1 A modification of the DIRECT method for Lipschitz global optimization for a symmetric function Ratko Grbić Faculty of Electrical Engineering, University of Osijek Kneza Trpimira 2b, HR – 31000 Osijek, Croatia e-mail: rgrbic@etfos.hr Emmanuel Karlo Nyarko Faculty of Electrical Engineering, University of Osijek Kneza Trpimira 2b, HR – 31000 Osijek, Croatia e-mail: nyarko@etfos.hr Rudolf Scitovski 1 Department of Mathematics, University of Osijek Trg Lj. Gaja 6, HR – 31000 Osijek, Croatia e-mail: scitowsk@mathos.hr Abstract. In this paper, we consider a global optimization problem for a symmetric Lipschitz continuous function. An efficient modification of the well-known DIRECT (DI- viding RECTangles) method called SymDIRECT is proposed for solving this problem. The method is illustrated and tested on several standard test functions. The application of this method to solving complex center-based clustering problems for the data having only one feature is particularly presented. Key words: Lipschitz continuous function; Global optimization; DIRECT; Symmet- ric function; Center-based clustering MSC2010: 65K05, 05E05, 90C26, 90C27, 90C56 1 Introduction A real symmetric function g :[a, b] n → R, of n variables is the one whose value at any n-tuple of arguments is the same as its value at any permutation of that n-tuple. These functions are often the subject of research in different applications. In this paper, we shall consider symmetric functions that occur naturally when solving the k-means problem and in cluster analysis (see, e.g., [20, 24, 47]), whereby special importance is attached to searching for a globally optimal partition of the data that have only one feature. If the function g attains its global minimum on [a, b] n , then generally there exist at least n! points from [a, b] n where this global minimum is attained. Namely, if the point (ξ ⋆ 1 ,...,ξ ⋆ n ) ∈ 1 Corresponding author: Rudolf Scitovski, e-mail: scitowsk@mathos.hr, telephone number: +385-31- 224-800, fax number: +385-31-224-801