The Auxiliary Function Method For Nonlinear Global Optimization

The optimization problem is a widely used discipline,which discusses the characters of optimal choice on decision problems and constructs computing approaches to find the optimal solution. Most of the problems that abstracted from the society can be classified to the problems that finding global optimal solution, so the optimization has been a focus in the field for study of optimization. During these years,many new algorithms have been proposed,such as the penalty function method,the tunneling function method and the filled function method(A common auxiliary function method) which we mainly discussed in this paper, etc. Now it has been developed into an independent branch in the optimization field.The main idea of the filled function method can be described as follows: First ,we can use some classical local methods (For example: radient method, quasi-Newton method, etc) to find a local minimizer of the original problem, and then construct an auxiliary function (the filled function) at the local minimizer of the original problem. Next we minimizing the auxiliary function to get a more better minimizer of the original problem. Then we construct a new auxiliary function in the new local minimizer of the original problem, continue to find a better local minimizer by the new filled function. These two phases run by turns and will not stop until no better local minimizer can be found anymore. In the end the last local minimizer is regarded as the approximate global minimizer of the original problem. The purpose of researching filled function method is to construct a filled function which has simple form and few parameters, so that we can reduce the computational steps and save many time. So to construct a superior auxiliary function is one of the crucial of the filled function method.The present thesis is organized as follows: In the first chapter, The background of optimization theories and some methods for global optimization problems are briefly presented, including the filled function methods and the tunneling function method. In the second chapter, A new definition of convex filled function method was proposed and a convex filled function was constructed. We modified an algorithm which has been used in many papers and get the numerical results. In the third chapter, We construct a non-parameter filled function, discuss its properties should meet some of the analysis and give the numerical results and a conclusion. In the fourth chapter, We constructed an free-parameter filled function and discussed its analytical properties. The numerical results was given by the modified algorithm in chapterâ…¡. In the fifth chapter, A general conclusion is given for this paper.