# An Innovative Hybrid Approach Producing Trial Solutions for Global Optimization
## 2 The Overall Algorithm  
Modern optimization methods have been developed to solve complex problems. The GA, inspired by natural selection, uses operators like crossover and mutation but often suffers from slow convergence and a high computational cost in high-dimensional problems. PSO leverages particle collaboration to search for optimal solutions, but it can get trapped in local optima. The Improved PSO (IPSO) [34] addresses this issue with dynamic adjustments, enhancing both speed and accuracy. DE stands out for its simplicity and efficiency, achieved through mutation and recombination, but its performance relies heavily on proper parameter tuning. The Grey Wolf Optimizer (GWO) [35] mimics the hunting behavior of wolves, balancing exploration and exploitation, but struggles with high-dimensional problems. The WOA, inspired by whales' hunting strategies, has proven effective, though its performance decreases in large-scale problems. The Multi-Enhanced WOA (MEWOA) introduces additional strategies to avoid local optima and accelerate convergence. Each method offers advantages and limitations, with the choice depending on the problem's requirements and the available resources. The proposed algorithm integrates features from different optimization methods to enhance efficiency. It begins with the initialization of the population, which is generated either through uniform distribution or the k-means method. Next, samples are computed based on [PERSON]'s linear search, incorporating elements from genetic algorithms while simultaneously applying the DE formula. Specifically, the nearest sample \(c_{i}\) is identified from the initial set \(x_{i}\), as shown in step 2 of the used algorithm. Then, a second sample is generated using the [PERSON] linear search, following the termination criterion of the method, considering both the initial and the nearest sample. A third sample is subsequently computed through the \"crossover\" mechanism of genetic algorithms, combining the initial sample with the best one obtained from the entire search space. These samples are used to create a new trial sample based on Formula (5), leveraging both local information and the position of the best sample. The periodic application of a local optimization method further improves the algorithm's performance by ensuring solution refinement. The termination criterion is controlled through an adaptive process, determining whether the search should continue or stop based on the achieved progress. In the conducted tests, the algorithm demonstrated high efficiency by significantly reducing the number of objective function evaluations required and the overall computational cost. The algorithm is as follows:  
1. **Initialization step. ** 1. **Set** the population size \(N\geq 4\). 2. **Set**\(n\) the dimension of the benchmark function. 3. **Initialize** the samples \(x_{i},i=1,\ldots,N\) using uniform or k-means [36] distribution. 2. **Calculation step. ** 1. **For**\(i=1\ldots N\)**do** 1. **Obtain** sample \(x_{i}\). 2. **Find** nearest sample \(c_{i}\) from \(x_{i}\): \[d(x_{i},c_{i})=\sqrt{\sum_{j=1}^{n}\bigl{(}x_{i,j}-c_{i,j}\bigr{)}^{2}}\] (2) 3. **Set** direction vectors: \(p_{1}=-\
abla f\text{s}(x_{i})\) end \(p_{2}=-\
abla f\text{s}(c_{i})\) 4.