Optimality Conditions And Optimization Methods For Weakly Concave Optimization Problems

Global optimization problems abound in image processing, chemical engineering design and control, network and transportation, molecular biology, finance, economic modeling, database and environmental engineering. Weakly concave programming problems consisting of minimizing a di?erence of a quadratic function and a convex function includes quadratic programming problems and concave minimization problems and we pay more attention to it in the fields of local optimality problems and global optimality problems.So, it is very necessary to research weakly concave programming problems. In this thesis,we would study some global optimality conditions and global optimization methods for this programming problems.The thesis is organized as follows.In the first chapter of this thesis, we briefly introduces the theory and method of global optimization.In the second chapter, we study global optimality conditions and global optimization methods for weakly concave programming problems with linear inequality and box constraints. We obtain a necessary global optimality condition for it by constructing a new box to replace its feasible set. Then a new local optimization method for it is designed based on the necessary global optimality conditions.And then a global optimization method is proposed by combining some auxiliary functions and the new local optimization method. Furthermore, numerical examples are also presented to show that the proposed global optimization method for weakly concave programming problems is e?cient.The third chapter based on the research on the second chapter, the further study optimality conditions for global optimization algorithm of weakly concave programming problem with quadratic constraints and box constraints.In the fourth chapter, we will further discuss some new necessary global optimality conditions for mixed weakly concave programming problems. Firstly, some new necessary global optimality conditions for it are proposed. A new local optimization method based on the necessary global optimality conditions is developed. Then a global optimization method based on auxiliary functions and the proposed new local optimization method is followed. Finally, several examples illustrate the e?ciency of the given methods.