Soft Computing

, Volume 23, Issue 21, pp 11331–11341 | Cite as

A nested particle swarm algorithm based on sphere mutation to solve bi-level optimization

  • Long Zhao
  • JingXuan WeiEmail author
Methodologies and Application


The problem of bi-level optimization has always been a hot topic due to its extensive application. Increasing size and complexity have prompted theoretical and practical interest in the design of effective algorithm. This paper adopts particle swarm algorithm (PSO) at both level. First, given the nested nature of bi-level problem, we introduce a hyper-sphere search into PSO as mutation operator to maintain the swarms diversity. Second, for complex constraints processing, the proposed algorithm adopts a dynamic constraint handling strategy, which makes the solution located on the constraint boundary easier to be obtained. Third, a quadratic approximation mutation is introduced into PSO, which guides particles to a better search area. Finally, the convergence is proved and the simulation results show that the proposed algorithm is effective.


Bi-level optimization Particle swarm algorithm Sphere search Radial basis function (RBF) 



This work is supported by the National Nature Science Foundation of China (No. 61203372).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interests.

Ethical approval

This article does not contain any studies with human participants performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Computer Science and TechnologyXidian UniversityXi’anChina

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