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  • Research Article
  • Open Access

Generalized Augmented Lagrangian Problem and Approximate Optimal Solutions in Nonlinear Programming

Journal of Inequalities and Applications20072007:019323

  • Received: 19 March 2007
  • Accepted: 29 August 2007
  • Published:


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.


  • Optimality Condition
  • Stationary Point
  • Original Problem
  • Dual Problem
  • Dual Function


Authors’ Affiliations

Department of Mathematics and Computer Science, Chongqing Normal University, Chongqing, 400047, China


  1. Rockafellar RT, Wets RJ-B: Variational Analysis, Fundamental Principles of Mathematical Sciences. Volume 317. Springer, Berlin, Germany; 1998:xiv+733.Google Scholar
  2. 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.MATHGoogle Scholar
  3. 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:1012654128753MathSciNetView ArticleMATHGoogle Scholar
  4. 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.28.3.533.16395MathSciNetView ArticleMATHGoogle Scholar
  5. Di Pillo G, Lucidi S: An augmented Lagrangian function with improved exactness properties. SIAM Journal on Optimization 2001,12(2):376–406.MathSciNetView ArticleMATHGoogle Scholar
  6. 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.27.4.775.295MathSciNetView ArticleMATHGoogle Scholar
  7. Rockafellar RT: Lagrange multipliers and optimality. SIAM Review 1993,35(2):183–238. 10.1137/1035044MathSciNetView ArticleMATHGoogle Scholar
  8. Rubinov AM, Glover BM, Yang XQ: Extended Lagrange and penalty functions in continuous optimization. Optimization 1999,46(4):327–351. 10.1080/02331939908844460MathSciNetView ArticleMATHGoogle Scholar
  9. Huang XX, Yang XQ: Further study on augmented Lagrangian duality theory. Journal of Global Optimization 2005,31(2):193–210. 10.1007/s10898-004-5695-7MathSciNetView ArticleMATHGoogle Scholar
  10. Liu JC: -duality theorem of nondifferentiable nonconvex multiobjective programming. Journal of Optimization Theory and Applications 1991,69(1):153–167. 10.1007/BF00940466MathSciNetView ArticleMATHGoogle Scholar
  11. Loridan P: Necessary conditions for-optimality. Mathematical Programming Study 1982, (19):140–152.Google Scholar
  12. Yokoyama K: -optimality criteria for convex programming problems via exact penalty functions. Mathematical Programming 1992,56(1–3):233–243.MathSciNetView ArticleMATHGoogle Scholar
  13. Ekeland I: On the variational principle. Journal of Mathematical Analysis and Applications 1974,47(2):324–353. 10.1016/0022-247X(74)90025-0MathSciNetView ArticleMATHGoogle Scholar
  14. Huang XX, Yang XQ: Approximate optimal solutions and nonlinear Lagrangian functions. Journal of Global Optimization 2001,21(1):51–65. 10.1023/A:1017960629124MathSciNetView ArticleMATHGoogle Scholar
  15. 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.Google Scholar


© Zhe Chen et al. 2007

This article is published under license to BioMed Central Ltd. 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.