# American Institute of Mathematical Sciences

July  2013, 9(3): 671-687. doi: 10.3934/jimo.2013.9.671

## 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

Received  June 2011 Revised  March 2013 Published  April 2013

By using properties of dualizing parametrization functions, Lagrangian functions and the epigraph technique, some sufficient and necessary conditions of the stable strong duality and strong total duality for a class of DC optimization problems are established.
Citation: Gang Li, Xiaoqi Yang, Yuying Zhou. Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces. Journal of Industrial and Management Optimization, 2013, 9 (3) : 671-687. doi: 10.3934/jimo.2013.9.671
##### 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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##### 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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