Filled Function Algorithms For Several Optimization Problems

Global optimization mainly studies the characters of the global optimal solution of multivariable nonlinear functions on some constrained region, constructs computing ap-proaches to find the global optimal solution, and discusses the theoretical characters and computational performances of solution methods. Global optimization problems are of-ten discovered in variety of fields, such as molecular biology, economics and finance, data mining and knowledge discovery, environmental engineering, transport network, image processing&&pattern recognition, chemical engineering design, commercial manufacture and so on. Therefore global optimization have attracted much attention.With the wide applications of global optimization, global optimization theories and algorithms are subsequently developed. Generally speaking, according to different properties of convergence, global optimization methods can be classified into two groups: deterministic and stochastic methods. Filled function method is one of the deterministic methods which has attracted much attention recently. So the dissertation mainly refers to the filled function method.The main idea of the filled function method is:if a local minimizer x*has been found, a filled function can be constructed over x*, such that the local minimizer of the filled function can make us leave x*and find a better local minimizer x**with f(x**)<f(x*). Repeat until a better local minimizer cannot be found.Filled function methods provide a way of using local optimization tools to solve global optimization problems. Therefore much attention have been paid to them by researches. However, early definitions of filled functions require that the filled functions have their local minimizers on the line. Besides, deficiencies existing in filled function-s, including complex forms, too many parameters, undesirable properties, may require heavier computation. Therefore, further research is worthy of continuing on improve-ment of the definition of the filled function and construction of filled functions with simple forms, better properties, less parameters and more efficient algorithms.The main task of the dissertation is to improve the definitions of the filled function, and then construct two new classes of the filled functions, finally analyse and discuss the properties of the new ones. Moreover, the filled function method can be used to solve several kind of optimization problems, such as the nonlinear complementarity problem, the variational inequality problem, the system of nonlinear equalities and inequalities and the multiobjective optimization problem. The main purpose of the dissertation is to enrich and improve the theories of the filled function methods, and to provide a new way to solve these optimization problems. The content in detail is as follows:The dissertation mainly consists of seven chapters. The first chapter is the intro-duction of the basic knowledge of global optimization problems, the development history of the filled function method and the main work of this dissertation.In Chapter2, the definition of the filled function is improved. Based on the new definition, we construct two classes of the filled functions, analyse and discuss the prop-erties of the new filled functions. Finally, a criterion is given to decide the point we have obtained is the global optimal solution.In Chapter3, Firstly, the original problem is converted into a corresponding un-constrained optimization problem by using the F-B function. Subsequently, a new filled function with one parameter for the unconstrained optimization problems is constructed based on the idea of filled function. And some relevant properties of this filled function are analyzed and discussed without the condition of Lipschitz continuous. Finally, an algorithm based on the proposed new filled function for solving the nonlinear comple-mentarity problem is presented. Numerical results show that the proposed filled function method is feasible.In Chapter4, a filled function method is suggested for solving finite dimensional variational inequality problems over sets defined by systems of equalities and inequal-ities. Firstly, based on the Karush-Kuhn-Tucker (KKT) conditions of the variational inequality problems, the original problem is converted into a corresponding constrained optimization problem. Subsequently, a new filled function with one parameter is pro-posed for solving the constrained optimization problem. Some properties of the filled function are studied and discussed. Finally, an algorithm based on the proposed filled function for solving variational inequality problems is presented. The implementation of the algorithm on several test problems is reported with numerical results.In Chapter5, Firstly, the original problem is reformulated into an equivalent con-strained global optimization problem. Subsequently, a new filled function with one parameter is constructed based on the special characteristics of the reformulated op-timization problem. Some properties of the filled function are studied and discussed. Finally, an algorithm based on the proposed filled function for solving nonlinear systems of equalities and inequalities is presented. The objective function value can be reduced by half in each iteration of our filled function algorithm. The implementation of the algorithm on several test problems is reported with numerical results.In Chapter6, a filled function algorithm is applied to compute one of the noniso-lated Pareto optimal points of an unconstrained multiobjective optimization problem. Firstly, the original problem is converted into an equivalent global optimization problem. Subsequently, a novel filled function algorithm is presented for solving the correspond-ing global optimization problem. The implementation of the algorithm on several test problems is reported with numerical results.Chapter7is the summary of the dissertation and prospect of the future research in this field. And the limitation of the paper is also mentioned for further discussion.