-
Previous Article
Homotopy method for a class of multiobjective optimization problems with equilibrium constraints
- JIMO Home
- This Issue
-
Next Article
Line search globalization of a semismooth Newton method for operator equations in Hilbert spaces with applications in optimal control
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.
|
[20] |
H. Tuy, A Note on Necessary and Sufficient Condition for Global Optimality, preprint, Institute of Mathematics, Hanoi, 1989. |
[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.
|
[20] |
H. Tuy, A Note on Necessary and Sufficient Condition for Global Optimality, preprint, Institute of Mathematics, Hanoi, 1989. |
[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. |
[1] |
Gang Li, Yinghong Xu, Zhenhua Qin. Optimality conditions of fenchel-lagrange duality and farkas-type results for composite dc infinite programs. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1275-1293. doi: 10.3934/jimo.2021019 |
[2] |
Yuying Zhou, Gang Li. The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions. Numerical Algebra, Control and Optimization, 2014, 4 (1) : 9-23. doi: 10.3934/naco.2014.4.9 |
[3] |
Annamaria Barbagallo, Rosalba Di Vincenzo, Stéphane Pia. On strong Lagrange duality for weighted traffic equilibrium problem. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1097-1113. doi: 10.3934/dcds.2011.31.1097 |
[4] |
Qinghong Zhang, Gang Chen, Ting Zhang. Duality formulations in semidefinite programming. Journal of Industrial and Management Optimization, 2010, 6 (4) : 881-893. doi: 10.3934/jimo.2010.6.881 |
[5] |
Regina S. Burachik, Xiaoqi Yang. Asymptotic strong duality. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 539-548. doi: 10.3934/naco.2011.1.539 |
[6] |
Regina Sandra Burachik, Alex Rubinov. On the absence of duality gap for Lagrange-type functions. Journal of Industrial and Management Optimization, 2005, 1 (1) : 33-38. doi: 10.3934/jimo.2005.1.33 |
[7] |
Yanqun Liu. Duality in linear programming: From trichotomy to quadrichotomy. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1003-1011. doi: 10.3934/jimo.2011.7.1003 |
[8] |
Xinmin Yang. On second order symmetric duality in nondifferentiable multiobjective programming. Journal of Industrial and Management Optimization, 2009, 5 (4) : 697-703. doi: 10.3934/jimo.2009.5.697 |
[9] |
Tone-Yau Huang, Tamaki Tanaka. Optimality and duality for complex multi-objective programming. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 121-134. doi: 10.3934/naco.2021055 |
[10] |
Xinmin Yang, Xiaoqi Yang, Kok Lay Teo. Higher-order symmetric duality in multiobjective programming with invexity. Journal of Industrial and Management Optimization, 2008, 4 (2) : 385-391. doi: 10.3934/jimo.2008.4.385 |
[11] |
Xinmin Yang, Xiaoqi Yang. A note on mixed type converse duality in multiobjective programming problems. Journal of Industrial and Management Optimization, 2010, 6 (3) : 497-500. doi: 10.3934/jimo.2010.6.497 |
[12] |
Deepak Singh, Bilal Ahmad Dar, Do Sang Kim. Sufficiency and duality in non-smooth interval valued programming problems. Journal of Industrial and Management Optimization, 2019, 15 (2) : 647-665. doi: 10.3934/jimo.2018063 |
[13] |
Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1873-1884. doi: 10.3934/jimo.2019033 |
[14] |
Yuhua Sun, Laisheng Wang. Optimality conditions and duality in nondifferentiable interval-valued programming. Journal of Industrial and Management Optimization, 2013, 9 (1) : 131-142. doi: 10.3934/jimo.2013.9.131 |
[15] |
Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361 |
[16] |
Xiao-Bing Li, Qi-Lin Wang, Zhi Lin. Optimality conditions and duality for minimax fractional programming problems with data uncertainty. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1133-1151. doi: 10.3934/jimo.2018089 |
[17] |
Xinmin Yang, Jin Yang, Heung Wing Joseph Lee. Strong duality theorem for multiobjective higher order nondifferentiable symmetric dual programs. Journal of Industrial and Management Optimization, 2013, 9 (3) : 525-530. doi: 10.3934/jimo.2013.9.525 |
[18] |
Cheng Lu, Zhenbo Wang, Wenxun Xing, Shu-Cherng Fang. Extended canonical duality and conic programming for solving 0-1 quadratic programming problems. Journal of Industrial and Management Optimization, 2010, 6 (4) : 779-793. doi: 10.3934/jimo.2010.6.779 |
[19] |
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 |
[20] |
Xiuhong Chen, Zhihua Li. On optimality conditions and duality for non-differentiable interval-valued programming problems with the generalized (F, ρ)-convexity. Journal of Industrial and Management Optimization, 2018, 14 (3) : 895-912. doi: 10.3934/jimo.2017081 |
2020 Impact Factor: 1.801
Tools
Metrics
Other articles
by authors
[Back to Top]