Abstract
The Particle Swarm Optimization (PSO) algorithm is a flexible heuristic optimizer that can be used for solving cardinality constrained binary optimization problems. In such problems, only K elements of the N-dimensional solution vector can be non-zero. The typical solution is to use a mapping function to enforce the cardinality constraint on the trial PSO solution. In this paper, we show that when K is small compared to N, the use of the mapped solution in the velocity vector tends to lead to early stagnation. As a solution, we recommend to use the untransformed solution as a direction in the velocity vector. We use numerical experiments to document the gains in performance when K is small compared to N.
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Notes
We are grateful to the referee for recommending us to use publicly available test data. The data is available at http://people.brunel.ac.uk/~mastjjb/jeb/orlib/files/port5.txt.
For this real-data setup, the best value of the objective function is unknown in practice. As a sanity check recommended by the reviewer, we have also optimized the function using an alternative heuristic, namely Differential Evolution, as implemented in the R package DEoptim ([2]), and using the sparse mapping function described at https://github.com/R-Finance/PortfolioAnalytics/blob/master/sandbox/testing_DEoptim_cardinality_constraint.R. The results obtained are similar as those found for the SV-PSO, both in terms of best value for the objective function and the shape of the trajectory.
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Acknowledgements
This paper was written while Chunlin Wan was visiting the finance department of the Solvay Business School at Vrije Universiteit Brussel. We are grateful to the Editor (Oleg Prokopyev), the associate editor, an anonymous referee, Xiang Deng, Paola Festa, Jiayin Gu, Nanjing Huang, Joao Miranda, Giang Nguyen, Wajid Raza and Marjan Wauters for helpful comments and suggestions. Financial support from the Chinese Scholarship Council and the National Natural Science Foundation of China (Grant Number 71742004) is gratefully acknowledged.
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Boudt, K., Wan, C. The effect of velocity sparsity on the performance of cardinality constrained particle swarm optimization. Optim Lett 14, 747–758 (2020). https://doi.org/10.1007/s11590-019-01398-w
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DOI: https://doi.org/10.1007/s11590-019-01398-w