Quadratic Surrogate Attractor for Particle Swarm Optimization

Maurizio Clemente; Marcello Canova

cs.NEMar 17, 2026

Quadratic Surrogate Attractor for Particle Swarm Optimization

Authors: Maurizio Clemente, Marcello Canova

Organizations: Center for Automotive Research, The Ohio State University, Columbus, OH 43212 USA

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 & tables

Fig. 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.

Panel of Figure 2 — Fig. 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%).

Panel of Figure 3 — Fig. 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%).

Panel of Figure 4 — Fig. 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%).

Panel of Figure 5 — Fig. 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%).

$\mathbf{\omega}$	$\mathbf{c_{1,0}}$	$\mathbf{c_{2,0}}$	$\mathbf{v_{\max,0}}$	$\mathbf{K}$	$\mathbf{S}$	$\mathbf{\tau}$
0.72984	2.8	2.05	2	200	52	1.2

TABLE I : Weights and parameters used [ 3 , 19 , 20 ] .

Function	N p	Bounds	Mean Q.S.	Mean Std.	Rel. Diff.	Time Q.S. [ $\mathrm{s}$ ]	Time Std. [ $\mathrm{s}$ ]	Rel. Diff.	IQR 25-75 Q.S.	IQR 25-75 Std.
Ackley 2D	6	[[-32.768, 32.768],[-32.768, 32.768]]	1.307e-2	3.294e-1	-96.03%	91.27	84.41	+ 8.12 %	(1.112e-11)–(3.568e-7)	(1.037e-7)–(1.478e-4)
Griewank 2D	6	[[-600, 600],[-600, 600]]	2.119e-1	7.927e-1	-73.27%	117.55	102.41	+ 14.78 %	(1.265e-12)–(1.300e-1)	(1.010e-8)–(2.610e-1)
Sphere 2D	6	[[-10, 10],[-10, 10]]	1.182e-7	1.235e-7	-4.292%	67.19	60.32	+ 11.38 %	(1.791e-18)–(5.523e-11)	(2.838e-14)–(7.071e-10)
Sphere 3D	10	[[-10, 10],[-10, 10],[-10, 10]]	3.122e-7	2.334e-6	-86.62%	75.09	67.59	+ 11.10 %	(3.306e-17)–(1.637e-10)	(1.525e-11)–(4.652e-8)
Flower 2D	6	[[-100, 100],[-100, 100]]	8.045e-4	4.776e-1	-99.83%	76.43	69.21	+ 10.43 %	(1.501e-12)–(7.327e-8)	(1.132e-6)–(7.560e-4)
Flower 3D	10	[[-100, 100],[-100, 100],[-100, 100]]	4.565e-4	5.374e-1	-99.92%	96.25	87.39	+ 10.14 %	(1.221e-9)–(2.168e-6)	(9.853e-5)–(3.932e-2)

TABLE II : Comparison of the quadratic surrogate attractor (Q.S.) and the standard PSO (Std.) across benchmark functions. Values report the mean, execution time, and inter-quartile ranges (IQR 25–75) over 400 independent runs of both algorithms for 200 iterations. Negative percentages denote a more accurate solution.

Particle Swarm Optimization

Particle Swarm Optimization (PSO) uses interacting candidate solutions to search complex spaces and tune models, designs, and control systems.

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Aug 7, 2026J/K move · Enter open · S save

Adaptive Hybrid Particle Swarm Optimization with Gradient Descent

Aryan Gurudeo

Black-Box Optimization Gradient Descent Particle Swarm Optimization

Open paper AskPDF arXiv

Gradient injection helps Particle Swarm Optimization (PSO) only when the swarm has identified a basin with smooth local structure, not universally. We propose Adaptive Hybrid PSO (AHPSO), which uses a sigmoid function on swarm diversity to automatically modulate gradient influence: near-zero during exploration, near-maximum during exploitation, with no manual phase-switching. Under budget-normalized comparison (PSO given equivalent total function evaluations), PSO wins 52.5% of 40 configurations versus AHPSO's 20% (p = 7.0e-5, Friedman). AHPSO retains advantage specifically on problems with smooth local basins (F8, F24-F27) where directed descent outperforms undirected sampling even at equal cost. Under iteration-matched comparison across 29 functions (42 configurations, 14,700 runs), AHPSO-Adadelta ranks first of 9 methods including CMA-ES (p = 9.75e-4). The contribution is a principled characterization of when gradient injection provides value in swarm-based search, not a claim of universal superiority.

Abstract

Particle Swarm Optimization

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