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Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces
| 1. | Harbin Institute of Technology Shenzhen Graduate School, Shenzhen 518055, China |
| 2. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong |
| 3. | Department of Mathematics, Soochow University, Suzhou, 215006 |
References:
| [1] |
J. M. Borwein and A. S. Lewis, Partially finite convex programming, part I: Quasi relative interiors and duality theory, Math. Program., 57 (1992), 15-48.
doi: 10.1007/BF01581072. |
| [2] |
R. I. Boţ, E. R. Csetnek and G. Wanka, Regularity conditions via quasi-relative interior in convex programming, SIAM J. Optim., 19 (2008), 217-233.
doi: 10.1137/07068432X. |
| [3] |
R. I. Boţ, S. M. Grad and G. Wanka, On strong and total Lagrange duality for convex optimization problems, J. Math. Anal. Appl., 337 (2008), 1315-1325.
doi: 10.1016/j.jmaa.2007.04.071. |
| [4] |
R. I. Boţ, S. M. Grad and G. Wanka, New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces, Nonlinear Anal., 69 (2008), 323-336.
doi: 10.1016/j.na.2007.05.021. |
| [5] |
R. S. Burachik and V. Jeyakumar, A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal., 12 (2005), 279-290. |
| [6] |
R. S. Burachik and V. Jeyakumar, A simple closure condition for the normal cone intersection formula, Proc. Amer. Math. Soc., 133 (2005), 1741-1748.
doi: 10.1090/S0002-9939-04-07844-X. |
| [7] |
R. S. Burachik, V. Jeyakumar and Z. Y. Wu, Necessary and sufficient conditions for stable conjugate duality, Nonlinear Anal., 64 (2006), 1998-2006.
doi: 10.1016/j.na.2005.07.034. |
| [8] |
B. D. Craven, "Mathematical Programming and Control Theory," Chapman and Hall, London, 1978. |
| [9] |
N. Dinh, T. T. A. Nghia and G. Vallet, A closedness condition and its applications to DC programs with convex constraints, Optimization, 59 (2010), 541-560.
doi: 10.1080/02331930801951348. |
| [10] |
D. H. Fang, C. Li and X. Q. Yang, Stable and total Fenchel duality for DC optimization problems in locally convex spaces, SIAM J. Optim., 21 (2011), 730-760.
doi: 10.1137/100789749. |
| [11] |
M. S. Gowda and M. Teboulle, A comparison of constraint qualifications in infinite dimensional convex programming, SIAM J. Control Optim., 28 (1990), 925-935.
doi: 10.1137/0328051. |
| [12] |
V. Jeyakumar and B. M. Glover, Characterizing global optimality for DC optimization problems under convex inequality constraints, J. Global Optim., 8 (1996), 171-187.
doi: 10.1007/BF00138691. |
| [13] |
V. Jeyakumar and G. M. Lee, Complete characterizations of stable Farkas' lemma and cone-convex programming duality, Math. Program., Ser. A, 114 (2008), 335-347.
doi: 10.1007/s10107-007-0104-x. |
| [14] |
B. Lemaire and M. Volle, Duality in DC programming, in "Nonconvex Optim. Appl., 27" (eds. J. P. Crouzeix, J. E. Martinez-Legaz and M. Volle), Kluwer Academic, Dordrecht, (1998), 331-345.
doi: 10.1007/978-1-4613-3341-8_15. |
| [15] |
C. Li, D. H. Fang, G. López and M. A. López, Stable and total Fenchel duality for convex optimization problems in locally convex spaces, SIAM, J. Optim., 20 (2009), 1032-1051.
doi: 10.1137/080734352. |
| [16] |
C. Li, F. Ng and T. K. Pong, The SECQ, linear regularity and the strong CHIP for infinite system of closed convex sets in normed linear spaces, SIAM J. Optim., 18 (2007), 643-665.
doi: 10.1137/060652087. |
| [17] |
J. E. Martinez-Legaz and M. Volle, Duality in DC programming: The case of several constraints, J. Math. Anal. Appl., 237 (1999), 657-671.
doi: 10.1006/jmaa.1999.6496. |
| [18] |
R. T. Rockafellar, "Conjuagate Duality and Optimization," Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics 16, Society for Industrial and Applied Mathematics, Philadelphia, 1974. |
| [19] |
X. T. Xiao, J. Gu, L. W. Zhang and S. W. Zhang, A sequential convex program method to DC program with joint chance constraints, J. Ind. Manag. Optim., 8 (2012), 733-747.
doi: 10.3934/jimo.2012.8.733. |
| [20] |
C. Zălinescu, "Convex Analysis in General Vector Space," World Sciencetific Publishing, Singapore, 2002.
doi: 10.1142/9789812777096. |
show all references
References:
| [1] |
J. M. Borwein and A. S. Lewis, Partially finite convex programming, part I: Quasi relative interiors and duality theory, Math. Program., 57 (1992), 15-48.
doi: 10.1007/BF01581072. |
| [2] |
R. I. Boţ, E. R. Csetnek and G. Wanka, Regularity conditions via quasi-relative interior in convex programming, SIAM J. Optim., 19 (2008), 217-233.
doi: 10.1137/07068432X. |
| [3] |
R. I. Boţ, S. M. Grad and G. Wanka, On strong and total Lagrange duality for convex optimization problems, J. Math. Anal. Appl., 337 (2008), 1315-1325.
doi: 10.1016/j.jmaa.2007.04.071. |
| [4] |
R. I. Boţ, S. M. Grad and G. Wanka, New regularity conditions for strong and total Fenchel-Lagrange duality in infinite dimensional spaces, Nonlinear Anal., 69 (2008), 323-336.
doi: 10.1016/j.na.2007.05.021. |
| [5] |
R. S. Burachik and V. Jeyakumar, A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal., 12 (2005), 279-290. |
| [6] |
R. S. Burachik and V. Jeyakumar, A simple closure condition for the normal cone intersection formula, Proc. Amer. Math. Soc., 133 (2005), 1741-1748.
doi: 10.1090/S0002-9939-04-07844-X. |
| [7] |
R. S. Burachik, V. Jeyakumar and Z. Y. Wu, Necessary and sufficient conditions for stable conjugate duality, Nonlinear Anal., 64 (2006), 1998-2006.
doi: 10.1016/j.na.2005.07.034. |
| [8] |
B. D. Craven, "Mathematical Programming and Control Theory," Chapman and Hall, London, 1978. |
| [9] |
N. Dinh, T. T. A. Nghia and G. Vallet, A closedness condition and its applications to DC programs with convex constraints, Optimization, 59 (2010), 541-560.
doi: 10.1080/02331930801951348. |
| [10] |
D. H. Fang, C. Li and X. Q. Yang, Stable and total Fenchel duality for DC optimization problems in locally convex spaces, SIAM J. Optim., 21 (2011), 730-760.
doi: 10.1137/100789749. |
| [11] |
M. S. Gowda and M. Teboulle, A comparison of constraint qualifications in infinite dimensional convex programming, SIAM J. Control Optim., 28 (1990), 925-935.
doi: 10.1137/0328051. |
| [12] |
V. Jeyakumar and B. M. Glover, Characterizing global optimality for DC optimization problems under convex inequality constraints, J. Global Optim., 8 (1996), 171-187.
doi: 10.1007/BF00138691. |
| [13] |
V. Jeyakumar and G. M. Lee, Complete characterizations of stable Farkas' lemma and cone-convex programming duality, Math. Program., Ser. A, 114 (2008), 335-347.
doi: 10.1007/s10107-007-0104-x. |
| [14] |
B. Lemaire and M. Volle, Duality in DC programming, in "Nonconvex Optim. Appl., 27" (eds. J. P. Crouzeix, J. E. Martinez-Legaz and M. Volle), Kluwer Academic, Dordrecht, (1998), 331-345.
doi: 10.1007/978-1-4613-3341-8_15. |
| [15] |
C. Li, D. H. Fang, G. López and M. A. López, Stable and total Fenchel duality for convex optimization problems in locally convex spaces, SIAM, J. Optim., 20 (2009), 1032-1051.
doi: 10.1137/080734352. |
| [16] |
C. Li, F. Ng and T. K. Pong, The SECQ, linear regularity and the strong CHIP for infinite system of closed convex sets in normed linear spaces, SIAM J. Optim., 18 (2007), 643-665.
doi: 10.1137/060652087. |
| [17] |
J. E. Martinez-Legaz and M. Volle, Duality in DC programming: The case of several constraints, J. Math. Anal. Appl., 237 (1999), 657-671.
doi: 10.1006/jmaa.1999.6496. |
| [18] |
R. T. Rockafellar, "Conjuagate Duality and Optimization," Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics 16, Society for Industrial and Applied Mathematics, Philadelphia, 1974. |
| [19] |
X. T. Xiao, J. Gu, L. W. Zhang and S. W. Zhang, A sequential convex program method to DC program with joint chance constraints, J. Ind. Manag. Optim., 8 (2012), 733-747.
doi: 10.3934/jimo.2012.8.733. |
| [20] |
C. Zălinescu, "Convex Analysis in General Vector Space," World Sciencetific Publishing, Singapore, 2002.
doi: 10.1142/9789812777096. |
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