The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions

Previous Article
Partial $S$-goodness for partially sparse signal recovery
NACO Home
This Issue
Next Article
On the Hermite--Hadamard inequality for convex functions of two variables

2014, 4(1): 9-23. doi: 10.3934/naco.2014.4.9

The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions

Yuying Zhou ^1, and Gang Li ^2,

1.	Department of Mathematics, Soochow University, Suzhou, 215006
2.	School of Sciences, Zhejiang A & F University, Hangzhou 311300, China

Received March 2013 Revised October 2013 Published December 2013

Abstract
Related Papers

In this paper, by virtue of the epigraph technique, we construct a new kind of closedness-type constraint qualification, which is the sufficient and necessary condition to guarantee the strong duality between a cone constraint composite optimization problem and its dual problem holds. Under this closedness-type constraint qualification condition, we obtain a formula of subdifferential for composite functions and study a cone constraint composite DC optimization problem in locally convex Hausdorff topological vector spaces.

Keywords: constraint qualification., duality, Epigraph, composite.

Mathematics Subject Classification: Primary: 90C26, 90C46; Secondary: 90C3.

Citation: 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

References:

[1]	L. T. H. An, An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints, Math. Program., 87 (2000), 401-426. doi: 10.1007/s101070050003.
[2]	L. T. H. An and P. D. Tao, The DC (difference of convex functions) programming and DCA revisited with DC models of real world non-convex optimization problems, Ann. Oper. Res., 133 (2005), 23-46. doi: 10.1007/s10479-004-5022-1.
[3]	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.
[4]	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.
[5]	R. I. Boţ, 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., 281 (2008), 1088-1107. doi: 10.1002/mana.200510662.
[6]	R. I. Boţ, S. M. Grad and G. Wanka, Generalized Moreau-Rockafellar results for composed convex functions, Optimization, 58 (2009), 917-933. doi: 10.1080/02331930902945082.
[7]	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.
[8]	R. I. Boţ, 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.
[9]	R. I. Boţ and G. 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.
[10]	R. I. Boţ and G. Wanka, An alternative formulation for a new closed cone constraint qualification, Nonlinear Anal., 64 (2006), 1367-1381. doi: 10.1016/j.na.2005.06.041.
[11]	R. S. Burachik and V. Jeyakumar, A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal., 12 (2005), 279-290.
[12]	R. S. Burachik and V. Jeyakumar, A new geometric condition for Fenchel duality in infinite dimensional spaces, Math. Program., 104 (2005), 229-233. doi: 10.1007/s10107-005-0614-3.
[13]	B. D. Craven, Mathematical Programming and Control Theory, Chapman and Hall, London, 1978.
[14]	N. Dinh, M. A. Goberna and M. A. López, From linear to convex systems: consistency, Farkas lemma and applications, J. Convex Anal., 13 (2006), 113-133.
[15]	N. Dinh, M. A. Goberna and 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.
[16]	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.
[17]	N. Dinh, G. Vallet and T. T. A. Nghia, Farkas-type results and duality for DC programs with convex constraints, J. Convex Anal., 15 (2008), 253-262.
[18]	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.
[19]	S. P. Fitzpatrick and S. Simons, The conjugates, compositions and marginals of convex functions, J. Convex Anal., 8 (2001), 423-446.
[20]	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.
[21]	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.
[22]	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 space, SIAM J. Optim., 18 (2007), 643-665. doi: 10.1137/060652087.
[23]	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), 669-685. doi: 10.3934/jimo.2013.9.669.
[24]	J. F. Toland, Duality in non-convex optimization, J. Math. Anal. Appl., 66 (1978), 399-415.
[25]	H. Tuy, Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht, 1998.
[26]	C. Zălinescu, Convex Analysis in General Vector Space, World Sciencetific Publishing, Singapore, 2002. doi: 10.1142/9789812777096.

show all references

References:

[1]	L. T. H. An, An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints, Math. Program., 87 (2000), 401-426. doi: 10.1007/s101070050003.
[2]	L. T. H. An and P. D. Tao, The DC (difference of convex functions) programming and DCA revisited with DC models of real world non-convex optimization problems, Ann. Oper. Res., 133 (2005), 23-46. doi: 10.1007/s10479-004-5022-1.
[3]	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.
[4]	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.
[5]	R. I. Boţ, 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., 281 (2008), 1088-1107. doi: 10.1002/mana.200510662.
[6]	R. I. Boţ, S. M. Grad and G. Wanka, Generalized Moreau-Rockafellar results for composed convex functions, Optimization, 58 (2009), 917-933. doi: 10.1080/02331930902945082.
[7]	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.
[8]	R. I. Boţ, 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.
[9]	R. I. Boţ and G. 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.
[10]	R. I. Boţ and G. Wanka, An alternative formulation for a new closed cone constraint qualification, Nonlinear Anal., 64 (2006), 1367-1381. doi: 10.1016/j.na.2005.06.041.
[11]	R. S. Burachik and V. Jeyakumar, A dual condition for the convex subdifferential sum formula with applications, J. Convex Anal., 12 (2005), 279-290.
[12]	R. S. Burachik and V. Jeyakumar, A new geometric condition for Fenchel duality in infinite dimensional spaces, Math. Program., 104 (2005), 229-233. doi: 10.1007/s10107-005-0614-3.
[13]	B. D. Craven, Mathematical Programming and Control Theory, Chapman and Hall, London, 1978.
[14]	N. Dinh, M. A. Goberna and M. A. López, From linear to convex systems: consistency, Farkas lemma and applications, J. Convex Anal., 13 (2006), 113-133.
[15]	N. Dinh, M. A. Goberna and 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.
[16]	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.
[17]	N. Dinh, G. Vallet and T. T. A. Nghia, Farkas-type results and duality for DC programs with convex constraints, J. Convex Anal., 15 (2008), 253-262.
[18]	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.
[19]	S. P. Fitzpatrick and S. Simons, The conjugates, compositions and marginals of convex functions, J. Convex Anal., 8 (2001), 423-446.
[20]	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.
[21]	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.
[22]	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 space, SIAM J. Optim., 18 (2007), 643-665. doi: 10.1137/060652087.
[23]	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), 669-685. doi: 10.3934/jimo.2013.9.669.
[24]	J. F. Toland, Duality in non-convex optimization, J. Math. Anal. Appl., 66 (1978), 399-415.
[25]	H. Tuy, Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht, 1998.
[26]	C. Zălinescu, Convex Analysis in General Vector Space, World Sciencetific Publishing, Singapore, 2002. doi: 10.1142/9789812777096.

[1]	Adela Capătă. Optimality conditions for strong vector equilibrium problems under a weak constraint qualification. Journal of Industrial and Management Optimization, 2015, 11 (2) : 563-574. doi: 10.3934/jimo.2015.11.563
[2]	Gang Li, Lipu Zhang, Zhe Liu. The stable duality of DC programs for composite convex functions. Journal of Industrial and Management Optimization, 2017, 13 (1) : 63-79. doi: 10.3934/jimo.2016004
[3]	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
[4]	Kanghui Guo, Demetrio Labate, Wang-Q Lim, Guido Weiss and Edward Wilson. Wavelets with composite dilations. Electronic Research Announcements, 2004, 10: 78-87.
[5]	Sylvain Sorin, Cheng Wan. Finite composite games: Equilibria and dynamics. Journal of Dynamics and Games, 2016, 3 (1) : 101-120. doi: 10.3934/jdg.2016005
[6]	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
[7]	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
[8]	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
[9]	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
[10]	Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437
[11]	Radu Strugariu, Mircea D. Voisei, Constantin Zălinescu. Counter-examples in bi-duality, triality and tri-duality. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1453-1468. doi: 10.3934/dcds.2011.31.1453
[12]	Haibo Yi. Efficient systolic multiplications in composite fields for cryptographic systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1135-1145. doi: 10.3934/dcdss.2019078
[13]	Giuseppina Autuori, Patrizia Pucci. Entire solutions of nonlocal elasticity models for composite materials. Discrete and Continuous Dynamical Systems - S, 2018, 11 (3) : 357-377. doi: 10.3934/dcdss.2018020
[14]	Paolo Luzzini, Paolo Musolino. Perturbation analysis of the effective conductivity of a periodic composite. Networks and Heterogeneous Media, 2020, 15 (4) : 581-603. doi: 10.3934/nhm.2020015
[15]	Roberto Triggiani. The coupled PDE system of a composite (sandwich) beam revisited. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 285-298. doi: 10.3934/dcdsb.2003.3.285
[16]	Gang Li, Yinghong Xu, Zhenhua Qin. Optimality conditions for composite DC infinite programming problems. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022064
[17]	Ziteng Wang, Shu-Cherng Fang, Wenxun Xing. On constraint qualifications: Motivation, design and inter-relations. Journal of Industrial and Management Optimization, 2013, 9 (4) : 983-1001. doi: 10.3934/jimo.2013.9.983
[18]	Jingzhen Liu, Lihua Bai, Ka-Fai Cedric Yiu. Optimal investment with a value-at-risk constraint. Journal of Industrial and Management Optimization, 2012, 8 (3) : 531-547. doi: 10.3934/jimo.2012.8.531
[19]	Albert Fathi. An Urysohn-type theorem under a dynamical constraint. Journal of Modern Dynamics, 2016, 10: 331-338. doi: 10.3934/jmd.2016.10.331
[20]	Jingzhen Liu, Ka-Fai Cedric Yiu, Kok Lay Teo. Optimal investment-consumption problem with constraint. Journal of Industrial and Management Optimization, 2013, 9 (4) : 743-768. doi: 10.3934/jimo.2013.9.743

Impact Factor:

Tools

Metrics

Other articles
by authors

on AIMS
Yuying Zhou Gang Li
on Google Scholar
Yuying Zhou Gang Li

[Back to Top]