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Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Generalized Augmented Lagrangian Problem and Approximate Optimal Solutions in Nonlinear Programming | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 19323 12 pages doi 10.1155 2007 19323 Research Article Generalized Augmented Lagrangian Problem and Approximate Optimal Solutions in Nonlinear Programming Zhe Chen Kequan Zhao and Yuke Chen Received 19 March 2007 Accepted 29 August 2007 Recommended by Yeol Je Cho We introduce some approximate optimal solutions and a generalized augmented Lagrangian in nonlinear programming establish dual function and dual problem based on the generalized augmented Lagrangian obtain approximate KKT necessary optimality condition of the generalized augmented Lagrangian dual problem prove that the approximate stationary points of generalized augmented Lagrangian problem converge to that of the original problem. Our results improve and generalize some known results. Copyright 2007 Zhe Chen et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction It is well known that dual method and penalty function method are popular methods in solving nonlinear optimization problems. Many constrained optimization problems can be formulated as an unconstrained optimization problem by dual method or penalty function method. Recently a general class of nonconvex constrained optimization problem has been reformulated as unconstrained optimization problem via augmented Lagrangian 1 . In 1 Rockafellar and Wets introduced an augmented Lagrangian for minimizing an extended real-valued function. Based on the augmented Lagrangian a strong duality result without any convexity requirement in the primal problem was obtained under mild conditions. A necessary and sufficient condition for the exact penalization based on the augment Lagrangian function was given 1 . Chen et al. 2 and Huang and Yang 3 used augmented Lagrangian functions to construct .
