American Institute of Mathematical Sciences

December  2018, 8(4): 461-479. doi: 10.3934/naco.2018029

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

 Department of electrical and computer engineering, University of Tabriz, Tabriz, Iran

Received  August 2017 Revised  March 2018 Published  September 2018

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.

Citation: Soheil Dolatabadi. Weighted vertices optimizer (WVO): A novel metaheuristic optimization algorithm. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 461-479. doi: 10.3934/naco.2018029
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a) the bisection method b) the raw concept of WVO algorithm using two vertices
the graphical description of proposed method for five vertices mode
the flowchart of WVO algorithm
2D plot of test functions
3D sketch of Shurb's function (F1)
the positions of WVO vertices in first iteration
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
the cost value versus iteration
cost value of F2 function in each iteration
cost value of F3 function in each iteration
cost value of F4 function in each iteration
cost value of F5 function in each iteration
cost value of F6 function in each iteration
]">Figure 14.  block diagram of AVR along with PID controller [17]
cost value of each method for AVR's PID
the step response of without PID controller and with optimized gains
the logarithmic plot of cost function versus iteration for F5 function and different N${}_{V}$
the logarithmic plot of cost function versus iteration for F6 function and different N${}_{V}$
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
 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
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 $$+x_1x_2+(-4+4x^2_2)x^2_2 {\left[\text{-5.5}\right]}^{\text{2}} -1.0316285 F3 Sphere \sqrt{\sum^D_{i=1}{x^2_i}} {\left[\text{-32.32}\right]}^{\text{10}} 0 F4 Ackley -A\times exp\left(-0.02\sqrt{\frac{\sum^D_{i=1}{x^2_i}}{D}}\right) -{\text{exp} \left(\frac{\sum^D_{i=1}{{\text{cos} \left(2\pi x_i\right)\ }}}{D}\right)\ }+A\ ;A=20 {\left[\text{-100.100}\right]}^{\text{10}} 0 F5 Griewank 1+\frac{1}{4000}\sum^D_{i=1}{x^2_i-\prod^D_{i=1}{\text{cos}\text{}(\frac{x_i}{\sqrt{i}})}} {\left[\text{-600.600}\right]}^{\text{10}} 0 F6 Rastrigin 10D+\sum^D_{i=1}{(x^2_i-10\text{cos}\text{}(2\pi x_i))} {\left[\text{-5.12.5.12}\right]}^{\text{10}} 0  ID Name Function Bound Global Min F1 Shubert (\sum^5_{i=1}{icos(\left(i+1\right)x_1+1)}) (\sum^5_{i=1}{icos(\left(i+1\right)x_2+1)}) {\left[\text{-2.12.2.12}\right]}^{\text{2}} -186.7309 F2 Six-hump camel back \left(4-2.1x^2_1+\frac{x^4_1}{3}\right)x^2_1$$+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
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
 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
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
 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
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$
 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$
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
 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
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
 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
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
 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
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
 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
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
 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
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