On The Use Of Augmented Lagrangians In The Generalized Semi-Infinite Programming

Semi-infinite programming has wide applications in many fields such as engineering design,optimal control,information technology and economic equilibrium. It has become an active field of research in optimization.Recently,with the development of high technology and the profound research on the social economy,a large number of mathematical models of generalized semi-infinite programming emerge in above fields.Therefore,it is very significant to study generalized semi-infinite programming.Because many important recults in theoretical and numerical aspacts of common semi-infinite programming problems are obtained,we can transform the generalized semi-infinite problems into common semi-infinite problems or finite problems.In particular,using augmented Lagrangian functions or penalty funtions is the main method to complete the equivalent transformation.In this paper,we study the use of augmented Lagrangian functions in generalized semi-infinite programming.This paper is composed of three chapters.Chapter 1 is the introduction of this paper,which introduces the development of semi-infinite programming and the main results in this paper.In Chapter 2,we study the first-order optimality conditions for a class of generalized semi-infinite programming.First,we transform the generalized semiinfinite programming into common semi-infinite programming by using a class of augmented Lagrangian fimction in[35],which is the generalization of the essentially quadratic augmented Lagrangian function in[34].A necessary and sufficient condition is given for the equivelent transformation.This necessary and sufficient condition is different from the one in[34]and its structure is much simpler and it is easy to check in practice.In addition,it does not require the compactness of Y.Therefore,under this equivalent condition,we can solve the generalized programming by using the feasible algorithms for the common semi-infinite programming. Furthermore,under the assumption that Y is compact,wc establish two new first-order optimality conditions for the generalized programming.The latter is obtained by adding the condition that the Abadie constraint qualification holds,which is much weaker than the Mangasarian-Fromowitz constraint qualification in[34].Finally,we verify it with an example.In Chapter 3,we study the use of the modified barrier augmented Lagrangian function in generalized semi-infinite programming.The generalized problem in this chapter is obtained by adding the equality constraints in the set-valued map of the generalized problem in Chapter 2.In order to transform it into common semi-infinite programming,we present two conditions.One is a necessary and sufficient condition,the other is a sufficient condition which can be verified easily.