Global Optimality Conditions And Global Optimization Methods For Nonconvex Optimization Problem

In many practical problems, a great quantity of important fields such as finan-cial economic, engineering, production management, location problem, network de-sign, traffic transportation, structural optimization, agricultural prediction, molec-ular biology, national defense, military matters and nuclear mechanical design, chemical engineering design and control, need the global optimality solutions of some mathematical programming problems expressed in optimization mathemat-ical model. But these mathematical programming problems are generally related to non-convex function, the mature traditional local optimization algorithm can not be used to find the global optimal solution of such a smooth mathematical programming without a hitch. Now we are facing two difficulties for finding the global optimization solutions of such mathematical programming problems, one is how to escape the current local optimal solution and to find a better local optimal solution; Another one is how to determine the point we are founding is the global one. Therefore global optimality conditions and global optimization method are the two important aspects needed to solve for global optimization problems. In this thesis, some important issues of mathematical programming are to be discussed, the global optimality conditions and global optimization methods for the global solution of those mathematical programming problems are mainly discussed.In the first chapter, the study of optimization problems at home and abroad and optimality conditions and optimization methods, including local optimality conditions and the global optimality conditions, local optimization methods and global optimization methods are briefly introduced.In the following chapter, the global optimality conditions and global optimiza-tion methods for the mixed weak concave programming problems with the objective functions given as the difference of quadratic functions and convex functions are considered. Firstly some necessary global optimality conditions and some sufficient global optimality conditions for mixed 0-1 weak concave programming problem are established. Then a local optimization method for mixed 0-1 weak concave pro-gramming problem is designed based on its necessary global optimality conditions. After that a global optimization method is proposed by combining some auxiliary functions, the local optimization method and sufficient global optimality conditions. Numerical experiments indicate that the proposed global optimization methods for the mixed 0-1 weakly concave programming problems is effective. In addition the necessary global optimality conditions as well as the sufficient conditions for global optimality for the mixed integer weak concave programming problem are also pre-sented.In the third chapter global optimality conditions and global optimization meth-ods for a general cubic programming problems with mixed variables are considered. Firstly some necessary local optimality conditions for a given local minimizer and some necessary global optimality conditions for a given global minimizer of cubic programming problems with mixed variables are established. Then strongly local optimization methods for cubic programming problems by exploiting the necessary global optimality conditions and global optimization methods for cubic program-ming problems by combining the strongly local optimization methods and weakly local optimization methods with respect to some auxiliary functions are proposed. Furthermore some numerical examples are given to illustrate our approaches are very efficient.In the fourth chapter, a class of polynomial integer programming problem is studied. By using the L-normal Cones and L-subdifferential, the necessary op-timality conditions and sufficient optimality conditions for a global minimizer of such programming problem are discussed. The relationships between these neces-sary global optimality conditions and these sufficient global optimality conditions are also discussed. The given example shows that our optimality conditions can bo used to determine a given feasible point is the global optimal solution or not.In Chapter five, the mixed integer nonlinear programming problems with bound constrained are discussed. Necessary global optimality conditions and sufficient global optimality conditions of these programming problems are given. In addition, very easy verifiable necessary global optimality conditions and sufficient global optimality conditions of some mixed integer polynomial programming problems such as four plans, three programming, quadratic programming problems are given.In Chapter six, the mixed-integer nonlinear programming problems with equal-ity and inequality constraints are considered. Some sufficient global optimality conditions of the mixed-integer nonlinear programming problems with equality and inequality constraints are given, and the sufficient global optimality conditions for mixed integer quadratic programming problem with equality and inequality quadratic constraints are also given.In Chapter seven, some sufficient global optimality condition for a given feasible point to be a global minimizer of programming problems with nonlinear objects and LMI (Linear Matrix Inequality) and bounded constraints of mixed integer variables are developed. The sufficient global optimality conditions for semidefinite programming problem (SDP) with linear objective function over linear equality constraints with a positive semidefinite matrix condition and box constraints are presented.