A Hybrid Generalized Reduced Gradient-Based Particle Swarm Optimizer for Constrained Engineering Optimization Problems

Document Type : Regular Article


1 Assistant Professor, Department of Civil Engineering, Ale Taha Institute of Higher Education, Tehran, Iran

2 Assistant Professor, Department of Civil Engineering, Bozorgmehr University of Qaenat, Qaen, Iran

3 MSc, Department of Civil Engineering, Ale Taha Institute of Higher Education, Tehran, Iran


A hybrid algorithm is presented that combines strong points of Particle Swarm Optimization (PSO) and Generalized Reduced Gradient (GRG) algorithm to keep a good compromise between exploration and exploitation. The hybrid PSO-GRG quickly approximates the optimum solution using PSO as a global search engine in the first phase of the search process. The solution accuracy is then improved during the second phase of the search process using the GRG algorithm to probe locally for a proper solution(s) in the vicinity of the current best position obtained by PSO. The k-nearest neighbors (k-NN)-based Purely Uniform Distributed (PUD) initial swarm is also applied to increase the convergence speed and reduce the number of function evaluations (NFEs). Hybridization between both algorithms allows the proposed algorithm to accelerate throughout the early stages of optimization using the high exploration power of PSO whereas, promising solutions will possess a high probability to be exploited in the second phase of optimization using the high exploitation ability of GRG. This prevents PUD-based hybrid PSO-GRG from becoming trapped in local optima while maintaining a balance between exploration and exploitation. The competence of the algorithm is compared with other state-of-the-art algorithms on benchmark optimization problems having a wide range of dimensions and varied complexities. Appraising offered algorithm performance revealed great competitive results on the Multiple Comparison Test (MCT) and Analysis of Variance (ANOVA) test. Results demonstrate the superiority of hybrid PSO-GRG compared to standard PSO in terms of fewer NFEs, fast convergence speed, and high escaping ability from local optima.

Graphical Abstract

A Hybrid Generalized Reduced Gradient-Based Particle Swarm Optimizer for Constrained Engineering Optimization Problems


  • The modified PSO algorithm has been hybridized with the Generalized Reduced Gradient (GRG) algorithm.
  • A novel method to enhance the initialization has been proposed by the k-NN algorithm.
  • The GRG algorithm has been employed to enhance local search in the hybrid PSO-GRG algorithm.
  • The exploration and exploitation capabilities of the PSO algorithm are balanced.
  • The suggested PSO-GRG algorithms are evaluated on benchmark mathematical and constrained engineering optimization problems including a wide range of dimensions and varied complexities.


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