Filled Function Methods For Solving Nonlinear Global Optimization Problem

The optimization is a widely used discipline, which discusses the characters of optimal choice on decision problems and constructs computing approaches to find the optimal solution. Due to the advancement of society and the development of science and technology, the optimization problems are often discovered in the field of economic planning administration, engineering design, production management, traffic transportation, national defence and so on. They are so important that meet. with much recognition。With the speedy development of computer and the hard work of scientists, the theoretic analysis and computational methods on optimization have been highly improved.To find the effective methods for finding the global optimal solutions of a general multi-minimizers function is one of the hot topics. There are two difficulties in global optimization. One is how to leave from a local minimizer to a smaller one and the other is how to judge that the current minimizer is global. The filled function algorithm introduced by Ge (1990) is one of the well-known and practical methods for settling the first difficulty。The main idea of filled function method is: If a local minimizer x~* has been found, we can make a filled function, such that iterative sequential points leave the valley in which x~* lies to find a better point x′in the lower valley (i.e. f(x′)＜f(x~*)). Then let x′be a new initial point to search a better minimizer.This paper mainly consists of five chapters. In the first chapter, several methods for global optimization problems are briefly presented.Chapter 2, a family of new filled function with two parameters and a class of new filled function with one parameter are proposed for finding a global minimizer for nonlinear programming problems with a closed bounded domain. Two corresponding algorithms are presented according to the theoretical analysis. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.Chapter 3, the ideas of filled function for continuous global optimization are extended to discrete cases. The definition of discrete filled function is given, two classes of discrete filled functions with two parameters and one parameter are presented respectively, two corresponding algorithms are proposed according to the theoretical analysis. The impiementation of the algorithms on several test problems is reported with satisfactory numerical results.Chapter 4, a new auxiliary function with one parameter with box constrained on R~n for escaping the current local minimizer of global optimization problem is proposed. Under some mild assumptions, we prove it is a filled function. A new algorithm is presented according to the theoretical analysis. And some numerical results demonstrate the efficiency of this global method for unconstrained global optimization. Chapter 5, the idea of filled function for unconstrained global optirnization is extended to nonlinear global problems with constraints. Firstly, we give a definition of filled function for constrained problem and under some mild assurnptions, we prove it is a filled function. Then, a new algorithm is presented according to the theoretical analysis.