## 3 Benchmark Functions
### Categorization and Justification  
**Group 1: Multi-dimensional unimodal functions:** These are defined by their smooth landscapes and a singular global optimum. The absence of local optima in these functions presents a straightforward optimization challenge, providing a clear and unobstructed path to the global minimum. The primary utility of these unimodal functions lies in their assessment of an algorithm's foundational competence in convergence and accuracy. These functions are essential for the initial testing phase of algorithms, ensuring that the basic search and optimization capabilities are efficient and effective.  
**Group 2: Fixed-dimensional unimodal functions:** The functions belonging to this group generally have an intermediate level of complexity. These functions are characterized by a small number of local optima, fixed dimensionality, and landscapes that demand a nuanced approach to exploration and exploitation. These functions serve as a critical test bed for evaluating an algorithm's capabilities with regard to efficiently navigating towards the global optimum amidst local optima.  
**Group 3: Multi-dimensional multimodal functions:** This group comprises functions that are multi-dimensional and pose a significant challenge due to the vast number of local optima. These functions feature intricate landscapes that necessitate advanced navigation strategies.  
**Group 4: Fixed-dimensional multimodal functions:** These functions are distinguished from the others by their fixed dimensionality and vast number of local optima, including the presence of chaotic, irregular landscapes. The functions in group 4 push the optimization algorithms to their limits, testing their scalability, adaptability, and robustness in the face of extreme complexity. These benchmarks are critical for advanced testing, assessing an algorithm's ability to maintain diversity in search strategies, avoid premature convergence, and efficiently explore extensive and rugged search spaces.  
These functions are presented in Table 2. A list of all the functions, with their unique identifying numbers, the name of the function, its mathematical description, the bounds, the dimensionality, and the value of the global optimum, is presented in Tables A1 and A2 in Appendix A. Overall, 50 functions are shown, with dimensionalities ranging from 2 to 30. In Table 2, the functions are listed according to the group to which they belong, with groups 1, 2, 3, and 4 containing twelve, seven, eight, and twenty-three functions, respectively. Table 3 presents the configuration of the control parameters for the metaheuristic algorithms evaluated in this study, which represent typical values used in previous studies, as indicated in the reference column. In the case of the proposed Deephive algorithm, the secondary control parameters are predominantly static given that the particles' behaviors are governed by an underlying neural network that serves as a decision-making policy. This design choice is predicated on the capability of the neural network to adaptively regulate the exploration and exploitation processes, obviating the need for extensive parameter tuning.  
For all the algorithms, including Deephive, the standard control parameters--the population size and maximum number of function evaluations--were uniformly set to 40 and 4000, respectively. This uniformity ensures a fair comparative basis for evaluating the efficacy of each algorithm. To ensure robust statistical analysis, each algorithm was executed 20 times independently on every benchmark problem. This approach aims to mitigate the influence of stochastic variations and provides a reliable set of performance metrics.