SCIENTIA SINICA Informationis, Volume 50 , Issue 3 : 396-407(2020) https://doi.org/10.1360/SSI-2019-0154

## A hybrid swarm intelligence with improved ring topology for nonlinear equations

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• ReceivedJul 20, 2019
• AcceptedAug 23, 2019
• PublishedFeb 26, 2020
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### References

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• Figure 1

Comparison between subscript-based ring topology and improved ring topology

• Figure 2

(Color online) Comparison of RR convergence curves of different algorithms. (a) F04; (b) F07.

• Table 1   Parameter settings of different algorithms
 Algorithm Parameter settings IHABC ${\rm~NP}=100$, $c1=c2=2.05$, $F=0.5$, ${\rm~CR}=0.9$, $\rm~{limit}=50$ CADE [24] $~{\rm~NP}=100$, $F=0.5$, ${\rm~CR}=0.9$, $T=10$ MONES [12] ${\rm~NP}=100$, $H_m~=~{\rm~NP}$ A-WeB [13] ${\rm~NP}=100$, $H_m~=~{\rm~NP}$ RADE [8] ${\rm~NP}=100$, $H_m~=~200$ DREA [9] ${\rm~NP}=10$, $u_{\rm~CR}=0.5$, $u_F=0.5$, $c=0.1$ MODFA [22] ${\rm~NP}=100$, $\alpha=0.23$, $\beta_0=1$, $\delta=0.98$, $\gamma=1$ FONDE [10] ${\rm~NP}=100$, $F=0.5$, ${\rm~CR}=0.9$, $m=11$
•

Algorithm 1 The improved ring topology

Input:Initial population $\mathcal{P}$.

Select the best nectar source, set it as the first node of the ring topology, and remove it from $\mathcal{P}$;

for $i=2$ to NP

Find the nectar source in $\mathcal{P}$ nearest to node ${\boldsymbol~x}_{i-1}$ in the topological structure of the ring topology, set it as ${\boldsymbol~x}_i$, and remove it from $\mathcal{P}$;

end for

Output: the improved ring topology.

• Table 2   Comparison of different algorithms with respect to root ratio (RR)
 Problem IHABC CADE MONES A-WeB RADE DREA MODFA FONDE F01 0.97 0.93 0.59 1.00 0.90 0.72 0.90 0.96 F02 0.99 1.00 1.00 0.94 0.99 1.00 0.59 0.99 F03 1.00 1.00 0.77 0.83 0.99 1.00 0.95 1.00 F04 0.89 0.70 0.43 0.88 0.63 0.84 0.82 0.86 F05 1.00 1.00 0.19 0.97 0.98 0.77 0.86 1.00 F06 1.00 1.00 1.00 1.00 0.99 1.00 1.00 1.00 F07 0.98 0.93 0.14 0.15 0.56 0.87 0.00 0.91 F08 0.98 0.95 0.31 0.23 0.83 1.00 0.00 1.00 Average 0.97 0.94 0.55 0.75 0.86 0.90 0.64 0.96
•

Algorithm 2 A hybrid swarm intelligence algorithm with improved ring topology

Initialize population $\mathcal{P}$ and evaluate the fitness;

${\rm~localBest}={\rm~nBest}=\mathcal{P}$, ${\rm~Iter}=1$;

while ${\rm~FEs}<{\rm~FEs}_{\max}$ do

Create a ring topology ${\boldsymbol~x}$ via Algorithm 1;

for $i=1$ to NP

Find the nearest nectar source ${\rm~localBest}({\rm~index})$ to ${\boldsymbol~x}_i$ in ${\rm~localBest}$;

if ${\rm~fit}({\boldsymbol~x}_i)<{\rm~fit}({\rm~localBest}({\rm~index}))$ then

${\rm~localBest}({\rm~index})={\boldsymbol~x}_i$;

end if

end for

for $i=1$ to NP

${\rm~nBest}(i)~\leftarrow~{\rm~neighborhoodBest}({\rm~localBest}(i-1),{\rm~localBest}(i),{\rm~localBest}(i+1))~$;

end for

for $i=1$ to NP

Update nectar source location via Eq. (6);

${\rm~FEs}={\rm~FEs}+{\rm~NP}$;

end for

for $j=1$ to NP

Generate a new position via Eq. (6), and calculate its fitness;

${\rm~FEs}={\rm~FEs}+1$;

if the root is found then

Store the root to an archive;

Generate a new nectar source via Eq. (3) and calculate the fitness;

${\rm~FEs}={\rm~FEs}+1$;

end if

end for

If nectar source has not been updated for more than limit, generate a new nectar source via Eq. (3);

${\rm~FEs}={\rm~FEs}+1$;

${\rm~Iter}={\rm~Iter}+1;$

end while

Return all the found roots.

• Table 3   Comparison of different algorithms with respect to success rate (SR)
 Problem IHABC CADE MONES A-WeB RADE DREA MODFA FONDE F01 0.73 0.46 0.00 1.00 0.31 0.00 0.06 0.52 F02 0.96 1.00 1.00 0.60 0.93 1.00 0.00 0.98 F03 1.00 1.00 0.00 0.12 0.98 1.00 0.80 1.00 F04 0.43 0.06 0.00 0.28 0.00 0.20 0.20 0.28 F05 1.00 1.00 0.00 0.76 0.89 0.00 0.40 1.00 F06 1.00 1.00 1.00 1.00 0.94 1.00 1.00 1.00 F07 0.80 0.40 0.00 0.00 0.00 0.00 0.00 0.28 F08 0.96 0.90 0.07 0.02 0.67 1.00 0.00 1.00 Average 0.86 0.72 0.25 0.47 0.59 0.53 0.30 0.75
• Table 4   Comparison of different components in IHABC with respect to root ratio (RR)
 Problem IHABC-1 IHABC-2 IHABC-3 IHABC-4 IHABC-5 IHABC-6 IHABC-7 IHABC F01 0.65 0.77 0.06 0.83 0.98 0.87 0.68 0.97 F02 0.57 0.81 0.00 0.97 0.98 0.99 0.80 0.99 F03 0.60 0.84 0.00 0.95 1.00 1.00 0.93 1.00 F04 0.69 0.64 0.18 0.63 0.71 0.60 0.51 0.88 F05 0.67 0.97 0.16 0.98 1.00 1.00 1.00 1.00 F06 0.69 0.72 0.10 0.73 1.00 1.00 0.99 1.00 F07 0.13 0.14 0.05 0.45 0.56 0.83 0.56 0.98 F08 0.91 0.90 0.00 0.93 0.78 0.98 0.58 0.98 Average 0.61 0.72 0.07 0.81 0.87 0.91 0.76 0.97
• Table 5   Comparison of different components in IHABC with respect to success rate (SR)
 Problem IHABC-1 IHABC-2 IHABC-3 IHABC-4 IHABC-5 IHABC-6 IHABC-7 IHABC F01 0.00 0.00 0.00 0.00 0.76 0.10 0.00 0.73 F02 0.03 0.06 0.00 0.76 0.90 0.96 0.06 0.96 F03 0.00 0.20 0.00 0.70 1.00 1.00 0.56 1.00 F04 0.06 0.06 0.00 0.03 0.06 0.00 0.00 0.43 F05 0.00 0.73 0.00 0.90 1.00 1.00 1.00 1.00 F06 0.00 0.00 0.00 0.00 1.00 1.00 0.96 1.00 F07 0.00 0.00 0.00 0.00 0.00 0.03 0.00 0.80 F08 0.86 0.80 0.00 0.86 0.57 0.96 0.16 0.96 Average 0.12 0.23 0.00 0.40 0.66 0.63 0.34 0.86
• Table 6   Influence of different neighborhood sizes on IHABC
 Problem RR SR $n=5$ $n=10$ $n=15$ $n=20$ $n=25$ $n=5$ $n=10$ $n=15$ $n=20$ $n=25$ F01 0.97 0.99 0.98 0.97 0.94 0.73 0.87 0.80 0.67 0.40 F02 0.99 1.00 1.00 1.00 1.00 0.96 1.00 1.00 1.00 1.00 F03 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 F04 0.88 0.91 0.70 0.76 0.81 0.43 0.60 0.07 0.20 0.33 F05 1.00 1.00 1.00 1.00 0.99 1.00 1.00 1.00 1.00 0.93 F06 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 F07 0.98 0.96 0.86 0.75 0.69 0.80 0.47 0.00 0.00 0.00 F08 0.98 0.90 0.93 0.90 0.87 0.96 0.80 0.87 0.80 0.73 Average 0.97 0.96 0.94 0.92 0.91 0.86 0.84 0.72 0.71 0.68

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