Global Optimally Conditions And Optimization Algorithms For Some Optimization Problems

Optimization theory provides decision makers the scientific basis of solving problems,which uses mathematical methods to study various systems' ways and programs ofoptimization. In practice of recent years, with the improvement of widely used of theoptimization and the living standard, there is a growing need of global optimal solution forsome programming problems.However, due to more than one local minimum point existed in the non-convexoptimization problem, which makes the global optimization very difficult. The functionvalue of these local minima points is not the same as the global minimum, a lot of classicoptimization methods are not necessarily directly to find the global minimum points. Even ifthere are feasible algorithms, we are still lack of the optimality conditions to judge whetherthe obtained point is the global minimum point, which may lead to the solving of thealgorithm becomes very complicated. So the work of this paper is to study globaloptimality conditions for some global optimization problems.The idea of this paper is to use the abstract convex as a theoretical tool,which is putforward in recent years, to establish conditions of global optimality and optimizationalgorithm for mixed {0,1} quadratic programming problem, programming problem of "cubicpolynomial and concave function" with mixed variables, and D.C. global optimizationproblem.This paper is organized into six sections: the first chapter is a brief introduction tooptimization problems, the research status of the optimality conditions and optimizationalgorithms; The second chapter begins with a introduction to several concepts of abstractconvex tools and its comparison with the classical convex analysis, and then give twoimportant lemmas; The third chapter is the first part of the main work of this paper: give anew algorithm for mixed {0,1} quadratic programming problem; The forth chapter is thesecond part of the main work o f this paper: for programming problem of "cubic polynomial and concave function" with mixed variables, give a necessary global optimality condition anda new global optimization algorithm; The fifth chapter is the third part of the main work ofthis paper: based on abstract convexity theory and the lemmas in the second section,gives global optimality sufficient conditions and necessary conditions for the D.C. functionin some cases; The sixth chapter is a summary of this paper.