# Combining Parallel Stochastic Methods and Mixed Termination Rules in Optimization
###### Abstract
Therefore, parallel processing of these methods is essential. Recently, various methods have been proposed to leverage parallel processing. Specifically, one study details the parallel implementation of the Particle Swarm Optimization (PSO) algorithm, which improves performance for load-balanced problems but shows reduced efficiency for load-imbalanced ones. The study suggests using larger particle populations and implementing asynchronous processing to enhance performance [26]. Another review provides a comprehensive overview of derivative-free optimization methods, useful for problems where objective and constraint functions come only from a black-box or simulator interface. The review focuses on recent developments and categorizes methods based on assumptions about functions and their characteristics [27]. Lastly, the PDoublePop version 1.0 software is presented, which implements parallel genetic algorithms with advanced features, such as an enhanced stopping rule and advanced mutation schemes. The software allows for coding the objective function in C++ or Fortran77 and has been tested on well-known benchmark functions, showing promising results [28]. Moreover, there are methods that utilize GPU architectures for solving complex optimization problems. In the case of the Traveling Salesman Problem (TSP), a parallel GPU implementation has been developed to achieve high performance on large scales, solving problems of up to 6000 cities with great efficiency [29]. Additionally, the use of the multi-start model in local search algorithms has proven to be particularly effective when combined with GPUs, reducing computation time and improving solution quality for large and time-intensive problems [30]. Finally, a parallel algorithm using Peano curves for dimension reduction has demonstrated significant improvements in speed and efficiency when run on GPUs, compared to CPU-only implementations, for multidimensional problems with multiple local minima [31]. Parallel optimization represents a significant approach in the field of optimization problem-solving and is applied across a broad range of applications, including the optimization of machine learning model parameters. In the case of GraphLab, a platform is presented that enhancesexisting abstract methods, such as MapReduce, for parallel execution of machine learning algorithms with high performance and accuracy [32]. Additionally, the new MARSAOP method for hyperparameter optimization uses multidimensional spirals as surrogates and dynamic coordinate search to achieve high-quality solutions with limited computational costs [33]. Lastly, the study proposes a processing time-estimation system using machine learning models to adapt to complex distributions and improve production scheduling, achieving significant reduction in completion time [34]. The use of parallel optimization has brought significant advancements in solving complex control and design problems [35]. In one case, the parallel application of the Genetic Algorithm with a Fair Competitive Strategy (HFCGA) has been used for the design of an optimized cascade controller for ball and beam systems. This approach allows for the automation of controller parameter tuning, improving performance compared to traditional methods, with parallel computations helping to avoid premature convergence to suboptimal solutions [36]. In another case, the advanced Teaching-Learning-Based Optimization (TLBO) method has been applied to optimize the design of propulsion control systems with hydrogen peroxide. This method, which includes improvements in the teaching and learning phases as well as in the search process, has proven highly effective in addressing uncertainties in real-world design problems, enhancing both accuracy and convergence speed of the optimization process [37]. Similarly, parallel optimization has significant applications in energy and resource management.