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New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces. (English) Zbl 1142.49015
Summary: We give new regularity conditions for convex optimization problems in separated locally convex spaces. We completely characterize the stable strong and strong Fenchel-Lagrange duality. Then we give similar statements for the case when a solution of the primal problem is assumed as known, obtaining complete characterizations for the so-called total and stable total Fenchel-Lagrange duality, respectively. For particular settings the conditions that we consider turn into some constraint qualifications already used by different authors, like Farkas-Minkowski $CQ$, locally Farkas-Minkowski $CQ$ and basic $CQ$, and we rediscover and improve some recent results from the literature.

MSC:
49N15Duality theory (optimization)
90C25Convex programming
90C34Semi-infinite programming
49N60Regularity of solutions in calculus of variations
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References:
[1] R.I. Boţ, S.M. Grad, G. Wanka, New regularity conditions for Lagrange and Fenchel--Lagrange duality in infinite dimensional spaces, Preprint 13/2006, Fakultät für Mathematik, Technische Universität Chemnitz, 2006 · Zbl 1169.49036
[2] R.I. Boţ, S.M. Grad, G. Wanka, On strong and total Lagrange duality for convex optimization problems, Journal of Mathematical Analysis and Applications (in press) · Zbl 1160.90004
[3] Boţ, R. I.; Wanka, G.: A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces, Nonlinear analysis: theory, methods & applications 64, No. 12, 2787-2804 (2006) · Zbl 1087.49026 · doi:10.1016/j.na.2005.09.017
[4] Boţ, R. I.; Wanka, G.: An alternative formulation for a new closed cone constraint qualification, Nonlinear analysis: theory, methods & applications 64, No. 6, 1367-1381 (2006) · Zbl 1105.46052 · doi:10.1016/j.na.2005.06.041
[5] Burachik, R. S.; Jeyakumar, V.: A dual condition for the convex subdifferential sum formula with applications, Journal of convex analysis 12, No. 2, 279-290 (2005) · Zbl 1098.49017
[6] Combari, C.; Laghdir, M.; Thibault, L.: Sous-différentiels de fonctions convexes composées, Annales des sciences mathématiques du Québec 18, No. 2, 119-148 (1994) · Zbl 0840.49009 · http://www.lacim.uqam.ca/~annales/volumes/18-2/119.html
[7] Dinh, N.; Goberna, M. A.; López, M. A.: From linear to convex systems: consistency, farkas’ lemma and applications, Journal of convex analysis 13, No. 1, 113-133 (2006) · Zbl 1137.90684
[8] N. Dinh, M.A. Goberna, M.A. López, T.Q. Son, New Farkas-type constraint qualifications in convex infinite programming, ESAIM: Control, Optimisation and Calculus of Variations (in press)
[9] Fajardo, M. D.; López, M. A.: Locally farkas--Minkowski systems in convex semi-infinite programming, Journal of optimization theory and applications 103, No. 2, 313-335 (1999) · Zbl 0945.90069 · doi:10.1023/A:1021700702376
[10] M.A. Goberna, V. Jeyakumar, M.A. López, Necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities, Nonlinear Analysis: Theory, Methods & Applications, in press (doi:10.1016/j.na.2006.12.014) · Zbl 1145.90051 · doi:10.1016/j.na.2006.12.014
[11] Goberna, M. A.; López, M. A.; Pastor, J.: Farkas--Minkowski systems in semi-infinite programming, Applied mathematics and optimization 7, 295-308 (1981) · Zbl 0438.90054 · doi:10.1007/BF01442122
[12] Hiriart-Urruty, J. -B.; Lemaréchal, C.: Convex analysis and minimization algorithms I: Fundamentals, (1993)
[13] Hu, H.: Characterizations of the strong basic constraint qualifications, Mathematics of operations research 30, No. 4, 956-965 (2005) · Zbl 1278.90310
[14] V. Jeyakumar, N. Dinh, G.M. Lee, A new closed cone constraint qualification for convex optimization, Applied Mathematics Report AMR 04/8, University of New South Wales, 2004
[15] V. Jeyakumar, W. Song, N. Dinh, G.M. Lee, Stable strong duality in convex optimization, Applied Mathematics Report AMR 05/22, University of New South Wales, 2005
[16] Li, C.; Ng, K. F.: On constraint qualification for an infinite system of convex inequalities in a Banach space, SIAM journal on optimization 15, No. 2, 488-512 (2005) · Zbl 1114.90142 · doi:10.1137/S1052623403434693
[17] Li, W.; Nahak, C.; Singer, I.: Constraint qualifications for semi-infinite systems of convex inequalities, SIAM journal on optimization 11, No. 1, 31-52 (2000) · Zbl 0999.90045 · doi:10.1137/S1052623499355247
[18] Luc, D. T.: Theory of vector optimization, (1989) · Zbl 0688.90051
[19] Penot, J. P.; Théra, M.: Semi-continuous mappings in general topology, Archiv der Mathematik 38, 158-166 (1982) · Zbl 0465.54019 · doi:10.1007/BF01304771
[20] Simons, S.: The Hahn--Banach--Lagrange theorem, Optimization 56, No. 1--2, 149-169 (2007) · Zbl 1128.46004 · doi:10.1080/02331930600819969
[21] Tiba, D.; Zălinescu, C.: On the necessity of some constraint qualification conditions in convex programming, Journal of convex analysis 11, No. 1, 95-110 (2004) · Zbl 1082.49028 · http://www.heldermann.de/JCA/JCA11/jca11007.htm
[22] Zălinescu, C.: Convex analysis in general vector spaces, (2002) · Zbl 1023.46003