×

New dual constraint qualifications characterizing zero duality gaps of convex programs and semidefinite programs. (English) Zbl 1239.90084

Summary: We present new dual constraint qualifications which completely characterize the zero duality gap property for convex programming problems in general Banach spaces. As an application, we derive constraint qualifications, characterizing zero duality gaps of semidefinite programs. Our approach makes use of convex majorants of support functions together with conjugate analysis and approximate subdifferentials of convex functions.

MSC:

90C25 Convex programming
90C22 Semidefinite programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bertsekas, D. P.; Nedić, A.; Ozdaglar, A. E., Convex Analysis and Optimization (2003), Athena Scientific: Athena Scientific Belmont, MA · Zbl 1140.90001
[2] Jeyakumar, V.; Lee, G. M.; Dinh, N., New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs, SIAM J. Optim., 14, 2, 534-547 (2003) · Zbl 1046.90059
[3] Rockafellar, R. T., Convex Analysis (1970), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0229.90020
[4] Rockafellar, T., (Conjugate Duality and Optimization. Conjugate Duality and Optimization, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, vol. 16 (1974), SIAM: SIAM Philadelphia)
[5] Zalinescu, C., Convex Analysis in General Vector Spaces (2002), World Scientific · Zbl 1023.46003
[6] Jeyakumar, V., Constraint qualifications characterizing Lagrangian duality in convex optimization, J. Optim. Theory Appl., 135, 3, 31-41 (2008) · Zbl 1194.90069
[7] Jeyakumar, V., The strong conical hull intersection property for convex programming, Math. Program, Ser. A, 106, 81-92 (2006) · Zbl 1134.90462
[8] Jeyakumar, V.; Wolkowicz, H., Zero duality gaps in infinite dimensional programming, J. Optim. Theory. Appl., 67, 87-108 (1990) · Zbl 0687.90077
[9] Tseng, P., Some convex programs without a duality gap, Math. Progam. Ser. B, 553-578 (2009) · Zbl 1176.90464
[10] Li, W.; Nahak, C.; Singer, I., Constraint qualifications for semi-infinite systems of convex inequalities, SIAM J. Optim., 11, 1, 31-52 (2000) · Zbl 0999.90045
[11] Li, C.; Jin, X. Q., Nonlinearly constrained best approximation in Hilbert spaces: The strong CHIP and the basic constraint qualification, SIAM J. Optim., 13, 1, 228-239 (2002) · Zbl 1012.41028
[12] Jeyakumar, V., A note on strong duality in convex semidefinite optimization: Necessary and sufficient conditions, Optim. Lett., 2, 15-25 (2008) · Zbl 1147.90038
[13] Ramana, M. V.; Tunel, L.; Wolkowicz, H., Strong duality for semidefinite programming, SIAM J. Optim., 7, 644662 (1997)
[14] Wolkowicz, H.; Saigal, R.; Vandenberghe, L., Handbook of Semidefinite Programming: International Series in Operations Research and Management Science, vol. 27 (2000), Kluwer: Kluwer Dordrecht
[15] Boţ, R. I.; Grad, S. M.; Wanka, G., On strong and total Lagrange duality for convex optimization problems, J. Math. Anal. Appl., 337, 2, 1315-1325 (2008) · Zbl 1160.90004
[16] Jahn, J., (Mathematical Vector Optimization in Partially Ordered Linear Spaces. Mathematical Vector Optimization in Partially Ordered Linear Spaces, Methoden und verfahren der mathematischen Physik, vol. 31 (1986), Verlag Peter Lang) · Zbl 0578.90048
[17] Burachik, R. S.; Jeyakumar, V., A new geometric condition for Fenchel’s duality in infinite dimensional spaces, Math. Program. Ser. B, 104, 229-233 (2005) · Zbl 1093.90077
[18] Dinh, N.; Vallet, G.; Nghia, T. T.A., Farkas-type results and duality for DC programs with convex constraints, J. Convex Anal., 15, 235-262 (2008) · Zbl 1145.49016
[19] N. Dinh, T.T.A. Nghia, G. Vallet, A closedness condition and its applications to DC programs with convex constraints, Optimization, in press (doi:10.1080/02331930801951348); N. Dinh, T.T.A. Nghia, G. Vallet, A closedness condition and its applications to DC programs with convex constraints, Optimization, in press (doi:10.1080/02331930801951348) · Zbl 1218.90155
[20] Precupanu, T., Closedness conditions for the optimality of a family of non-convex optimization problems, Math. Oper. Statist. Ser. Optim., 15, 339-346 (1984) · Zbl 0555.90104
[21] Burachik, R. S.; Jeyakumar, V., A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal., 12, 2, 279-290 (2005) · Zbl 1098.49017
[22] Goberna, M. A.; López, M. A., (Linear Semi-infinite Optimization. Linear Semi-infinite Optimization, Wiley Series in Mathematical Methods in Practice, vol. 2 (1998), John Wiley and Sons, Ltd.: John Wiley and Sons, Ltd. Chichester) · Zbl 0909.90257
[23] Phelps, R. R., (Convex Functions, Monotone Operators and Differentiability. Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Mathematics, vol. 1364 (1993), Springer-Verlag) · Zbl 0921.46039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.