Abstract

A necessary and sufficient sequential optimality condition without a constraint qualification for a general convex optimization problem is given in terms of the ε-subdifferential. Further, a sequential characterization of optimal solutions involving the convex subdifferential is derived using a version of the Brøndsted-Rockafellar Theorem. We prove that some results from the literature concerning sequential generalizations of the Pshenichnyi-Rockafellar Lemma are obtained as particular cases of our results. Moreover, by this general approach we succeed to improve some sequential Lagrange multiplier conditions given in the past.

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Keywords

  • Convex programming
  • conjugate function
  • epsilon-subdifferential
  • sequential optimality conditions

Mathematics Subject Classification

90C25, 90C46, 47A55, 42A50

References

  1. A. Ben-Israel, A. Ben-Tal, S. Zlobec: Optimality conditions in convex programming, in: Survey of Mathematical Programming, Proc. Int. Symp, A. Prékopa (ed.), Hungarian Academy of Sciences, Budapest and North-Holland, Amsterdam (1979) 137–156.Google Scholar
  2. A. Ben-Israel, A. Ben-Tal, S. Zlobec: Characterization of optimality in convex programming without a constraint qualification, J. Optimization Theory Appl. 20(4) (1976) 417–437.https://doi.org/10.1007/BF00933129Google Scholar
  3. J. M. Borwein, H. Wolkowicz: Characterizations of optimality without constraint qualification for the abstract convex program, Math. Program. Study 19 (1982) 77–100.https://doi.org/10.1007/BFb0120983Google Scholar
  4. J. M. Borwein, H. Wolkowicz: Characterization of optimality for the abstract convex program with finite dimensional range, J. Aust. Math. Soc., Ser. A 30(4) (1981) 390–411.https://doi.org/10.1017/S1446788700017882Google Scholar
  5. A. Brøndsted, R. T. Rockafellar: On the subdifferential of convex functions, Proc. Amer. Math. Soc. 16 (1965) 605–611.https://doi.org/10.1090/S0002-9939-1965-0178103-8Google Scholar
  6. B. D. Craven, S. Zlobec: Complete characterization of optimality of convex programming in Banach spaces, Appl. Anal. 11(1) (1980) 61–78.https://doi.org/10.1080/00036818008839319Google Scholar
  7. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland, Amsterdam (1976).Google Scholar
  8. V. Jeyakumar: Asymptotic dual conditions characterizing optimality for infinite convex programs, J. Optimization Theory Appl. 93(1) (1997) 153–165.https://doi.org/10.1023/A:1022606002804Google Scholar
  9. V. Jeyakumar, G. M. Lee, N. Dinh: New sequential Lagrange multiplier conditions characterizing optimality without constraint qualifications for convex programs, SIAM J. Optim. 14(2) (2003) 534–547.https://doi.org/10.1137/S1052623402417699Google Scholar
  10. V. Jeyakumar, Z. Y. Wu: A qualification free sequential Pshenichnyi-Rockafellar Lemma and convex semidefinite programming, J. Convex Analysis 13(3–4) (2006) 773–784.https://doi.org/10.68381/jca13058Google Scholar
  11. V. Jeyakumar, Z. Y. Wu, G. M. Lee, N. Dinh: Liberating the subgradient optimality conditions from constraint qualifications, J. Global Optim. 36(1) (2006) 127–137.https://doi.org/10.1007/s10898-006-9003-6Google Scholar
  12. V. Jeyakumar, A. Zaffaroni: Asymptotic conditions for weak and proper optimality in infinite dimensional convex vector optimization, Numer. Funct. Anal. Optimization 17(3–4) (1996) 323–343.https://doi.org/10.1080/01630569608816697Google Scholar
  13. J.-P. Penot: Subdifferential calculus without qualification assumptions, J. Convex Analysis 3(2) (1996) 207–219.https://doi.org/10.68381/jca03014Google Scholar
  14. B. N. Pshenichnyi: Necessary Conditions for an Extremum, Marcel Dekker, New York (1971).Google Scholar
  15. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  16. L. Thibault: A short note on sequential convex subdifferential calculus, unpublished paper (1994).Google Scholar
  17. L. Thibault: Sequential convex subdifferential calculus and sequential Lagrange multipliers, SIAM J. Control Optimization 35(4) (1997) 1434–1444.https://doi.org/10.1137/S0363012995287714Google Scholar
  18. L. Thibault: Limiting convex subdifferential calculus with applications to integration and maximal monotonicity of subdifferential, in: Constructive, Experimental, and Nonlinear Analysis, M. Théra (ed.), CMS Conference Proceedings 27, American Mathematical Society, Providence (2000) 279–289.Google Scholar
  19. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021Google Scholar

Suggested citation

R. I. Boţ, E. R. Csetnek, G. Wanka. “Sequential Optimality Conditions in Convex Programming via Perturbation Approach.” Journal of Convex Analysis 15 (2008), No. 1, 149–164. https://doi.org/10.68381/jca15010

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Published by Heldermann Verlag, 2008. Rights now held by Banach Press LLC.