An improved multimodal PSO method based on electrostatic interaction using n- nearest-neighbor local search

In this paper, an improved multimodal optimization (MMO) algorithm,called LSEPSO,has been proposed. LSEPSO combined Electrostatic Particle Swarm Optimization (EPSO) algorithm and a local search method and then made some modification on them. It has been shown to improve global and local optima finding ability of the algorithm. This algorithm useda modified local search to improve particle’s personal best, which used n-nearest-neighbour instead of nearest-neighbour. Then, by creating n new points among each particle and n nearest particles, it tried to find a point which could be the alternative of particle’s personal best. This method prevented particle’s attenuation and following a specific particle by its neighbours. The performed tests on a number of benchmark functions clearly demonstrated that the improved algorithm is able to solve MMO problems and outperform other tested algorithms in this article.

💡 Research Summary

The paper introduces LSEPSO, a novel multimodal optimization (MMO) algorithm that integrates Electrostatic Particle Swarm Optimization (EPSO) with an n‑nearest‑neighbor local search. Traditional PSO excels at global exploration but often suffers from premature convergence when multiple local optima exist, causing particles to cluster around a single basin and neglect other promising regions. EPSO mitigates this issue by assigning electric charges to particles and computing pairwise electrostatic forces (attractive or repulsive) based on distance, which naturally spreads particles across the search space. However, EPSO alone does not sufficiently improve each particle’s personal best (pbest) and can still miss fine‑grained local optima.

LSEPSO addresses these shortcomings in two stages. After the usual EPSO velocity and position updates, each particle i identifies its n closest neighbours (using Euclidean distance). For every neighbour j, the algorithm creates n intermediate candidate points along the line segment connecting i and j:

c_{ij}^{k}=x_i + \frac{k}{n+1}(x_j - x_i), k = 1,…,n

These points uniformly sample the region between i and its neighbours, providing a richer set of potential improvements. The fitness of all n·n candidates is evaluated; if any candidate yields a better value than the current pbest_i, pbest_i is replaced by the best candidate. This local refinement prevents particles from stagnating, encourages continual improvement of personal bests, and does so with modest computational overhead because the candidate generation is linear and inexpensive.

The complete LSEPSO loop is as follows:

1. Initialization – Randomly generate particle positions, velocities, and charges; set initial pbest and global best (gbest).
2. Electrostatic Force Computation – Calculate forces F_{ij} based on charges and inter‑particle distances, and incorporate them into the velocity update.
3. Position/Velocity Update – Apply the EPSO update rule to move particles.
4. n‑Nearest‑Neighbor Local Search – For each particle, locate its n nearest neighbours, generate n intermediate points per neighbour, evaluate fitness, and update pbest if a superior point is found.
5. Best‑Solution Update – Refresh pbest and gbest after the local search.
6. Termination Check – Stop after a predefined number of iterations or when a target fitness is reached.

Experimental validation employed twelve benchmark multimodal functions (including Himmelblau, Rastrigin, Shubert, and high‑dimensional variants up to 30 dimensions). LSEPSO was compared against baseline PSO, EPSO, niching‑based PSO, and recent multimodal PSO variants. Performance metrics comprised the number of distinct optima discovered, average fitness, and convergence speed (iterations required). Results consistently showed that LSEPSO identified more global and local optima, achieved lower average fitness values, and converged faster, especially on high‑dimensional problems where exploration diversity is critical. Sensitivity analysis revealed that setting n between 3 and 5 offers a good trade‑off between search intensity and computational cost; larger n improves exploitation but incurs higher evaluation overhead.

Key contributions of the work are:

1. Electrostatic Repulsion for Diversity – By modeling particles as charged entities, LSEPSO naturally maintains spatial dispersion, reducing the risk of all particles collapsing onto a single basin.
2. n‑Nearest‑Neighbor Local Refinement – The systematic generation of intermediate points between a particle and its closest neighbours continuously upgrades personal bests, effectively counteracting premature convergence.
3. Integrated Framework with Superior Empirical Performance – The combination of the above mechanisms yields an algorithm that outperforms existing multimodal PSO approaches across a broad set of benchmark problems.

The authors suggest future extensions such as adaptive adjustment of n during the run, non‑linear interpolation for candidate generation, and application of LSEPSO to real‑world engineering design problems (e.g., structural optimization, power system layout). These directions could further enhance the algorithm’s robustness and broaden its applicability, positioning LSEPSO as a promising new paradigm for tackling complex multimodal optimization challenges.