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The stable duality of DC programs for composite convex functions
1. | School of Sciences, Zhejiang Agriculture and Forestry University, Hangzhou, Zhejiang 311300, China |
2. | Institute of Digital Media and Communication Technology, Zhejiang University of Media and Communications, Hangzhou, Zhejiang 310018, China |
In this paper, we consider a composite DC optimization problem with a cone-convex system in locally convex Hausdorff topological vector spaces. By using the properties of the epigraph of the conjugate functions, some necessary and sufficient conditions which characterize the strong Fenchel-Lagrange duality and the stable strong Fenchel-Lagrange duality are given. We apply the results obtained to study the minmax optimization problem and $l_1$ penalty problem.
References:
[1] |
R. I. Boƫ, 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. |
[2] |
R. I. Boƫ, S. M. Grad and G. Wanka,
Generalized Moreau-Rockafellar results for composed convex functions, Optimization(7), 58 (2009), 917-933.
doi: 10.1080/02331930902945082. |
[3] |
R. I. Boƫ, S. M. Grad and G. Wanka,
A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces, Math. Nachr.(8), 281 (2008), 1088-1107.
doi: 10.1002/mana.200510662. |
[4] |
R. I. Boƫ, I. B. Hodrea and G. Wanka,
Farkas-type results for inequality systems with composed convex functions via conjugate duality, J. Math. Anal. Appl., 322 (2006), 316-328.
doi: 10.1016/j.jmaa.2005.09.007. |
[5] |
R. I. Boƫ, E. Varcyas and C. Wanka,
A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces, Nonlinear Anal., 64 (2006), 2787-2804.
doi: 10.1016/j.na.2005.09.017. |
[6] |
R. I. Boƫ and G. Wanka,
Farkas-type results with conjugate functions, SIAM J. Optim., 15 (2005), 540-554.
doi: 10.1137/030602332. |
[7] |
R. S. Burachik, V. Jeyakumar and Z. Y. Wu,
Necessary and sufficient conditions for stable conjugate duality, Nonlinear Anal.(9), 64 (2006), 1998-2006.
doi: 10.1016/j.na.2005.07.034. |
[8] |
R. S. Burachik and V. Jeyakumar,
A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal., 12 (2005), 279-290.
|
[9] |
N. Dinh, M. A. Goberna, M. A. López and T. Q. Son,
New Farkas-type constraint qualifications in convex infinite programming, ESAIM Control Optim. Calc. Var., 13 (2007), 580-597.
doi: 10.1051/cocv:2007027. |
[10] |
N. Dinh, B. S. Mordukhovich and T. T. A. Nghia,
Qualification and optimality conditions for DC programs with infinite constraints, Acta Mathematica Vietnamica, 34 (2009), 125-155.
|
[11] |
N. Dinh, T. T. A. Nghia and G. Vallet,
A closedness condition and its applications to DC programs with convex constraints, Optimization(4), 59 (2010), 541-560.
doi: 10.1080/02331930801951348. |
[12] |
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.(3), 21 (2011), 730-760.
doi: 10.1137/100789749. |
[13] |
J. -B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms Ⅱ, Advanced Theory and Bundle Methods, Springer-Verlag, Berlin, 1993.
![]() |
[14] |
V. Jeyakumar,
Asymptotic dual conditions characterizing optimality for convex programs, J. Optim. Theory Appl., 93 (1997), 153-165.
doi: 10.1023/A:1022606002804. |
[15] |
V. Jeyakumar, A. Rubinov, B. M. Glover and Y. Ishizuka,
Inequality systems and global optimization, J. Math. Anal. Appl., 202 (1996), 900-919.
doi: 10.1006/jmaa.1996.0353. |
[16] |
M. Laghdir,
Optimality conditions and Toland's duality for a non-convex minimization problem, Mat. Versn., 55 (2003), 21-30.
|
[17] |
G. Li, X. Q. Yang and Y. Y. Zhou,
Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces, J. Ind. Manag. Optim., 9 (2013), 671-687.
doi: 10.3934/jimo.2013.9.671. |
[18] |
J. E. Martínez-Legaz and M. Volle,
Duality in DC programming: the case of several DC constraints, J. Math. Anal. Appl., 237 (1999), 657-671.
doi: 10.1006/jmaa.1999.6496. |
[19] |
J.F Toland, Duality in non-convex optimization, J. Math. Anal. Appl., 66 (1978), 399-415. Google Scholar |
[20] |
H. Tuy, A Note on Necessary and Sufficient Condition for Global Optimality, preprint, Institute of Mathematics, Hanoi, 1989. Google Scholar |
[21] |
C. Zălinescu, Convex Analysis in General Vector Space, World Sciencetific Publishing, Singapore, 2002.
doi: 10.1142/9789812777096.![]() ![]() |
[22] |
Y. Y. Zhou and G. Li,
The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions, Numerical Algebra, Control and Optimization, 4 (2014), 9-23.
doi: 10.3934/naco.2014.4.9. |
show all references
References:
[1] |
R. I. Boƫ, 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. |
[2] |
R. I. Boƫ, S. M. Grad and G. Wanka,
Generalized Moreau-Rockafellar results for composed convex functions, Optimization(7), 58 (2009), 917-933.
doi: 10.1080/02331930902945082. |
[3] |
R. I. Boƫ, S. M. Grad and G. Wanka,
A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces, Math. Nachr.(8), 281 (2008), 1088-1107.
doi: 10.1002/mana.200510662. |
[4] |
R. I. Boƫ, I. B. Hodrea and G. Wanka,
Farkas-type results for inequality systems with composed convex functions via conjugate duality, J. Math. Anal. Appl., 322 (2006), 316-328.
doi: 10.1016/j.jmaa.2005.09.007. |
[5] |
R. I. Boƫ, E. Varcyas and C. Wanka,
A weaker regularity condition for subdifferential calculus and Fenchel duality in infinite dimensional spaces, Nonlinear Anal., 64 (2006), 2787-2804.
doi: 10.1016/j.na.2005.09.017. |
[6] |
R. I. Boƫ and G. Wanka,
Farkas-type results with conjugate functions, SIAM J. Optim., 15 (2005), 540-554.
doi: 10.1137/030602332. |
[7] |
R. S. Burachik, V. Jeyakumar and Z. Y. Wu,
Necessary and sufficient conditions for stable conjugate duality, Nonlinear Anal.(9), 64 (2006), 1998-2006.
doi: 10.1016/j.na.2005.07.034. |
[8] |
R. S. Burachik and V. Jeyakumar,
A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal., 12 (2005), 279-290.
|
[9] |
N. Dinh, M. A. Goberna, M. A. López and T. Q. Son,
New Farkas-type constraint qualifications in convex infinite programming, ESAIM Control Optim. Calc. Var., 13 (2007), 580-597.
doi: 10.1051/cocv:2007027. |
[10] |
N. Dinh, B. S. Mordukhovich and T. T. A. Nghia,
Qualification and optimality conditions for DC programs with infinite constraints, Acta Mathematica Vietnamica, 34 (2009), 125-155.
|
[11] |
N. Dinh, T. T. A. Nghia and G. Vallet,
A closedness condition and its applications to DC programs with convex constraints, Optimization(4), 59 (2010), 541-560.
doi: 10.1080/02331930801951348. |
[12] |
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.(3), 21 (2011), 730-760.
doi: 10.1137/100789749. |
[13] |
J. -B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms Ⅱ, Advanced Theory and Bundle Methods, Springer-Verlag, Berlin, 1993.
![]() |
[14] |
V. Jeyakumar,
Asymptotic dual conditions characterizing optimality for convex programs, J. Optim. Theory Appl., 93 (1997), 153-165.
doi: 10.1023/A:1022606002804. |
[15] |
V. Jeyakumar, A. Rubinov, B. M. Glover and Y. Ishizuka,
Inequality systems and global optimization, J. Math. Anal. Appl., 202 (1996), 900-919.
doi: 10.1006/jmaa.1996.0353. |
[16] |
M. Laghdir,
Optimality conditions and Toland's duality for a non-convex minimization problem, Mat. Versn., 55 (2003), 21-30.
|
[17] |
G. Li, X. Q. Yang and Y. Y. Zhou,
Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces, J. Ind. Manag. Optim., 9 (2013), 671-687.
doi: 10.3934/jimo.2013.9.671. |
[18] |
J. E. Martínez-Legaz and M. Volle,
Duality in DC programming: the case of several DC constraints, J. Math. Anal. Appl., 237 (1999), 657-671.
doi: 10.1006/jmaa.1999.6496. |
[19] |
J.F Toland, Duality in non-convex optimization, J. Math. Anal. Appl., 66 (1978), 399-415. Google Scholar |
[20] |
H. Tuy, A Note on Necessary and Sufficient Condition for Global Optimality, preprint, Institute of Mathematics, Hanoi, 1989. Google Scholar |
[21] |
C. Zălinescu, Convex Analysis in General Vector Space, World Sciencetific Publishing, Singapore, 2002.
doi: 10.1142/9789812777096.![]() ![]() |
[22] |
Y. Y. Zhou and G. Li,
The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions, Numerical Algebra, Control and Optimization, 4 (2014), 9-23.
doi: 10.3934/naco.2014.4.9. |
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