V${}_{Speed}$ W${}_{GB}$ W${}_{GW}$
    0.6 0.3 0.085 2 0.6 10 1
     | Show Table
    DownLoad: CSV

    Table 2.  benchmark optimization functions

    ID Name Function Bound Global Min
    F1 Shubert $(\sum^5_{i=1}{icos(\left(i+1\right)x_1+1)})$
    $(\sum^5_{i=1}{icos(\left(i+1\right)x_2+1)})$
    ${\left[\text{-2.12.2.12}\right]}^{\text{2}}$ -186.7309
    F2 Six-hump camel back $\left(4-2.1x^2_1+\frac{x^4_1}{3}\right)x^2_1 $$+x_1x_2+(-4+4x^2_2)x^2_2$ ${\left[\text{-5.5}\right]}^{\text{2}}$ -1.0316285
    F3 Sphere $\sqrt{\sum^D_{i=1}{x^2_i}}$ ${\left[\text{-32.32}\right]}^{\text{10}}$ 0
    F4 Ackley $-A\times exp\left(-0.02\sqrt{\frac{\sum^D_{i=1}{x^2_i}}{D}}\right)$ $-{\text{exp} \left(\frac{\sum^D_{i=1}{{\text{cos} \left(2\pi x_i\right)\ }}}{D}\right)\ }+A\ ;A=20$ ${\left[\text{-100.100}\right]}^{\text{10}}$ 0
    F5 Griewank $1+\frac{1}{4000}\sum^D_{i=1}{x^2_i-\prod^D_{i=1}{\text{cos}\text{}(\frac{x_i}{\sqrt{i}})}}$ ${\left[\text{-600.600}\right]}^{\text{10}}$ 0
    F6 Rastrigin $10D+\sum^D_{i=1}{(x^2_i-10\text{cos}\text{}(2\pi x_i))}$ ${\left[\text{-5.12.5.12}\right]}^{\text{10}}$ 0
     | Show Table
    DownLoad: CSV

    Table 3.  cost value of each optimization algorithm in 78th iteration

    Method Cost value
    WVO 1.78 E-15
    PSO 7.36 E-4
    GA 9.17 E-5
    IWO 2.157
    HS 2.56E-3
    CA 5.93E-6
    mIWO 1.23 E-5
    mHS 8.69 E-9
    mCA 5.12 E-10
     | Show Table
    DownLoad: CSV

    Table 4.  the performance of each optimization method

    WVO PSO GA IWO HS CA mIWO mHS mCA
    F1 N1 13 45 27 184 88 47 132 44 23
    B2 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309 -176.7309
    R3 1 5 3 9 7 6 8 4 2
    F2 N 19 50 24 54 60 24 39 45 20
    B -1.03163 -1.03163 -1.03163 -1.03162 -1.03162 -1.03162 -1.03162 -1.03162 -1.03162
    R 1 7 3 8 9 3 5 6 2
    F3 N 77 1158 1500 1498 1501 1500 1382 1500 1500
    B 1.71 E-58 0 7.64 E-128 2.43 E-6 2 E-10 1.89 E-143 1.13 E-12 3.12E-8 0
    R 5 1 4 9 7 3 6 8 2
    F4 N 67 233 199 198 1501 1336 173 1500 1363
    B 8.88 E-16 4.44 E-15 20 20 6.2 E-5 20.29 6.13 E-3 2.52 E-8 1.23 E-4
    R 1 2 7 8 4 9 6 3 5
    F5 N 41 116 251 744 1501 468 632 432 321
    B 0 0.09747 0 0.90271 1.52 E-8 20.25 2.38 E-3 1.58 E-32 9.78 E-2
    R 1 7 2 8 4 9 5 3 6
    F6 N 30 119 418 200 1501 1130 123 1245 1351
    B 0 5.9697 0 8.9552 4.06 E-10 22.94947 3.25 E-9 3.15 E-21 4.65 E-3
    R 1 7 2 8 4 9 5 3 6
    $\boldsymbol{\mathit{\boldsymbol{\sum}}}$ R 1 5 2 9 6 8 6 4 3
       1N:Number of iteration - 2B:Best cost value - 3R:Rank
     | Show Table
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    Table 5.  the understudy composition functions [12]

    CF1 CF2 CF3
    $f_1, f_2, \dots , f_{10}=F5$ $f_{1-2}\left(x\right)=F4$
    $f_{3-4}\left(x\right)=F6$
    $f_{5-6}\left(x\right)=F7$
    $f_{7-8}\left(x\right)=F5$
    $f_{9-10}\left(x\right)=F3$
    $f_{1-2}\left(x\right)=F6$
    $f_{3-4}\left(x\right)=F7$
    $f_{5-6}\left(x\right)=F5$
    $f_{7-8}\left(x\right)=F4$
    $f_{9-10}\left(x\right)=F3$
     | Show Table
    DownLoad: CSV

    Table 6.  results of optimization algorithms for three CFs

    PSO [12] DE [12] GA WVO
    CF1 Mean 1.7203 E2 1.4441 E2 1.3451 E2 1.1121 E2
    Std. deviation 3.2869 E1 1.9401 E1 1.9142 E1 1.4232 E1
    CF2 Mean 3.1430 E2 3.2486 E2 3.2314 E2 3.0021 E2
    Std. deviation 2.0006 E1 1.4784 E1 1.8154 E1 1.6823 E1
    CF3 Mean 8.3450 E1 1.0789 E1 7.5421 E1 3.8124 E1
    Std. deviation 1.0111 E2 2.6040 E0 1.0512 E1 8.5412 E1
     | Show Table
    DownLoad: CSV

    Table 7.  value of AVR's parameters [17]

    Parameter value
    K${}_{A}$ 10
    ${\tau _A}$ 0.1
    K${}_{E}$ 1
    ${\tau _E}$ 0.4
    K${}_{G}$ 1
    ${\tau _G}$ 1
    K${}_{R}$ 1
    ${\tau _R}$ 0.01
     | Show Table
    DownLoad: CSV

    Table 8.  obtained values and the result for each optimization method

    KP KI KD RT ST
    (sec)
    OS
    (%)
    Final
    error
    (%)
    Cost value
    WVO 0.600518 0.41376 0.20136 0.3101 0.5013 0.0003 0 3.11706
    PSO 0.600532 0.41386 0.20137 0.3141 0.5013 0.0017 0 3.11752
    GA 0.610065 0.42965 0.20784 0.3226 0.5005 0.1522 0.018 3.17785
     | Show Table
    DownLoad: CSV

    Table 9.  effect of C${}_{F}$, C${}_{B}$, C${}_{G}$, W${}_{GB}$ and W${}_{GW}$ on performance of WVO

    Function C${}_{F}$ The best cost Iteration C${}_{B}$ The best cost Iteration C${}_{G}$ The best cost Iteration W${}_{GB}$ The best cost Iteration W${}_{GW}$ The best cost Iteration
    F5 0.2 3.12E-12 46 0.2 0 43 0.02 0 73 2 0 47 2 0 41
    0.4 2.02E-19 42 0.4 0 41 0.04 0 45 5 0 43 5 0 45
    0.6 0 42 0.6 0 42 0.06 0 45 10 0 42 10 0 47
    0.8 0 42 0.8 2.31E-26 43 0.08 0 41 15 0 42 15 0 47
    1 3.12E-30 44 1 8.64E-23 45 0.1 0 53 20 0 43 20 0 48
    F6 0.2 2.31E-6 45 0.2 0 41 0.02 2.31E-26 45 2 0 45 2 0 43
    0.4 0 42 0.4 0 42 0.04 0 43 5 0 43 5 0 43
    0.6 0 42 0.6 1.12E-28 45 0.06 0 43 10 0 43 10 0 43
    0.8 0 45 0.8 6.78E-25 49 0.08 0 41 15 0 44 15 0 45
    1 0 45 1 1.32E-24 51 0.1 0 44 20 0 44 20 0 48
     | Show Table
    DownLoad: CSV

    Table 10.  the effect of N${}_{V}$ value on speed of algorithm

    Function N${}_{V}$ The best cost Iteration
    F5 2 0 47
    3 0 43
    4 0 42
    5 0 41
    10 0 43
    15 4.55E-8 116
    F5 2 0 45
    3 0 43
    4 0 43
    5 0 44
    10 0 44
    15 0.406497 116
     | Show Table
    DownLoad: CSV
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