Augmented Lagrangian For Semi-infinite Programming

Semi-infinite programming problem deals with optimization problems in which either the number of decision variables or the number of constraints is infinite,it is an important research topic in mathematical programming.It is not only has been applied widely and directly in engineering design,optimal control,information technology,economic balance and other fields,but also plays an important role in some academic aspects,such as Chebyshev approximation theory,mathematical physics,fuzzy set,the robust optimization and so on.Therefore,the research of theoretical knowledge and effective numerical algorithms on semi-infinite programming has strong application value,and has aroused great attention and research from scholars in the world.In this paper,using the augmented Lagrange function method studied saddle points and augmented Lagrange multiplier of semi-infinite programming problem.Including first order necessary condition,first order sufficient condition,the existence of local saddle points,the necessary and sufficient conditions for the existence of augmented Lagrange multipliers.The main content of this paper is arranged as follows:The first chapter is the introduction part.First of all,the research background and current situation of semi-infinite programming problem are introduced.Furthermore,we introduce the significance and main contents of this paper.Finally,to derive the desired result successfully,some fundamental definitions and lemmas are introduced in the preliminary knowledge part.The second chapter mainly adopts an augmented Lagrange function method,and under the condition of Reduction Approach,discussing the relationship between local saddle point and local optimal solution of semi-infinite programming problem.First of all,the local optimal solution of the problem is obtained by the existence of saddle points.Secondly,under some appropriate conditions,we got that the local optimal solution is the sufficient condition for the existence of the local saddle point.The third chapter consider a class of augmented Lagrange function of semiinfinite programming problem under the condition of finite discrete,and obtained the first order necessary conditions and the first order sufficient condition of existence of augmented Lagrange multiplier.It is worth mentioning that augmented function in this paper is not only to meet the augmented function of a class of special functions,but the general augmented function.Therefore,the condition of this paper is weaker,and the conclusion is more general and general.The fourth chapter mainly given the necessary and sufficient conditions and some sufficient conditions for the existence of the augmented Lagrange multiplier.First of all,the relationship between the zero duality gap,the multiplier and the saddle points is given.Secondly,give necessary and sufficient conditions for the existence of the augmented Lagrange multiplier.Finally,some sufficient conditions are obtained for the existence of augmented Lagrange multiplier.