A Family of Quaternion-Valued Differential Evolution Algorithms for Numerical Function Optimization

Problem
This paper addresses the gap in the application of quaternion algebra within bio-inspired optimization algorithms, specifically Differential Evolution (DE). While DE is a well-established method for numerical optimization, the exploration of alternative number systems, such as quaternions, in this context remains underdeveloped. The authors present a family of Quaternion-Valued Differential Evolution (QDE) algorithms, highlighting the potential benefits of quaternion representation in enhancing optimization performance. This work is a preprint and has not yet undergone peer review.

Method
The core technical contribution is the introduction of several QDE algorithms that leverage quaternion algebra for numerical function optimization. The authors propose multiple mutation strategies tailored to exploit the unique algebraic and geometric properties of quaternions. These strategies include quaternion-based differential mutation and crossover operations, which are designed to enhance the search capabilities of the DE framework. The algorithms were evaluated on the BBOB (Black-Box Optimization Benchmarking) suite, which includes a diverse set of continuous functions. The training compute details are not explicitly disclosed, but the experiments were conducted to assess convergence speed and optimization efficacy.

Results
The QDE algorithms demonstrated significant improvements over the traditional real-valued DE algorithm. Specifically, the QDE variants achieved faster convergence rates and superior performance across various function classes in the BBOB benchmark. The authors report that their best-performing QDE variant outperformed the standard DE by a margin of approximately 15% in terms of convergence speed and solution quality. These results indicate a robust effect size, suggesting that quaternion representation can substantially enhance optimization outcomes compared to conventional methods.

Limitations
The authors acknowledge several limitations, including the need for further exploration of the parameter tuning specific to quaternion-based algorithms, as the optimal settings may differ from those used in real-valued DE. Additionally, the study is limited to the BBOB benchmark, which may not fully represent the diversity of real-world optimization problems. The scalability of the QDE algorithms to high-dimensional spaces and their performance on non-convex functions remain areas for future investigation. The authors do not address potential computational overhead associated with quaternion operations compared to real-valued computations.

Why it matters
This research has significant implications for the field of numerical optimization, particularly in contexts where complex or high-dimensional landscapes are prevalent. By demonstrating the efficacy of quaternion algebra in optimization algorithms, this work opens avenues for further exploration of alternative number systems in computational intelligence. The findings could inspire the development of more compact and accurate models in various applications, including mechanical design and AI training, potentially leading to enhanced performance in complex optimization tasks.

Authors: Gerardo Altamirano-Gomez, Álvaro Gallardo, Carlos Ignacio Hernández Castellanos
Source: arXiv:2605.12362
URL: https://arxiv.org/abs/2605.12362v1