Optimality Condition Of Constrained Nonlinear Programming Research

Optimization is an important branch of applied mathematics and it is a widely used subject. It shows us how to choose the most reasonable way in much feasibility in order to achieve the optimal objective. Optimization is a study on various problems including the optimality conditions of research, numerical solution method of determination and so on, as well as this method between experimental condition and realistic question computer experiment. For a long time, People discussion and study the optimization problem, making the optimization theory and the algorithm rapid development, so that formed a new subject. The optimality conditions are the optimal solution of the optimization problem of satisfies necessary and sufficient conditions. Various programming problem to the optimality conditions of research, domestic and overseas scholars pay a hard labor also have made great achievements.Non-smooth programming in the study is from the 1960s has been developed rapidly, and the concept of generalized convex function and promoted the development of non-smooth programming. In recent years, many scholars on non-smooth generalized convexity and in non-smooth programming and the application of the theory of optimization have done a lot of research, and are committed to non-smooth generalized convex function concept of promotion. In recent decades, non-smooth analysis becomes a hot spot of modern mathematics. In many applications, the practical problems and math subject it more and more involved non-smooth optimization problems, thus, this field is increasingly concerned. At present, non-smooth programming has become study and solves many of the natural sciences and the engineering of complex issues a powerful tool, and widely used in engineering technology, production management and national defense construction, etc.This paper structure as follows: the first chapter presents a non-smooth programming background and the developing situation, F-J, KKToptimality conditions and Lagrange dual problem of the nonlinear programming ;the second chapter in view of the objective function and constraint functions is the regular weak Lipchitz under certain conditions, gave and had proven the optimality condition of non-smooth programming about B-(p, r) question which had the equality and the inequality constraints, discussed the condition KKT and local optimal solution relationship; the third chapter in view of the objective function and the restraint function is the regular weak Lipchitz, the objective function and the constraint functions are generalized (F,ρ)-convexity, discuss and prove optimality condition of the programming, and discuss the Wolfe dual; the fourth chapter has given this article conclusion.