Enhancing Hyper-To-Real Space Projections Through Euclidean Norm Meta-Heuristic Optimization

TL;DR

This paper proposes enhancing hypercomplex meta-heuristic optimization by replacing the Euclidean norm with the Minkowski p-norm, leading to improved solutions in benchmark functions without extra computational cost.

Contribution

It introduces the idea of using Minkowski p-norm instead of Euclidean norm in hypercomplex optimization, enabling better solutions through a simple, fine-tuned adjustment.

Findings

Minkowski p-norm improves solution quality over Euclidean norm.

The approach is effective across multiple benchmark functions.

Fine-tuning p is cost-free and does not add hyperparameters.

Abstract

The continuous computational power growth in the last decades has made solving several optimization problems significant to humankind a tractable task; however, tackling some of them remains a challenge due to the overwhelming amount of candidate solutions to be evaluated, even by using sophisticated algorithms. In such a context, a set of nature-inspired stochastic methods, called meta-heuristic optimization, can provide robust approximate solutions to different kinds of problems with a small computational burden, such as derivative-free real function optimization. Nevertheless, these methods may converge to inadequate solutions if the function landscape is too harsh, e.g., enclosing too many local optima. Previous works addressed this issue by employing a hypercomplex representation of the search space, like quaternions, where the landscape becomes smoother and supposedly easier to optimize. Under this approach, meta-heuristic computations happen in the hypercomplex space, whereas variables are mapped back to the real domain before function evaluation. Despite this latter operation being performed by the Euclidean norm, we have found that after the optimization procedure has finished, it is usually possible to obtain even better solutions by employing the Minkowski $p$-norm instead and fine-tuning $p$ through an auxiliary sub-problem with neglecting additional cost and no hyperparameters. Such behavior was observed in eight well-established benchmarking functions, thus fostering a new research direction for hypercomplex meta-heuristic optimization.