The stable duality of DC programs for composite convex functions

Gang Li; Lipu Zhang; Zhe Liu

doi:10.3934/jimo.2016004

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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 & 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

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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

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