Opposition-based antlion optimizer using Cauchy distribution and its application to data clustering problem

  • Shail Kumar DinkarEmail author
  • Kusum Deep
Original Article


This paper proposes an improved version of antlion optimizer (ALO) to solve data clustering problem. In this work, Cauchy distribution-based random walk is employed in place of uniform distribution to jump out of local optima as a first strategy. Then opposition-based learning model is utilized in conjunction with acceleration coefficient to overcome the slow convergence of classical ALO as second strategy to propose opposition-based ALO using Cauchy distribution (OB-C-ALO). The performance of the proposed OB-C-ALO is evaluated over a set of benchmark problems of different varieties of characteristics and analysed statistically by performing Wilcoxon rank-sum test. The proposed version then utilizes K-means clustering by refining the clusters formed using K-means as objective function. The algorithm is evaluated on six data sets of UCI machine learning repository and compared with classical ALO and recently developed version of ALO, namely OB-L-ALO, over benchmark test problems as well as data clustering problem and proved to be better in terms of performance achieved.


Optimization Cauchy distribution Opposition-based learning Data clustering Intra-cluster variance 

List of symbols

\(S_{\text{ant}} = \left( {S_{A,1} ,S_{A,2} , \ldots S_{A,n} , \ldots ,S_{A,N} } \right)^{T}\)

Initial population of ant

\(S_{A,n} = \left( {S_{A,n}^{1} , \ldots S_{A,n}^{d} , \ldots ,S_{A,n}^{D} } \right)\)

nth ant


dth variable of the nth ant

\(T_{\text{ant}} = \left( {T_{A,1} ,T_{A,2} \ldots T_{A,n} , \ldots T_{A,N} } \right)\)T

Fitness matrix of ant

\(T_{A,n} = f\left( {S_{A,n}^{1} , \ldots ,S_{A,n}^{d} , \ldots ,S_{A,n}^{D} } \right)\)

Fitness value of nth ant

\(T_{\text{antlion}} = \left( {S_{AL,1} ,S_{AL,2} , \ldots ,S_{AL,n} , \ldots ,S_{AL,N} } \right)^{T}\)

Antlion population

\(T_{AL,n} = \left( {S_{AL,n}^{1} , \ldots S_{AL,n}^{d} , \ldots S_{AL,n}^{D} } \right)\)

nth antlion


dth variable of the nth antlion

\(T_{\text{antlion}} = \left( {T_{AL,1} , \ldots ,T_{AL,n} , \ldots ,T_{AL,N} } \right)\)

Fitness matrix of antlion

\(T_{AL,n} = f\left( {S_{AL,n}^{1} , \ldots S_{AL,n}^{d} , \ldots S_{AL,n}^{D} } \right)\)

Fitness value of nth antlion

\(it_{\text{curr}} ,it_{ \hbox{max} }\)

Current and maximum iteration

L, U

Lower and upper bounds


Selected antlion


Elite (best) antlion


Random walk around \(S_{\text{sel}}\)


Random walk around \(S_{\text{elite}}\)


Acceleration coefficient

\(p_{\hbox{max} } = 1,\,p_{\hbox{min} } - 0.00001\)

Max and min values of constant

\(X(x_{1} ,x_{2} , \ldots ,x_{D} )\)

Point in D-dimensional space

\(Y(y_{1} ,y_{2} , \ldots ,y_{D} )\)

Point in D-dimensional space

\(d\left( {X,Y} \right)\)

Distance between two points

\(T = \left( {t_{1} ,t_{2} , \ldots ,t_{n} } \right)\)

n data objects

\(C = \left\{ {c_{1} ,c_{1} , \ldots ,c_{k} } \right\}\)

Set of k-clusters

\(F_{i} = \left\{ {f_{1} , \ldots ,f_{p} ,f_{p + 1} , \ldots ,f_{k*p} } \right\}\)

Set of cluster centres


Number of features


Number of clusters



The first author is thankful to All India Council of Technical Education (AICTE), Government of India, for funding this research.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoorkeeRoorkeeIndia

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