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.

翻译：近几十年来计算能力的持续增长，使得解决对人类社会具有重要意义的多个优化问题成为可处理的任务；然而，由于需要评估的海量候选解（即使使用复杂算法），部分问题依然难以攻克。在此背景下，一类称为元启发式优化的自然启发随机方法，能够以较小的计算代价为不同类型问题提供鲁棒的近似解，例如无导数实函数优化。然而，若函数地形过于复杂（如包含大量局部最优解），这些方法可能收敛至非理想解。先前研究通过采用超复数表示搜索空间（如四元数）来应对该问题，在此方法下函数地貌趋于平滑且更易优化。该方案中，元启发式计算在超复数空间进行，变量在函数评估前映射回实数域。尽管此映射操作通常由欧几里得范数完成，但我们发现优化流程结束后，通过采用闵可夫斯基$p$-范数并借助辅助子问题微调$p$值（几乎不增加额外代价且无超参数），通常能获得更优解。该现象在八个经典基准函数中得到验证，从而为超复数元启发式优化开辟了新研究方向。