## 1 Introduction
### Derivative-Based Methods
The inquiry described in [50] introduces the hybridization of the particle swarm optimization with variable neighborhood search and simulated annealing to tackle permutation flow-shop scheduling problems. The findings detailed in [51] highlight an original hybrid algorithm integrating principles from the particle swarm optimization and puffer fish algorithms, aiming to accurately estimate parameters related to fuel cells. The study in [52] showcases the fusion of the brainstorm optimization algorithm with the chaotic accelerated particle swarm optimization algorithm. The purpose is to explore the potential enhancements that this amalgamated approach could offer over using the individual algorithms independently. The research elucidated in [53] presents an innovative hybrid learning moth search algorithm. This algorithm uniquely integrates two distinct learning mechanisms: global-best harmony search learning and Baldwinian learning. The objective is to effectively address the multidimensional knapsack problem, harnessing the benefits of these combined learning approaches. The research described in this paper proposes a novel contribution by integrating the opposition Nelder-Mead algorithm into the selection phase of genetic algorithms to address the premature convergence problem and enhance exploration capabilities. This integration offers several significant advantages and advancements to the field of optimization and evolutionary computation, including:  
1. **Prevention of premature convergence:** Premature convergence is a common issue in genetic algorithms [54; 55; 56], where the algorithm converges to suboptimal solutions without adequately exploring the search space. By incorporating the opposition Nelder-Mead algorithm into the selection phase, our research provides a solution to this problem. The opposition Nelder-Mead algorithm, known for its effectiveness in local search and optimization [57], brings its exploratory power to the genetic algorithm. This integration ensures that the algorithm can avoid premature convergence by continuously exploring and exploiting promising regions of the search space. 2. **Enhanced exploration capabilities:** The integration of the opposition Nelder-Mead algorithm into the selection phase enhances the exploration capabilities. Genetic algorithms traditionally rely on genetic operators such as crossover and mutation for exploration. However, these operators may not be sufficient to thoroughly explore complex search spaces [58]. By incorporating the opposition Nelder-Mead algorithm, which excels in local exploration, our methodology enhances the exploration capabilities of genetic algorithms. This integration enables a more comprehensive search of the search space, leading to the discovery of diverse and potentially better solutions. 3. **Improved convergence speed and solution quality:** The integration of the opposition Nelder-Mead algorithm offers the potential for improved convergence speed and solution quality. The opposition Nelder-Mead algorithm is known for its efficiency in converging toward local optima. By utilizing this algorithm during the selection phase, our methodology aims to guide the genetic algorithm toward better solutions at a faster rate. This combination of global exploration from genetic algorithms and local optimization from the Nelder-Mead algorithm results in an algorithm that can converge faster and produce high-quality solutions. It is worth pointing out that the convergence speed in metaheuristics refers to the rate at which an algorithm approaches a solution of acceptable quality. It indicates how quickly an algorithm narrows down its search space and refines its solutions, ultimately aiming to find an optimal or near-optimal solution. A faster convergence speed implies that the algorithm reaches promising solutions in fewer iterations, whereas a slower convergence speed suggests that more iterations are needed to achieve comparable results.