Studies On Optimality Conditions And Convexification Methods For A Class Of Global Optimization

Global optimization is to study the solutions of global optimization problem. A large number of problems encountered in practice can be abstracted as global optimization problems, it is widely used in economy, finance, communication, military,image processing. In recent years, theory and algorithms of global optimization have achieved certain progress, some new theory and algorithms spring up successively.However, theory and algorithm of global optimization are not yet matured. Therefore,it is of important theoretical and actual application to study of it. The aim of the thesis is to develop convexification methods, then several optimality conditions are proposed for global optimization.In the first chapter, some background, significance and the development relevant to the global optimality conditions and optimization methods are introduced.In the second chapter, some fundamental definitions and conclusions relevant to the study of global optimization are discussed.In the third chapter, some optimality conditions are proposed for a general class of nonconvex global optimization problem. The auxiliary function is proposed for original objective function, the integral global optimality conditions are obtained with a limiting computation process of the auxiliary function.In the fourth chapter, by using method of function transformations with parameter,a new method of convexification is proposed in the thesis, a class of convexification schems is presented for solving global optimization problem with positive sub-definite objective function, then convexity of the transformed function on the domain is investigated and proved. Furthermore, the general class of global optimization with positive sub-definite objective function can be solved to global optimality, then the global optimization problem can be converted into equivalent convex programming problem. Finally, the conclusions of the theorems are verified by several numerical examples.In the fifth chapter, new convexification and concavification methods are proposed for a general class of global optimization problem. Transformations withparameter are proposed for solving global optimization problem in which the objective function is nonconvex or nonconcave. It is shown that the original objective function can be transformed into convex or concave function, which extends applications of convexification and concavification schems in solving global optimization problems.