Abstract

We show that separable convex functions enjoy ε-subdifferential sum formula as well as the Fenchel duality without a regularity assumption, and establish that for convex programs with separable convex constraints a new partially asymptotic Lagrange multiplier conditions hold without a constraint qualification. Examples are given to illustrate the results

Author information

Contact details are reproduced from the original publication and may be historical.

Guo Yin Li

Dept. of Applied Mathematics, University of New South Wales, Sydney 2052, Australia

g.li@unsw.edu.au

Vaithilingam Jeyakumar

Dept. of Applied Mathematics, University of New South Wales, Sydney 2052, Australia

v.jeyakumar@unsw.edu.au

References

  1. A. Auslender, M. Teboulle: Asymptotic Cones and Functions in Optimization and Variational Inequalities, Springer Monographs in Mathematics, Springer, New York (2003).Google Scholar
  2. D. P. Bertsekas: Extended monotropic programming and duality, J. Optim. Theor. Appl. 139 (2008) 209–226.https://doi.org/10.1007/s10957-008-9393-3Google Scholar
  3. R. I. Bot, S. M. Grad, G. Wanka: On strong and total Lagrange duality for convex optimization problems, J. Math. Anal. Appl 337 (2008) 1315–1325.https://doi.org/10.1016/j.jmaa.2007.04.071Google Scholar
  4. R. S. Burachik, V. Jeyakumar, Z. Y. Wu: Necessary and sufficient conditions for stable conjugate duality, Nonlinear Anal., Theory Methods Appl. 64A(9) (2006) 1998–2006.https://doi.org/10.1016/j.na.2005.07.034Google Scholar
  5. R. S. Burachik, V. Jeyakumar: A new geometric condition for Fenchel’s duality in infinite dimensional spaces, Math. Program. B 104(2–3) (2005) 229–233.https://doi.org/10.1007/s10107-005-0614-3Google Scholar
  6. R. S. Burachik, V. Jeyakumar: A dual condition for the convex subdifferential sum formula with applications, J. Convex Analysis 12(2) (2005) 279–290.https://doi.org/10.68381/jca12020Google Scholar
  7. V. Jeyakumar: Constraint qualifications characterizing Lagrangian duality in convex optimization, J. Optim. Theor. Appl. 136 (2008) 31-41.https://doi.org/10.1007/s10957-007-9294-xGoogle Scholar
  8. V. Jeyakumar, G. M. Lee, N. Dinh: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs, SIAM J. Optim. 14(2) (2003) 534–547.https://doi.org/10.1137/S1052623402417699Google Scholar
  9. V. Jeyakumar, Z. Y. Wu, G. M. Lee, N. Dinh: Liberating the subgradient optimality conditions from constraint qualifications, J. Glob. Optim. 36 (2006) 127–137.https://doi.org/10.1007/s10898-006-9003-6Google Scholar
  10. D. Kalpana, K. S. Surjeet, V. Vanita: Convex programming with single separable constraint and bounded variables, Comput. Optim. Appl. 36 (2007) 67–82.https://doi.org/10.1007/s10589-006-0396-4Google Scholar
  11. G. Y. Li, K. F. Ng: On extension of Fenchel duality and its application, SIAM J. Optim. 19 (2008) 1489–1509.https://doi.org/10.1137/080716803Google Scholar
  12. S. Simons: Minimax and Monotonicity, Lecture Notes in Mathematics 1693, Springer, Berlin (1998).https://doi.org/10.1007/BFb0093633Google Scholar
  13. R. T. Rockafellar: Convex Analysis, Princeton Univ. Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  14. S. M. Stefanov: Separable Programming. Theory and Methods, Applied Optimization 53, Kluwer, Dordrecht (2001).https://doi.org/10.1007/978-1-4757-3417-1Google Scholar
  15. L. Thibault: On subdifferentials of optimal value functions, SIAM J. Control Optimization 29 (1991) 1019–1036.https://doi.org/10.1137/0329056Google Scholar
  16. P. Tseng: Some convex programs without a duality gap, Math. Progam. B 116 (2009) 553–578.https://doi.org/10.1007/s10107-007-0110-zGoogle Scholar

Suggested citation

G. Y. Li, V. Jeyakumar. “Qualification-Free Optimality Conditions for Convex Programs with Separable Inequality Constraints.” Journal of Convex Analysis 16 (2009), No. 3&4, 845–856. https://doi.org/10.68381/jca16051

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.