# Combining Parallel Stochastic Methods and Mixed Termination Rules in Optimization
###### Abstract  
Parallel optimization enables faster and more efficient problem-solving by reducing computational resource consumption and time. By simultaneously combining multiple methods, such as evolutionary algorithms and swarm-based optimization, effective exploration of the search space and achievement of optimal solutions in shorter time frames are realized. In this study, a combination of termination criteria is proposed, utilizing three different criteria to end the algorithmic process. These criteria include measuring the difference between optimal values in successive iterations, calculating the mean value of the cost function in each iteration, and the so-called \"DoubleBox\" criterion, which is based on the relative variance of the best value of the objective cost function over a specific number of iterations. The problem is addressed through the parallel execution of three different optimization methods (PSO, Differential Evolution, and Multistart). Each method operates independently on separate computational units with the goal of faster discovery of the optimal solution and more efficient use of computational resources. The optimal solution identified in each iteration is transferred to the other computational units. The proposed enhancements were tested on a series of well-known optimization problems from the relevant literature, demonstrating significant improvements in convergence speed and solution quality compared to traditional approaches. global optimization; parallel techniques; termination rules; evolutionary techniques +
Footnote †: journal: _Algorithms_  
computational framework combining variance-based methods with artificial neural networks has significantly improved accuracy and reduced computational cost in sensitivity analyses [6]. Even in the field of economics, optimization plays a crucial role. The particle swarm optimization (PSO) method is shown to be highly effective in solving the economic dispatch problem in power systems, providing higher quality solutions with greater efficiency compared to genetic algorithms [7]. Additionally, a multi-objective optimization approach for economic load dispatch and emission reduction in hydroelectric and thermal plants achieves optimal solutions that align with the user's preferences and goals [8]. Finally, optimization is also applied in biological process models [9] and medical data classification, achieving high accuracy in diagnosing diseases and analyzing genomes [10]. A series of methods have been proposed in the recent literature to handle problems described by Equation (1). Methods used to handle problems described by Equation (1) are usually divided into two categories: deterministic and stochastic methods. In the first category, the most common method is the interval method [11; 12], where the set \(S\) is divided through a series of steps into subregions and some subregions that do not contain the global solution can be removed using some pre-defined criteria. The second category contains stochastic techniques that do not guarantee finding the total minimum, and they are easier to program since they do not rely on some assumptions about the objective function. In addition, they also constitute the vast majority of global optimization methods. Among them, there are methods based on considerations derived from Physics, such as the Simulated Annealing method [13], the Henry's Gas Solubility Optimization (HGSO) [14], the Gravitational Search Algorithm (GSA) [15], the Small World Optimization Algorithm (SWOA) [16], etc. Also, a series of evolutionary-based techniques have also been suggested for Global Optimization problems, such as Genetic Algorithms [17], the Differential Evolution method [18; 19], Particle Swarm Optimization (PSO) [20; 21], Ant Colony Optimization (ACO) [22], Bat Algorithm (BA) [23], Whale Optimization Algorithm (WOA) [24], Grasshopper Optimization Algorithm (GOA) [25], etc. However, the above optimization methods require significant computational power and time.