zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

49N15Duality theory (optimization)
90C25Convex programming
90C34Semi-infinite programming
49N60Regularity of solutions in calculus of variations
Full Text: DOI
[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