A Filled Function Approach for Nonsmooth Constrained Global Optimization

A novel filled function is given in this paper to find a global minima for a nonsmooth constrained optimization problem. First, a modified concept of the filled function for nonsmooth constrained global optimization is introduced, and a filled function, which makes use of the idea of the filled function for unconstrained optimization and penalty function for constrained optimization, is proposed. Then, a solution algorithm based on the proposed filled function is developed. At last, some preliminary numerical results are reported. The results show that the proposed approach is promising. 1. Introduction Recently, since more accurate precisions demanded by real-world problems, studies on global optimization have become a hot topic. Many theories and algorithms for global optimization have been proposed. Among these methods, filled function method is a particularly popular one. The filled function method was originally introduced in [1, 2] for smooth unconstrained global optimization. Its idea is to construct a filled function via it the objective function leaves the current local minimum to find a better one. The filled function method consists of two phase: local minimization and filling. The two phases are performed repeatedly until no better minimizer could be located. The filled function method was further developed in literature [3–9]. It should be noted that these filled function methods deal only with smooth unconstrained or box constrained optimization problem. However, many practical problems could only be modelled as nonsmooth constrained global optimization problems. To address this situation, in this paper, we generalize the filled function proposed in [10] and establish a novel filled function approach for nonsmooth constrained global optimization. The key idea of this approach is to combine the concept of filled function for unconstrained global optimization with the penalty function for constrained optimization. In general, there are two difficulties in global optimization: the first is how to leave the current local minimizer of to go to a better one; the second is how to check whether the current minimizer is a global solution of the problem. Just like other GO methods, the filled function method has some weaknesses discussed in [11]. In particular, the filled function method cannot solve the second issue, so our paper focuses on the former issue. The rest of this paper is organized as follows. In Section 2, some preliminaries about nonsmooth optimization and filled function are listed. In Section 3, the concept of modified filled