Global Optimality Conditions And Optimization Methods For Several Special Polynomial Programming Problems

The polynomial programming problem which has a polynomial objective function, ei-ther with no constraints or with polynomial constraints. The polynomial programming prob-lem is an important part of the nonlinear programming problem since the nonlinear function can be expressed as a polynomial function approximately by Taylor series, then many non-linear programming problems can be expressed as polynomial programming problems. The polynomial programming problem includes some common programming problems, such as quadratic, cubic, normal quartic programming problems and so on, which have large number of practical applications. Frequently, practitioners need to solve the polynomial programming problem which widely appears in some important fields, such as engineering design, produc-tion management, financial and economic, molecular biology, chemical engineering design and control, national defense military. Therefore, the polynomial programming problem has attracted considerable attention and many researchers explore it by different ways. Hence, it is very necessary to research polynomial programming problems. In this thesis, we will con-sider some global optimality conditions and global optimization methods for the polynomial programming problem.In this thesis, we focus on both global optimality conditions and global optimization method for several kinds of special polynomial programming problems. The main contents of this paper are arranged as follows:In the first chapter, we introduce the optimality conditions and optimization methods for optimization problems briefly, including local optimality conditions, the global optimality conditions, local optimization methods and global optimization methods.In the second chapter, the global optimality conditions for mixed integer cubic program-ming problems is considered, we establish some necessary and sufficient global optimality conditions for mixed integer cubic programming problems via quadratic overestimators and underestimators. Firstly, by utilizing quadratic overestimators, we derive some necessary global optimality conditions for mixed integer cubic minimization problems where the cu- bic objective function contains no cross terms. Then, we obtain some sufficient conditions for global minimizers using quadratic underestimators. Finally, an example is given to demon-strate how to use the global optimality conditions to check a given point.In the third chapter, the quartic polynomial optimization problem with convex quadratic constraints is considered, which is denoted by (QPOPQ). The main idea is to construct a box set to instead of the original feasible set, and the box is a subset of original feasible set. Then we establish a necessary global optimality condition for the problem (QPOPQ), and a strongly local optimization method is designed by exploiting the necessary global optimality condition. Then, a global optimization method for the problem (QPOPQ) is designed by combining the new local optimization method with some auxiliary functions. The results in this chapter extend some corresponding results in some references. Finally, some numerical examples are given to illustrate that the optimization methods are efficient.In the fourth chapter, the general polynomial programming problems with convex quadratic constraints is considered, which is denoted by (GPQ), The main idea is also to con-struct a box set to instead of the original feasible set, and the box is a subset of original feasible set. The objective function can be simplified into a univariate polynomial function, then we get a necessary global optimality condition for the problem (GPQ) by exploiting the property of the univariate polynomial functions, this condition can improve the KKT points. A strongly local optimization method is proposed by using the necessary global optimality condition. Then, a global optimization method for the problem (GPQ) is designed by combining the local optimization method with some auxiliary functions. The numerical results demonstrate that the optimization methods are efficient.In chapter five, a class of polynomial integer programming problems with linear equality constraints is considered. This class of problems has a wide range of practical applications and is NP-hard. Using the penalty function method, some global optimality conditions for such problems are established, including the necessary conditions and sufficient conditions. Finally, some examples are given to demonstrate how to use the global optimality conditions to check a given point.In chapter six, we make a conclusion for this paper and make a prospect for further work.