American Institute of Mathematical Sciences

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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

Gang Li 1, , Xiaoqi Yang 2, and  Yuying Zhou 3,

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.
Keywords: epigraph, Conjugate functions, DC programming..
Mathematics Subject Classification: Primary: 90C26, 90C46; Secondary: 90C3.
Citation: Gang Li, Xiaoqi Yang, Yuying Zhou. Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces. Journal of Industrial & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

B. D. Craven, "Mathematical Programming and Control Theory," Chapman and Hall, London, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

C. Zălinescu, "Convex Analysis in General Vector Space," World Sciencetific Publishing, Singapore, 2002. doi: 10.1142/9789812777096.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

B. D. Craven, "Mathematical Programming and Control Theory," Chapman and Hall, London, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

C. Zălinescu, "Convex Analysis in General Vector Space," World Sciencetific Publishing, Singapore, 2002. doi: 10.1142/9789812777096.  Google Scholar

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