In this study, we propose a new hybrid algorithm consisting of two meta-heuristic algorithms; Differential Evolution (DE) and the Monarch Butterfly Optimization (MBO). This hybrid is called DEMBO. Both of the meta-heuristic algorithms are typically used to solve nonlinear systems and unconstrained optimization problems. DE is a common metaheuristic algorithm that searches large areas of candidate space. Unfortunately, it often requires more significant numbers of function evaluations to get the optimal solution. As for MBO, it is known for its time-consuming fitness functions, but it traps at the local minima. In order to overcome all of these disadvantages, we combine the DE with MBO and propose DEMBO which can obtain the optimal solutions for the majority of nonlinear systems as well as unconstrained optimization problems. We apply our proposed algorithm, DEMBO, on nine different, unconstrained optimization problems and eight well-known nonlinear systems. Our results, when compared with other existing algorithms in the literature, demonstrate that DEMBO gives the best results for the majority of the nonlinear systems and unconstrained optimization problems. As such, the experimental results demonstrate the efficiency of our hybrid algorithm in comparison to the known algorithms. Highlights This paper proposes a new hybridization of differential evolution and monarch butterfly optimization. Solve system of nonlinear equations and unconstrained optimization problem. The efficiency and effectiveness of our algorithm are provided. Experimental results prove the superiority of our algorithm over the state-of-the-arts.

We propose a novel method for adaptive K-means clustering. The proposed method overcomes the problems of the traditional K-means algorithm. Specifically, the proposed method does not require prior knowledge of the number of clusters. Additionally, the initial identification of the cluster elements has no negative impact on the final generated clusters. Inspired by cell cloning in microorganism cultures, each added data sample causes the existing cluster ‘colonies’ to evaluate, with the other clusters, various merging or splitting actions in order for reaching the optimum cluster set. The proposed algorithm is adequate for clustering data in isolated or overlapped compact spherical clusters. Experimental results support the effectiveness of this clustering algorithm.

Traditional K-means clustering assumes, to some extent, a uniform distribution of data around predefined centroids, which limits its effectiveness for many realistic datasets. In this paper, a new clustering technique, simulated-annealing-based ellipsoidal clustering (SAELLC), is proposed to automatically partition data into an optimal number of ellipsoidal clusters, a capability absent in traditional methods. SAELLC transforms each identified cluster into a hyperspherical cluster, where the diameter of the hypersphere equals the minor axis of the original ellipsoid, and the center is encoded to represent the entire cluster. During the assignment of points to clusters, local ellipsoidal properties are independently considered. For objective function evaluation, the method adaptively transforms these ellipsoidal clusters into a variable number of global clusters. Two objective functions are simultaneously optimized: one reflecting partition compactness using the silhouette function (SF) and Euclidean distance, and another addressing cluster connectedness through a nearest-neighbor algorithm. This optimization is achieved using a newly-developed multiobjective simulated annealing approach. SAELLC is designed to automatically determine the optimal number of clusters, achieve precise partitioning, and accommodate a wide range of cluster shapes, including spherical, ellipsoidal, and non-symmetric forms. Extensive experiments conducted on UCI datasets demonstrated SAELLC’s superior performance compared to six well-known clustering algorithms. The results highlight its remarkable ability to handle diverse data distributions and automatically identify the optimal number of clusters, making it a robust choice for advanced clustering analysis.