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A Unified Augmented Lagrangian Approach to Duality and Exact Penalization

Department of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China
mahuangx{at}polyu.edu.hk
mayangxq{at}polyu.edu.hk

In this paper, the existence of an optimal path and its convergence to the optimal set of a primal problem of minimizing an extended real-valued function are established via a generalized augmented Lagrangian and corresponding generalized augmented Lagrangian problems, in which no convexity is imposed on the augmenting function. These results further imply a zero duality gap property between the primal problem and the generalized augmented Lagrangian dual problem. A necessary and sufficient condition for the exact penalty representation in the framework of a generalized augmented Lagrangian is obtained. In the context of constrained programs, we show that generalized augmented Lagrangians present a unified approach to several classes of exact penalization results. Some equivalences among exact penalization results are obtained.

Key Words: Generalized augmented Lagrangian; constrained program; duality; exact penalty function; nonlinear Lagrangian
History: Received: April 18, 2001; revision received: August 30, 2002;

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