Research On The Existence Of Generalized Augmented Lagrangian Multipliers

The augmented Lagrangian function and its dual theory are an indispensable part of mathematical programming theory.In this paper,the new concept of generalized augmented Lagrangian multiplier is introduced to extend its support form from linear to nonlinear,which allows us to study duality theory in a richer framework.We focuses on the existence of the generalized augmented Lagrangian multiplier and its relationship with the global saddle points and zero duality gap property.Aiming at two different types of optimization models,nonlinear programming and cone constrained optimization,we discusses the conditions for the existence of generalized augmented Lagrangian multipliers and the related duality theory.Firstly,the relationship between the generalized augmented Lagrange multipliers and the zero duality gap property is established.The existence of the generalized augmented Lagrange multipliers does not require that the primal problem to be optimal.Secondly,the relationship between the global saddle points and the zero duality gap property is provided.Compared with the existence of the generalized augmented Lagrange multipliers,global saddle points require that the primal problem is solvable.Since then,a close relationship between the three has been developed.Finally,through the perturbation analysis of the primal problem,some sufficient conditions for the existence of the generalized augmented Lagrange multipliers are established.The structure of this paper is as follows.The first part mainly described the background source and significance of the research on generalized augmented Lagrange multipliers,and the basic knowledge used in this paper.The second and three parts explored the existence conditions of the generalized augmented Lagrange multipliers for nonlinear programming and cone constrained optimization problems,respectively.The fourth part is the summary and prospect,which comprehensively summarized the research results of this article,and proposed the direction of the next step of exploration.