×

Regularity conditions characterizing Fenchel-Lagrange duality and Farkas-type results in DC infinite programming. (English) Zbl 1312.49047

Summary: In this paper, we consider a DC infinite programming problem \((P)\) with inequality constraints. By using the properties of the epigraph of the conjugate functions, we introduce some new notions of regularity conditions for \((P)\). Under these new regularity conditions, we completely characterize the Fenchel-Lagrange duality and the stable Fenchel-Lagrange duality for \((P)\). Similarly, we also completely characterize the Farkas-type results and the stable Farkas-type results for \((P)\). As applications, we obtain the corresponding results for conic programming problems.

MSC:

49N15 Duality theory (optimization)
90C46 Optimality conditions and duality in mathematical programming
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] An, L. T.H.; Belghiti, M. T.; Tao, P. D., A new efficient algorithm based on DC programming and DCA for clustering, J. Global Optim., 37, 593-608 (2007) · Zbl 1198.90327
[2] An, L. T.H.; Tao, P. D., The DC (difference of convex functions) programming and DCA revisited with DC models of real world non-convex optimization problems, Ann. Oper. Res., 133, 23-46 (2005) · Zbl 1116.90122
[3] Boţ, R. I., Conjugate Duality in Convex Optimization (2010), Springer-Verlag: Springer-Verlag Berlin · Zbl 1190.90002
[4] Boţ, R. I.; Grad, S. M.; Wanka, G., On strong and total Lagrange duality for convex optimization problems, J. Math. Anal. Appl., 337, 1315-1325 (2008) · Zbl 1160.90004
[5] Boţ, R. I.; Grad, S. M.; Wanka, G., New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces, Nonlinear Anal., 69, 323-336 (2008) · Zbl 1142.49015
[6] Boţ, R. I.; Grad, S. M.; Wanka, G., A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces, Math. Nachr., 281, 1-20 (2008)
[7] Boţ, R. I.; Hodrea, I. B.; Wanka, G., Some new Farkas-type results for inequality system with DC functions, J. Global Optim., 39, 595-608 (2007) · Zbl 1182.90071
[8] Boţ, R. I.; Hodrea, I. B.; Wanka, G., Farkas-type results for fractional programming problems, Nonlinear Anal., 67, 1690-1703 (2007) · Zbl 1278.90395
[9] Boţ, R. I.; Wanka, G., Farkas-type results with conjugate functions, SIAM J. Optim., 15, 540-554 (2005) · Zbl 1114.90147
[10] Boţ, R. I.; Wanka, G., A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces, Nonlinear Anal., 64, 2787-2804 (2006) · Zbl 1087.49026
[11] Burachik, R. S.; Jeyakumar, V., A new geometric condition for Fenchel’s duality in infinite dimensional spaces, Math. Program., 104, 229-233 (2005) · Zbl 1093.90077
[12] Dinh, N.; Goberna, M. A.; López, M. A.; Song, T. Q., New Farkas-type constraint qualifications in convex infinite programming, ESAIM Control Optim. Calc. Var., 13, 580-597 (2007) · Zbl 1126.90059
[13] Dinh, N.; Mordukhovich, B. S.; Nghia, T. T.A., Qualification and optimality conditions for convex and DC programs with infinite constraints, Acta Math. Vietnam., 34, 123-153 (2009) · Zbl 1190.90264
[14] Dinh, N.; Mordukhovich, B. S.; Nghia, T. T.A., Subdifferentials of value functions and optimality conditions for DC and bilevel infinite and semi-infinite programs, Math. Program., 123, 101-138 (2010) · Zbl 1226.90102
[15] Dinh, N.; Nghia, T. T.A.; Vallet, G., A closedness condition and its applications to DC programs with convex constraints, Optimization, 59, 541-560 (2010) · Zbl 1218.90155
[16] 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
[17] Fang, D. H.; Li, C.; Ng, K. F., Constraint qualifications for extended Farkas’s lemmas and Lagrangian dualities in convex infinite programming, SIAM J. Optim., 20, 1311-1332 (2009) · Zbl 1206.90198
[18] Fang, D. H.; Li, C.; Ng, K. F., Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming, Nonlinear Anal., 73, 1143-1159 (2010) · Zbl 1218.90200
[19] Fang, D. H.; Li, C.; Yang, X. Q., Stable and total Fenchel duality for DC optimization problems in locally convex spaces, SIAM J. Optim., 21, 730-760 (2011) · Zbl 1236.90140
[20] Goberna, M. A.; Jeyakumar, V.; López, M. A., Necessary and sufficient conditions for solvability of systems of infinite convex inequalities, Nonlinear Anal., 68, 1184-1194 (2008) · Zbl 1145.90051
[21] Goberna, M.; López, M., Linear Semi-Infinite Optimization (1998), Wiley: Wiley Chichester · Zbl 0909.90257
[22] Horst, R.; Thoai, N. V., DC programming: overview, J. Optim. Theory Appl., 103, 1-43 (1999) · Zbl 1073.90537
[23] Jeyakumar, V., Constraint qualifications characterizing Lagrangian duality in convex optimization, J. Optim. Theory Appl., 136, 31-41 (2008) · Zbl 1194.90069
[24] Jeyakumar, V.; Dinh, N.; Lee, G. M., A new closed cone constraint qualification for convex optimization (2004), University of New South: University of New South Wales, Applied Mathematics Report AMR 04/8
[25] Jeyakumar, V.; Lee, G. M., Complete characterization of stable Farkas lemma and cone-convex programming duality, Math. Program., 114, 335-347 (2008) · Zbl 1145.90074
[26] Jeyakumar, V.; Rubinov, A.; Glover, B. M.; Ishizuka, Y., Inequality systems and global optimization, J. Math. Anal. Appl., 202, 900-919 (1996) · Zbl 0856.90128
[27] Laghdir, M., Optimality conditions and Toland’s duality for a non-convex minimization problem, Mat. Vers., 55, 21-30 (2003) · Zbl 1051.49020
[28] López, M.; Still, G., Semi-infinite programming, European J. Oper. Res., 180, 491-518 (2007) · Zbl 1124.90042
[29] Martinez-Legaz, J. E.; Volle, M., Duality in D.C. programming: The case of several D.C. constraints, J. Math. Anal. Appl., 237, 657-671 (1999) · Zbl 0946.90064
[30] Polak, E., Optimization. Algorithms and Consistent Approximations, Appl. Math. Sci. (1997), Springer-Verlag: Springer-Verlag New York · Zbl 0899.90148
[31] Sun, X. K.; Li, S. J., Duality and Farkas-type results for extended Ky Fan inequalities with DC functions, Optim. Lett., 7, 499-510 (2013) · Zbl 1287.90069
[32] Toland, J. F., Duality in nonconvex optimization, J. Math. Anal. Appl., 66, 399-415 (1978) · Zbl 0403.90066
[33] Tuy, H., Convex Analysis and Global Optimization (1998), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0904.90156
[34] Zǎlinescu, C., Convex Analysis in General Vector Spaces (2002), World Scientific: World Scientific River Edge, New Jersey · Zbl 1023.46003
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.