# Kernel Clustering with a Differential Harmony Search Algorithm for Scheme Classification
## 1 Introduction  
Metaheuristics are designed to find, generate, or select a heuristic that provides a good solution to an optimization problem [1]. By searching over a large set of feasible solutions, metaheuristics can often find good solutions with less computational effort [2]. There has been a widespread usage of metaheuristics and their applications in artificial intelligence, e.g., transit network design problems [3], sewer pipe networks [4], water distribution systems [5], sizing optimization of truss structures [6], ordinary differential equations [7], and so forth. Using metaheuristic algorithms, complex problems are not far from finding their solutions [7]. Metaheuristics are good at finding near-optimal solutions to numerical real-valued problems [8]. Harmony search (HS) is a new phenomenon-mimicking algorithm, proposed by [PERSON] in 2001 [9]. It is a relatively recent metaheuristic method based on musical performances. In the HS algorithm, a new solution is generated by pitch adjustment and random selection. HS has the ability to deal with discontinuous optimization problems, as well as continuous problems. Comparing with other artificial intelligent algorithms such as the genetic algorithm and its variants, the HS algorithm requires fewer parameters and these parameters are easy to set. Moreover, HS can overcome the drawback of GA's building block theory. The advantages of HS have resulted in much interest in recent years, and HS has been widely applied in many fields, such as email classification [10], single machine scheduling problems [11], and so on. Although HS has its advantages, i.e., it is good at identifying the high-performance regions of the solution space at a reasonable time, it gets into trouble in performing local searches for numerical applications [12]. In order to improve the optimization ability, different variants of HS have been developed. [PERSON] et al. presented an improved harmony search algorithm (IHS) which changes the parameters dynamically with the generation number [12]. The IHS algorithm presents a strategy of parameter adjustment which improves the performances of HS algorithm. However, the userneeds to specify the new parameters, which are not easy to set, and it still performs local searches for some numerical applications. Other modifications, such as the global-best harmony search algorithm (GHS) [13] and chaotic harmony search algorithms (CHS) [14], have shown better performance than the classical HS, but they still have their own shortcomings. The GHS algorithm generates new harmonies using the variable from the best harmony. However, the algorithm cannot be adopted when the variables have different value ranges. This drawback limits the scope of the application of this method. The CHS generates new solutions following the chaotic map, but simulations show that CHS still suffers from a local optimum when dealing with some numerical applications. Fuzzy clustering is a class of algorithms for cluster analysis which determine the affinities of samples using mathematical methods. It divides samples into classes or clusters so that items in the same class are as similar as possible, while items in different classes are dissimilar. It is a useful technique for analyzing statistical data in multi-dimensional space. Fuzzy clustering has gained a significant level of interest for its ability to classify the samples which are multi-attributed and difficult to analyze. In recent years, there have been a large number of studies on fuzzy clustering. Fuzzy c-means clustering (FCM) [14] and fuzzy k-means clustering (FKM) [15] are widely used to categorize similar data into clusters. The kernel fuzzy clustering algorithm (KFC) [16] and the weighted fuzzy kernel-clustering algorithm (WFKCA) [17] are also efficient ways for cluster analysis. Research has shown that WFKCA has good convergence properties and the prototypes obtained can be well represented in the original space. However, these clustering methods have the same problem: the iterative solution is not the optimal solution.