- Research Article
- Open Access
Generalized Augmented Lagrangian Problem and Approximate Optimal Solutions in Nonlinear Programming
Journal of Inequalities and Applications volume 2007, Article number: 019323 (2007)
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.
Rockafellar RT, Wets RJ-B: Variational Analysis, Fundamental Principles of Mathematical Sciences. Volume 317. Springer, Berlin, Germany; 1998:xiv+733.
Chen G-Y, Huang XX, Yang XQ: Vector Optimization: Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems. Volume 541. Springer, Berlin, Germany; 2005:x+306.
Huang XX, Yang XQ: Duality and exact penalization for vector optimization via augmented Lagrangian. Journal of Optimization Theory and Applications 2001,111(3):615–640. 10.1023/A:1012654128753
Huang XX, Yang XQ: A unified augmented Lagrangian approach to duality and exact penalization. Mathematics of Operations Research 2003,28(3):533–552. 10.1287/moor.28.3.533.16395
Di Pillo G, Lucidi S: An augmented Lagrangian function with improved exactness properties. SIAM Journal on Optimization 2001,12(2):376–406.
Rubinov AM, Huang XX, Yang XQ: The zero duality gap property and lower semicontinuity of the perturbation function. Mathematics of Operations Research 2002,27(4):775–791. 10.1287/moor.27.4.775.295
Rockafellar RT: Lagrange multipliers and optimality. SIAM Review 1993,35(2):183–238. 10.1137/1035044
Rubinov AM, Glover BM, Yang XQ: Extended Lagrange and penalty functions in continuous optimization. Optimization 1999,46(4):327–351. 10.1080/02331939908844460
Huang XX, Yang XQ: Further study on augmented Lagrangian duality theory. Journal of Global Optimization 2005,31(2):193–210. 10.1007/s10898-004-5695-7
Liu JC: -duality theorem of nondifferentiable nonconvex multiobjective programming. Journal of Optimization Theory and Applications 1991,69(1):153–167. 10.1007/BF00940466
Loridan P: Necessary conditions for-optimality. Mathematical Programming Study 1982, (19):140–152.
Yokoyama K: -optimality criteria for convex programming problems via exact penalty functions. Mathematical Programming 1992,56(1–3):233–243.
Ekeland I: On the variational principle. Journal of Mathematical Analysis and Applications 1974,47(2):324–353. 10.1016/0022-247X(74)90025-0
Huang XX, Yang XQ: Approximate optimal solutions and nonlinear Lagrangian functions. Journal of Global Optimization 2001,21(1):51–65. 10.1023/A:1017960629124
Clarke FH: Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, New York, NY, USA; 1983:xiii+308.
About this article
Cite this article
Chen, Z., Zhao, K. & Chen, Y. Generalized Augmented Lagrangian Problem and Approximate Optimal Solutions in Nonlinear Programming. J Inequal Appl 2007, 019323 (2007). https://doi.org/10.1155/2007/19323
- Optimality Condition
- Stationary Point
- Original Problem
- Dual Problem
- Dual Function