American Institute of Mathematical Sciences

Advanced
  • 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
January  2017, 13(1): 63-79. doi: 10.3934/jimo.2016004

The stable duality of DC programs for composite convex functions

Gang Li 1,, , Lipu Zhang 2, and  Zhe Liu 1,

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

* Corresponding author

Received  January 2015 Revised  June 2015 Published  March 2016

Fund Project: The work was supported by the Natural Science Foundation of China (11401533,11301484,11171247), the Scientific Research Foundation of Zhejiang Agriculture and Forestry University(2013FR080) and Nature science foundation of Zhejiang Province (LY14A010033)

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.

Keywords: Strong Fenchel-Lagrange duality, DC programming, stable duality.
Mathematics Subject Classification: Primary: 90C26, 90C30; Secondary: 90C46.
Citation: Gang Li, Lipu Zhang, Zhe Liu. The stable duality of DC programs for composite convex functions. Journal of Industrial & Management Optimization, 2017, 13 (1) : 63-79. doi: 10.3934/jimo.2016004
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. I. Boƫ and G. Wanka, Farkas-type results with conjugate functions, SIAM J. Optim., 15 (2005), 540-554.  doi: 10.1137/030602332.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13] J. -B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms Ⅱ, Advanced Theory and Bundle Methods, Springer-Verlag, Berlin, 1993.   Google Scholar
[14]

V. Jeyakumar, Asymptotic dual conditions characterizing optimality for convex programs, J. Optim. Theory Appl., 93 (1997), 153-165.  doi: 10.1023/A:1022606002804.  Google Scholar

[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.  Google Scholar

[16]

M. Laghdir, Optimality conditions and Toland's duality for a non-convex minimization problem, Mat. Versn., 55 (2003), 21-30.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

R. I. Boƫ and G. Wanka, Farkas-type results with conjugate functions, SIAM J. Optim., 15 (2005), 540-554.  doi: 10.1137/030602332.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13] J. -B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms Ⅱ, Advanced Theory and Bundle Methods, Springer-Verlag, Berlin, 1993.   Google Scholar
[14]

V. Jeyakumar, Asymptotic dual conditions characterizing optimality for convex programs, J. Optim. Theory Appl., 93 (1997), 153-165.  doi: 10.1023/A:1022606002804.  Google Scholar

[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.  Google Scholar

[16]

M. Laghdir, Optimality conditions and Toland's duality for a non-convex minimization problem, Mat. Versn., 55 (2003), 21-30.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[1]

Yuying Zhou, Gang Li. The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 9-23. doi: 10.3934/naco.2014.4.9
[2]

Annamaria Barbagallo, Rosalba Di Vincenzo, Stéphane Pia. On strong Lagrange duality for weighted traffic equilibrium problem. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1097-1113. doi: 10.3934/dcds.2011.31.1097
[3]

Regina S. Burachik, Xiaoqi Yang. Asymptotic strong duality. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 539-548. doi: 10.3934/naco.2011.1.539
[4]

Qinghong Zhang, Gang Chen, Ting Zhang. Duality formulations in semidefinite programming. Journal of Industrial & Management Optimization, 2010, 6 (4) : 881-893. doi: 10.3934/jimo.2010.6.881
[5]

Regina Sandra Burachik, Alex Rubinov. On the absence of duality gap for Lagrange-type functions. Journal of Industrial & Management Optimization, 2005, 1 (1) : 33-38. doi: 10.3934/jimo.2005.1.33
[6]

Yanqun Liu. Duality in linear programming: From trichotomy to quadrichotomy. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1003-1011. doi: 10.3934/jimo.2011.7.1003
[7]

Xinmin Yang. On second order symmetric duality in nondifferentiable multiobjective programming. Journal of Industrial & Management Optimization, 2009, 5 (4) : 697-703. doi: 10.3934/jimo.2009.5.697
[8]

Xinmin Yang, Jin Yang, Heung Wing Joseph Lee. Strong duality theorem for multiobjective higher order nondifferentiable symmetric dual programs. Journal of Industrial & Management Optimization, 2013, 9 (3) : 525-530. doi: 10.3934/jimo.2013.9.525
[9]

Xinmin Yang, Xiaoqi Yang, Kok Lay Teo. Higher-order symmetric duality in multiobjective programming with invexity. Journal of Industrial & Management Optimization, 2008, 4 (2) : 385-391. doi: 10.3934/jimo.2008.4.385
[10]

Xinmin Yang, Xiaoqi Yang. A note on mixed type converse duality in multiobjective programming problems. Journal of Industrial & Management Optimization, 2010, 6 (3) : 497-500. doi: 10.3934/jimo.2010.6.497
[11]

Yuhua Sun, Laisheng Wang. Optimality conditions and duality in nondifferentiable interval-valued programming. Journal of Industrial & Management Optimization, 2013, 9 (1) : 131-142. doi: 10.3934/jimo.2013.9.131
[12]

Xian-Jun Long, Jing Quan. Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 361-370. doi: 10.3934/naco.2011.1.361
[13]

Deepak Singh, Bilal Ahmad Dar, Do Sang Kim. Sufficiency and duality in non-smooth interval valued programming problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 647-665. doi: 10.3934/jimo.2018063
[14]

Xiao-Bing Li, Qi-Lin Wang, Zhi Lin. Optimality conditions and duality for minimax fractional programming problems with data uncertainty. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1133-1151. doi: 10.3934/jimo.2018089
[15]

Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019033
[16]

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 & Management Optimization, 2010, 6 (4) : 779-793. doi: 10.3934/jimo.2010.6.779
[17]

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
[18]

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 & Management Optimization, 2018, 14 (3) : 895-912. doi: 10.3934/jimo.2017081
[19]

Shiri Artstein-Avidan and Vitali Milman. A characterization of the concept of duality. Electronic Research Announcements, 2007, 14: 42-59. doi: 10.3934/era.2007.14.42
[20]

Adel Alahmadi, Steven Dougherty, André Leroy, Patrick Solé. On the duality and the direction of polycyclic codes. Advances in Mathematics of Communications, 2016, 10 (4) : 921-929. doi: 10.3934/amc.2016049

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (32)
  • HTML views (298)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences