Global Optimality Conditions And Optimization Methods For Several Classes Of Special Nonconvex Programming Problems

Global optimization problems are widely used in many fields such as environmen-tal engineering, national defense, financial economic, production management, location problem, traffic transportation and so on. Global optimization problems mainly by es-tablishing mathematical optimization models to solve practical problems. However, most of the functions in these mathematical optimization problems are nonconvex. Hence non-convex programming problems are essential in these optimization problems. Especially in recent decades, many experts have made some progress about the research of some spe-cial nonconvex programming problems, such as quadratic programming problems, weak-ly concave programming problems, cubic programming problems, quartic programming problems and so on. These achievements not only promote the research work of global optimization, but also promote the development of the society. Hence, it is meaningful to study the global optimality conditions and global optimization methods of some classes of the nonconvex programming problems in this paper.Throughout this paper, we consider the global optimality conditions and the global optimization methods for some kinds of the special nonconvex programming problems. This paper is organized as following:In chapter 1, an introduction about the research works of global optimization prob-lems is given in detail.In chapter 2, the global optimality conditions and the global optimization methods of weakly concave programming problems with convex quadratic inequality constraints the objective function is the difference between a quadratic function and a convex function are considered. First, we utilize a box set to replace the original feasible domain, and then give the global optimality necessary conditions for this problem. What’s more, a local optimization method is designed to solve the problem by using the necessary conditions. Then we are able to design a global optimization method to solve the problem by using the auxiliary function and the local optimization method. Finally, some numerical examples are used to demonstrate the effectiveness of the methods.In chapter 3, a special optimization problem that the objective function is the differ-ence between a cubic function and a convex function with linear constraints is considered. Similarity with the method in chapter 2, the global optimality necessary conditions for this problem is established. At the same time, a local optimization method and a glob-al optimization method for solving this kind of problem are designed. Finally, a few of numerical examples are given to illustrate that the global optimization method is efficient.In chapter 4, a special optimization problems that the objective function is the dif-ference between a cubic function and a convex function with convex quadratic inequality constraints is considered. Based on the research of chapter 2 and chapter 3, a necessary global optimality condition and a global optimization method for this problem are pro-posed. Finally, some numerical examples are given to show that the global optimization method is efficient.In chapter 5, considering the cubic programming problems with integer constrains, a global optimality necessary conditions is presented. Then a local optimization method is provided by employing the necessary global optimality conditions. Furthermore using the auxiliary function and the local optimization method, a global optimization method for solving this kind of problems is designed. At last, we use a few of numerical examples to illustrate the effectiveness of the global optimization method.In chapter 6, we give some conclusions and prospects.