# Fuzzy Clustering Approaches Based on Numerical Optimizations of Modified Objective Functions
## 1 Introduction  
Unsupervised learning is a sub-domain of machine learning where the algorithms operate on data without prior information being provided in the form of labeled responses or predefined classes. Several important forms of unsupervised learning are developed and successfully applied, including clustering, association rule mining, latent variable models, anomaly detection, etc. Clustering aims at finding unknown internal structures in datasets, grouping the data instances into groups based on their similarities with each other [1]. Besides the classical crisp clustering methods, another significant domain of clustering is soft clustering, where the boundaries between the clusters do not have to be crisply separated; thus, overlapping clusters and partial memberships of data instances into several clusters are naturally occurring. An important form of soft clustering is the Fuzzy C-Means (FCM), a clustering algorithm that extends the classical K-Means algorithm, allowingpartial memberships of data instances into multiple clusters with varying membership degrees [2]. The algorithm operates based on the minimization of an objective function expressed as the sum of weighted distances between data instances and cluster centroids. The modus operandi of the fuzzy clustering algorithm is an iterative procedure where the new centroids are re-evaluated based on the current membership degrees of each data instance to the clusters, and the memberships are updated based on the distances of each data instance from the new centroids [3]. This dual updating procedure continues until the changes fall below a predefined threshold, thus achieving convergence. Finally, each data instance is assigned a membership degree, which is a real value in the [0, 1] interval, in such a way allowing for a nuanced representation of the intrinsic structures of the dataset. Applications of fuzzy clustering extend to a myriad of disciplines, including image processing, customer profiling, bioinformatics, text organization, etc. [4]. Despite its overall utility, the classical Fuzzy C-Means algorithm may struggle with datasets that are characterized by complex distributions, such as clusters of varying sizes or densities, overlapping clusters, and the presence of noise and outliers. On such occasions, the application of the standard Fuzzy C-Means algorithm may lead to suboptimal clustering outcomes, where clusters are poorly defined and the membership values of data instances may not be properly assigned into the clusters [5]. To address these issues, ideas of modifying the objective function have been occasionally suggested in the machine learning research community, incorporating additional parameters, employing different distance measures, and adjusting penalization terms [6]. For instance, the usage of regularization parameters can diminish the influence of noise and outliers, thus improving the stability of the clustering procedure [7]. Furthermore, the incorporation of spatial characteristics or density-related information can assist in providing a better definition of the clusters and refine the adjustments for the overlapping clusters [8]. The idea of modifying the objective function of Fuzzy C-Means or K-Means in order to make new versions of the respective algorithms that perform better on specific datasets is known in the machine learning community. There is a variety of modification forms, and also there are several ways how each modification is processed into an iterative procedure [9]. Once the objective function is modified, the iterative procedure for updating the centers of the clusters and the membership values should be properly adapted. There are two main approaches towards its optimization: employing the Lagrange multipliers or using numerical optimization directly on the modified objective function.