## 1 Introduction
### Derivative-Based Methods
**Personal health and fitness:** Optimization algorithms are increasingly being used in personal health and fitness applications. Fitness trackers and mobile apps employ optimization techniques to provide personalized exercise and diet plans, optimizing the balance between calorie intake and expenditure. These algorithms consider individual goals, preferences, and constraints to help users achieve desired health and fitness outcomes [36, 37, 38, 39, 40]. 7. **Internet and e-commerce:** Optimization algorithms are utilized in internet-based applications and e-commerce platforms to enhance the user experience and optimize various processes. From search engine algorithms that rank search results to recommendation systems that personalize product suggestions, optimization is employed to improve relevance, efficiency, and customer satisfaction [41, 42, 43]. These are just a few examples that highlight the wide range of applications where optimization algorithms and techniques are utilized to improve efficiency, decision making, and resource allocation in our daily lives. Optimization continues to drive advancements and contribute to enhancing various aspects of our modern society. Hybrid metaheuristic algorithms represent a critical and evolving frontier in optimization research, addressing the inherent limitations of individual algorithms by harnessing the collective strengths of diverse optimization techniques. In a complex and dynamic problem landscape where no single algorithm universally excels, hybridization offers a compelling approach to achieving enhanced performance, increased robustness, and superior convergence rates. By fusing different algorithms, these hybrids can adapt to various problem characteristics, balance exploration and exploitation, and efficiently navigate high-dimensional solution spaces. As real-world challenges become more intricate, the ability of hybrid metaheuristics to provide innovative solutions is paramount, driving the progress of optimization methodologies across domains ranging from engineering and finance to artificial intelligence and beyond. Over the past few years, numerous hybrid algorithms have emerged in the literature, and we will examine a selection of these. The study outlined in [44] demonstrates two hybrid metaheuristic algorithms, specifically the genetic algorithm and the multiple population genetic algorithm, that are synergistically combined with variable neighborhood search to tackle some challenging NP-hard problems. The research highlighted in [45] portrays an innovative hybrid algorithm that merges genetic algorithms with the spotted hyena algorithm to effectively address the complexities of the production shop scheduling problem. The research work showcased in [46] describes an advanced hybrid metaheuristic approach for solving the traveling salesman problem with drones. This approach draws from two algorithms, namely the genetic algorithm and the ant colony optimization algorithm. The paper in [47] establishes an innovative approach known as the hybrid muddy soil fish optimization-based energy-aware routing scheme, designed to enhance the efficiency of routing in wireless sensor networks, facilitated by the Internet of Things. The research discussed in [48] introduces a novel metaheuristic approach, the hybrid brainstorm optimization algorithm, to effectively address the emergency relief routing problem. This innovative algorithm amalgamates concepts from the simulated annealing algorithm and the large neighborhood search algorithm into the foundation of the former, to significantly enhance its capacity to evade local optima and speed up the convergence process. The examination in [49] exposes a groundbreaking hybrid metaheuristic algorithm, termed the chaotic sand-cat swarm optimization, as a potent solution for intricate optimization problems that exhibit constraints. This algorithm seamlessly merges the attributes of the newly introduced technique with the innovative concept of chaos, promising enhanced performance in handling complex scenarios.