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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 and 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.

[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.

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