Modified T-F Function Method for Finding Global Minimizer on Unconstrained Optimization

This paper indicates that the filled function which appeared in one of the papers by Y. L. Shang et al. (2007) is also a tunneling function; that is, we prove that under some general assumptions this function has the characters of both tunneling function and filled function. A solution algorithm based on this T-F function is given and numerical tests from test functions show that our T-F function method is very effective in finding better minima. 1. Introduction Because of the advances in science, economics, and engineering, studies on global optimization for the multiminimum nonlinear programming problem have become a topic of great concern. The existence of multiple local minima of a general nonconvex objective function makes global optimization a great challenge [1–3]. Many deterministic methods using an auxiliary function have been proposed to search for a globally optimal solution of a given function of several variables, including filled function method [4] and tunneling method [5]. The filled function method was first introduced by Ge in the paper in [4]. The key idea of the filled function method is to leave from a local minimizer to a better minimizer of with the auxiliary function constructed at the local minimizer of . Geometrically, flattens in the higher basin of than . So a local minimizer of can be found, which lies in the lower basin of than . To minimize with initial point , one can find a lower minimizer of . with replacing , one can construct a new filled function and then find a much lower minimizer of in the same way. Repeating the above process, one can finally find the global minimizer of . The basin of at an isolated minimizer of , , is defined in the paper in [4] as a connected domain which contains and in which the steepest descent trajectory of converges to from any initial point. The hill of at is the basin of at its isolated minimizer . The concept of the filled function is introduced in the paper in [4]. Assume that is a local minimizer of . A function is called a filled function of at if has the following properties.(P1) is a maximizer of and the whole basin of at becomes a part of a hill of .(P2) has no minimizers or saddle points in any higher basin of than .(P3) if has a lower basin than , then there is a point in such a basin that minimizes on the line through and . The form of the filled function proposed in paper [4] is as follows: where and are two adjustable parameters. However, this function still has some unexpected features. First, this function has only a finite number of local minimizers. Second, the