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GuASPSO: a new approach to hold a better exploration–exploitation balance in PSO algorithm

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Abstract

This paper presents a new variant of particle swarm optimization (PSO) algorithm named guided adaptive search-based particle swarm optimizer (GuASPSO). In this algorithm, the personal best particles are all divided into a linearly decreasing number of clusters. Then, the unique global best guide of a given particle located at a cluster is obtained as the weighted average calculated over other clusters’ best particles. Since the clustered particles are being well-distributed over the whole search space in the clustering process, there would be a moderate distance between each particle and its unique global best guide, contributing the particles neither to be trapped in local optima nor engaged in a drift leading to lose diversity in the search space. In this approach, the number of clusters is high at the early iterations and is gradually decreased by lapse of iterations to less stress the diversity factor and further stress the fitness role to cause the particles to better converge to the optimal point. Holding this balance between global and personal bests’ role to attract the particles, on the one hand and between convergence and diversity, on the other hand, can hold a better exploration–exploitation balance in the proposed algorithm. To test the performance of GuASPSO, four popular meta-heuristic algorithms, including genetic algorithm, gravitational search algorithm, gray wolf optimizer, and PSO algorithm as well as 23 standard benchmark functions as the test beds, are employed. The experimental results validated GuASPSO as a robust well-designed algorithm to handle various optimization problems.

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Abbreviations

D :

Number of dimensions of the optimization problem

N :

Swarm size

t :

Number of iterations

w :

Inertia weight

r 1 :

First random vector

r 2 :

Second random vector

c 1 :

Cognitive acceleration coefficient

c 2 :

Social acceleration coefficient

χ :

Constriction coefficient

k max :

Maximum value of k (appearing in Eq. 4)

k min :

Minimum value of k (appearing in Eq. 4)

t max :

Maximum number of iterations

X :

Input vectors to the SOM network (Pbest particles)

W i :

Weight vectors of the SOM network neurons

M :

Number of the neurons/clusters

η(t):

Variable learning-rate parameter

τ 1 :

Maximum number of SOM iterations

Ncluster(t):

Number of the clusters in the tth iteration

\( W_{c}^{t} \) :

Weight of the cth active cluster

\( \left| {C_{c}^{t} } \right| \) :

Number of the Pbest particles collected in the cth active cluster at tth iteration

\( {\text{Gbest}}_{i}^{t} \) :

Unique Gbest particle of the ith particle at the tth iteration

\( {\text{Cbest}}_{j}^{t} \) :

Cluster best: the best Pbest particle in the jth cluster at the tth iteration

c(i):

Cluster the ith Pbest particle is belonging to

w i :

Weight assigned to the ith criteria in compromise programming

Z i :

Ith criterion

Z i,best :

The best ith criterion

Z i,worst :

The worst ith criterion

CPI:

Compromise programming index

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Authors and Affiliations

  1. Department of Civil Engineering, Isfahan University of Technology, Isfahan, Iran

    Farshad Rezaei & Hamid R. Safavi

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  1. Farshad Rezaei
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  2. Hamid R. Safavi
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Correspondence to Hamid R. Safavi.

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Appendix A

Appendix A

See Tables 10, 11, 12, 13, 14, 15, 16, 17.

Table 10 \( a_{ij} \) in \( F_{14} \)
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Table 11 \( a_{i} \) and \( b_{i} \) in \( F_{15} \)
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Table 12 \( a_{ij} \) and \( c_{i} \) in \( F_{19} \)
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Table 13 \( P_{ij} \) in \( F_{19} \)
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Table 14 \( a_{ij} \) and \( c_{i} \) in \( F_{20} \)
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Table 15 \( P_{ij} \) in \( F_{20} \)
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Table 16 \( a_{ij} \) and \( c_{i} \) in \( F_{21} \), \( F_{22} \) and \( F_{23} \)
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Table 17 Optima in functions presented in Table 3
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Rezaei, F., Safavi, H.R. GuASPSO: a new approach to hold a better exploration–exploitation balance in PSO algorithm. Soft Comput 24, 4855–4875 (2020). https://doi.org/10.1007/s00500-019-04240-8

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