## 1 Introduction
The use of graphics processing units (GPUs) as general-purpose computing devices has represented a noteworthy advancement in HPC and is fundamental to increasing the computational efficiency of algorithms that are inherently parallel. These hardware units are equipped with a multitude of cores, allowing for the execution of numerous parallel operations simultaneously, thus surpassing the capabilities of traditional central processing units (CPUs). This approach offers a cost-effective means of achieving scalability and efficiency, making it possible to accelerate computations across various domains, including scientific simulations, artificial intelligence, financial modeling, and data analytics. The GPU acceleration of population-based metaheuristics allows for the efficient resolution of large and complex optimization problems and facilitates the exploration of vast solution spaces that would otherwise be impractical with traditional sequential computing due to high computational costs. This paper provides a review and detailed description of metaphorless algorithms, including the Jaya optimization algorithm [12], the enhanced Jaya (EJAYA) algorithm [15], which is a variant of Jaya that has recently been shown to be effective in solving nonlinear equation systems [16], the three Rao algorithms [17], the best-worst-play (BWP) algorithm [18], and the recent max-min greedy interaction (MaGI) algorithm [14]. In addition to examining prior parallelizations of the Jaya optimization algorithm, this paper introduces enhanced GPU-based massively parallel implementations of these algorithms, employing a more efficient parallelization strategy. In particular, a novel GPU-accelerated version of the MaGI algorithm is proposed. The performance of the GPU-accelerated versions of the metaphorless algorithms is evaluated using a series of challenging and large-scale NESs across a range of GPU devices. NESs must be reformulated into optimization problems to solve them using metaheuristics. Considering a NES represented generically in vector form as \(f(x)=0\), with components \(f_{i}(x_{1},\ldots,x_{n})=0,\;\;i=1,\ldots,n\), this system of \(n\) nonlinear equations with \(n\) unknowns can be transformed into an \(n\)-dimensional nonlinear minimization problem. Transforming NESs into optimization problems involves defining an objective function to minimize. In this paper, the objective function is defined as the sum of the absolute values of the residuals, \(\sum_{i=1}^{n}|f_{i}(x_{1},\ldots,x_{n})|\). In population-based optimization algorithms, the initial population generation typically involves the random or strategic generation of a group of potential candidates to serve as the starting point for the optimization process. As a standard practice, the default initialization approach for the majority of metaheuristic algorithms involves the use of random number generation, or the Monte Carlo method [19], in which individuals within the initial population are generated randomly within predefined bounds or limitations. This approach does not guarantee an even distribution of candidate solutions across the search space [20] and can influence the performance of the algorithms. However, tests have shown that this behavior is contingent on both the test problems and the algorithm under consideration [19], indicating that certain combinations of problem domains and algorithms are almost impervious to this scenario. Considering the prevalent use of randomly generated starting populations and the limited number of experiments employing alternative initialization procedures, a stochastic population initialization approach was adopted in this study. In addition to proposing novel parallelization for the MaGI algorithm and more efficient parallelizations for the other metaphorless algorithms, the primary goal of this work is to investigate the efficacy, scalability, and performance characteristics of GPU-based implementations of metaphorless optimization algorithms within the framework of a standardized parallelization strategy.