Quadratic Surrogate Attractor for Particle Swarm Optimization

Abstract

This paper presents a particle swarm optimization algorithm that leverages surrogate modeling to replace the conventional global best solution with the minimum of an n-dimensional quadratic form, providing a better-conditioned dynamic attractor for the swarm. This refined convergence target, informed by the local landscape, enhances global convergence behavior and increases robustness against premature convergence and noise, while incurring only minimal computational overhead. The surrogate-augmented approach is evaluated against the standard algorithm through a numerical study on a set of benchmark optimization functions that exhibit diverse landscapes. To ensure statistical significance, 400 independent runs are conducted for each function and algorithm, and the results are analyzed based on their statistical characteristics and corresponding distributions. The quadratic surrogate attractor consistently outperforms the conventional algorithm across all tested functions. The improvement is particularly pronounced for quasi-convex functions, where the surrogate model can exploit the underlying convex-like structure of the landscape.

Figures (5)

• Figure 1: Schematic layout of the particle position $X_i^k$ update algorithm. In the standard approach, particles move following their speed $V_i^k$ and inertia $\omega$, the location of their own best evaluation $B_i^k$, and the overall best solution found. In the proposed method, the latter contribution is replaced by the minimum of a quadratic surrogate model $P^k$ constructed from multiple distinct best-performing locations.
• Figure 2: Ackley’s function is shown in the left panel. The right panel compares the quadratic surrogate attractor with the standard algorithm over 400 independent runs of 200 iterations each. The solid line represents the mean across runs, while the shaded area denotes the inter-quartile range (25–75%).
• Figure 3: Griewank’s function is shown in the left panel. The right panel compares the quadratic surrogate attractor with the standard algorithm over 400 independent runs of 200 iterations each. The solid line represents the mean across runs, while the shaded area denotes the inter-quartile range (25–75%).
• Figure 4: The Sphere function is shown in the left panel. The central panel illustrates algorithms performance for the two‑dimensional case with six particles, while the right panel presents the three‑dimensional case with ten particles. In both cases, the quadratic surrogate attractor is compared with the standard algorithm over 400 independent runs of 200 iterations each. The solid line represents the mean across runs, while the shaded area denotes the inter-quartile range (25–75%).
• Figure 5: The Flower function is shown in the left panel. The central panel illustrates algorithms performance for the two‑dimensional case with six particles, while the right panel presents the three‑dimensional case with ten particles. In both cases, the quadratic surrogate attractor is compared with the standard algorithm over 400 independent runs of 200 iterations each. The solid line represents the mean across runs, while the shaded area denotes the inter-quartile range (25–75%).