Abstract

This paper analyzes inequality systems with an arbitrary number of proper lower semicontinuous convex constraint functions and a closed convex constraint subset of a locally convex topological vector space. More in detail, starting from well-known results on linear systems (with no constraint set), the paper reviews and completes previous works on the above class of convex systems, providing consistency theorems, two new versions of Farkas' lemma, and optimality conditions in convex optimization. A new closed cone constraint qualification is proposed. Suitable counterparts of these results for cone-convex systems are also given

Author information

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

N. Dinh

Dept. of Mathematics-Informatics, University of Pedagogy, Ho Chi Minh City, Vietnam

Miguel A. Goberna

Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03080 Alicante, Spain

Marco Antonio López

Dept. of Statistics and Operations Research, University of Alicante, Apt. de Correos 99, 03080 Alicante, Spain

References

  1. F. Bohnenblust, S. Karlin, L. S. Shapley: Games with continuous pay-off, Ann. Math. Stud. 24 (1950) 181-192.https://doi.org/10.1515/9781400881727-016Google Scholar
  2. J. M. Borwein: The limiting Lagrangian as a consequence of Helly’s theorem, J. Optimization Theory Appl. 33 (1981) 497–513.https://doi.org/10.1007/BF00935755Google Scholar
  3. R. S. Burachik, V. Jeyakumar: A dual condition for the convex subdifferential sum formula with applications, J. Convex Analysis 12 (2005) 279–290.https://doi.org/10.68381/jca12020Google Scholar
  4. Yung-Chin Chu: Generalizations of some fundamental theorems on linear inequalities, Acta Math. Sin. 16 (1966) 25–40.Google Scholar
  5. K. Fan: Existence theorems and extreme solutions for inequalities concerning convex functions or linear transformations, Math. Z. 68 (1957) 205–217.https://doi.org/10.1007/BF01160340Google Scholar
  6. M. A. Goberna, V. Jeyakumar, N. Dinh: Dual characterizations of set containments with strict convex inequalities, J. Glob. Optim., to appear.Google Scholar
  7. M. A. Goberna, M. A. López: Linear Semi-Infinite Optimization, Wiley Series in Mathematical Methods in Practice, Wiley, Chichester (1998).Google Scholar
  8. M. A. Goberna, M. A. López, J. A. Mira, J. Valls: On the existence of solutions for linear inequality systems, J. Math. Anal. Appl. 192 (1995) 133–150.https://doi.org/10.1006/jmaa.1995.1164Google Scholar
  9. Ch.-W Ha: On systems of convex inequalities, J. Math. Anal. Appl. 68 (1979) 25–34.https://doi.org/10.1016/0022-247X(79)90098-2Google Scholar
  10. R. B. Holmes: Geometric Functional Analysis, Springer, Berlin (1975).Google Scholar
  11. V. Jeyakumar: Asymptotic dual conditions characterizing optimality for infinite convex programs, J. Optimization Theory Appl. 93 (1997) 153–165.https://doi.org/10.1023/A:1022606002804Google Scholar
  12. V. Jeyakumar: Characterizing set containments involving infinite convex constraints and reverse-convex constraints, SIAM J. Optimization 13 (2003) 947–959.https://doi.org/10.1137/S1052623402401944Google Scholar
  13. V. Jeyakumar, N. Dinh, G. M. Lee: A new closed cone constraint qualification for convex optimization, Applied Mathematics Research Report AMR04/8, UNSW (2004); see http://www2.maths.unsw.edu.au/applied/reports/2004/amr04-8.pdfGoogle Scholar
  14. V. Jeyakumar, A. M. Rubinov, B. M. Glover, Y. Ishizuka: Inequality systems and global optimization, J. Math. Anal. Appl. 202 (1996) 900–919.https://doi.org/10.1006/jmaa.1996.0353Google Scholar
  15. M. A. López, E. Vercher: Convex semi-infinite games, J. Optimization Theory Appl. 50 (1986) 289–312.https://doi.org/10.1007/BF00939275Google Scholar
  16. Z. Denkowski, S. Migórski, N. S. Papageorgiou: An Introduction to Nonlinear Analysis. Theory, Kluwer Academic Publishers, Boston (2003).https://doi.org/10.1007/978-1-4419-9158-4Google Scholar
  17. R. T. Rockafellar: Level sets and continuity of conjugate convex functions, Trans. Am. Math. Soc. 123 (1966) 46–63.https://doi.org/10.1090/S0002-9947-1966-0192318-XGoogle Scholar
  18. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  19. N. Shioji, W. Takahashi: Fan’s theorem concerning systems of convex inequalities and its applications, J. Math. Anal. Appl. 135 (1988) 383–398.https://doi.org/10.1016/0022-247X(88)90162-XGoogle Scholar
  20. Jan van Tiel: Convex Analysis, Wiley, Belfast (1984).Google Scholar
  21. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, New Jersey (2002).https://doi.org/10.1142/5021Google Scholar

Suggested citation

N. Dinh, M. A. Goberna, M. A. López. “From Linear to Convex Systems: Consistency, Farkas' Lemma and Applications.” Journal of Convex Analysis 13 (2006), No. 1, 113–133. https://doi.org/10.68381/jca13008

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2006. Rights now held by Banach Press LLC.