JOURNAL OF INDUSTRIAL AND doi:10.3934/jimo.2016.12.375 MANAGEMENT OPTIMIZATION Volume 12, Number 1, January 2016 pp. 375–387 A CRITERION FOR AN APPROXIMATION GLOBAL OPTIMAL SOLUTION BASED ON THE FILLED FUNCTIONS Liuyang Yuan College of Science, Wuhan University of Science and Technology Wuhan, Hubei, 430081, China Zhongping Wan School of Mathematics and Statistics, Wuhan University Wuhan, Hubei, 430072, China Qiuhua Tang Industrial Engineering Department, Wuhan University of Science and Technology Wuhan, Hubei, 430081, China (Communicated by Wenxun Xing) Abstract. In this paper, a new definition of the filled function is given. Based on the new definition, a new class of filled functions is constructed, and the properties of the new filled functions are analysed and discussed. Moreover, according to the new class of filled functions, a criterion is given to decide whether the point we have obtained is an approximate global optimal solu- tion. Finally, a global optimization algorithm based on the new class of filled functions is presented. The implementation of the algorithm on several test problems is reported with numerical results. 1. Introduction. In this paper, the following box-constrained global optimization problem is considered: min f (x) s.t. x ∈ X, (1) where f (x): R n → R is continuously differentiable and X is a box. When f : R n → R is coercive, i.e., f (x) → +∞ as ||x|| → +∞, then a global optimization problem: min x∈R n f (x) can be always reduced into an equivalent problem formulated in (1). Global optimization finds wide applications in almost all fields of science and technology. The existence of multiple local minima of a general nonconvex objective function makes global optimization a great challenge (see, e.g., [2, 6, 13]). Recently, rapid progress has been made both in theories and practical applications for global optimization. However, there are still two main difficulties in global optimization. The first one is to find the global minimizer of a given function f (x). The second 2010 Mathematics Subject Classification. Primary: 90C30; Secondary: 90C33. Key words and phrases. Global optimization, the filled function, a criterion, local minimizer, global minimizer. The first author is supported by NSF grant 11401450, 71471140 and 51275366. Corresponding author: Liuyang Yuan, yangly0601@126.com. 375