The Filled Function Method For Nonlinear Global Optimization

Global optimal theory and solution, which discusses the characters of optimalchoice on decision problems and constructs computing approaches to find the optimalsolution, is widely used in many fields. Most of the problems in science,economicsand engineering can be classified to the problems that finding global optimal solution,so the global optimization has been a focus in the field for study of optimization.With the speedy development of computer and the hard work of scientists,thetheoretic analysis and computational methods on the global optimization have beenhighly improved. Meanwhile，many new algorithms have been proposed，such as thetransforming function method, and the filled function method(A common auxiliaryfunction method) which we mainly discussed in this paper, etc. Nowadays the globaloptimization has been developed into an independent branch in the optimization field.The basic idea of the filled function method can be described as follows: First ofall,a local minimizer of the objective function can be found with some classicalmethods (For example: radient method, quasi-Newton method, etc,and we find thelocal minimizer with the fminsearch function in the matlab), and then construct a filledfunction at the local minimizer of the objective function. Next we minimize the filledfunction to get a more better minimizer of the objective function. Then we construct anew filled function at the new local minimizer of the objective function to find a betterlocal minimizer.The two courses run in turn and won’t stop until no better localminimizer can be found anymore. The local minimizer that found by us in the end isregarded as the approximate global minimizer of the objective function.We only use the classic method by the filled function method,so it has beenpopular by many workersWe have three purposes of researching filled function method:First, a filled function which is constructed has simple form and few parameters; Second, the filled function should be better properties;Third, the filled function should have more efficient algorithms.This paper mainly consists of four chapters: In the first chapter, The optimizationtheories and some methods for global optimization problems are briefly presented,first,we introduce the basic idea of the filled function and some definition of the filledfunction,including several filled functions. In the second chapter, A new filled functionis suggested in this paper for finding a global minimizer of continuous unconstrainedoptimization problems In the third chapter,based on the filled function which ispresented in Chapter2, A new filled function is suggested in this paper for finding aglobal minimizer of continuous unconstrained optimization problems. The propertiesof the new filled function are discussed in the paper and this properties fit thedefinition of the previously reported in paper [38].The filled function algorithm isdesigned according to the new filled function and the numerical results are given toshow that the algorithm is feasible and effective. In the forth chapter, A generalconclusion is given for this paper.