March  2022, 18(2): 1275-1293. doi: 10.3934/jimo.2021019

Optimality conditions of fenchel-lagrange duality and farkas-type results for composite dc infinite programs

1. 

Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

2. 

Department of Information Technology, Zhejiang Institute of Mechanical & Electrical Engineering, Hangzhou 310053, China

*Corresponding author

Received  May 2020 Revised  December 2020 Published  March 2022 Early access  January 2021

Fund Project: This work was supported by the Natural Science Foundation of China (11401533), the Zhejiang Provincial Natural Science Foundation of China (LY18A010030), the Scientific Research Fund of Zhejiang Provincial Education Department (19060042-F) and the Science Foundation of Zhejiang Sci-Tech University (19062150-Y)

This paper is concerned with a DC composite programs with infinite DC inequalities constraints. Without any topological assumptions and generalized increasing property, we first construct some new regularity conditions by virtue of the epigraph technique. Then we give some complete characterizations of the (stable) Fenchel-Lagrange duality and the (stable) Farkas-type assertions. As applications, corresponding assertions for the DC programs with infinite inequality constraints and the conic programs with DC composite function are also given.

Citation: 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
References:
[1]

R. I. BoţS.-M. Grad and G. Wanka, A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces, Mathematische Nachrichten, 281 (2008), 1088-1107.  doi: 10.1002/mana.200510662.

[2]

R. I. BoţS.-M. Grad and G. Wanka, Generalized Moreau-Rockafellar results for composed convex functions, Optimization, 58 (2009), 917-933.  doi: 10.1080/02331930902945082.

[3]

R. I. BoţS.-M. Grad and G. Wanka, On strong and total Lagrange duality for convex optimization problems, Journal of Mathematical Analysis and Applications, 337 (2008), 1315-1325.  doi: 10.1016/j.jmaa.2007.04.071.

[4]

R. I. BoţI. B. Hodrea and G. Wanka, Some new Farkas-type results for inequality system with DC functions, Journal of Global Optimization, 39 (2007), 595-608.  doi: 10.1007/s10898-007-9159-8.

[5]

N. DinhM. A. GobernaM. A. López and T. Q. Song, New Farkas-type constraint qualifications in convex infinite programming, ESAIM: Control Optimisation and Calculus of Variations, 13 (2007), 580-597.  doi: 10.1051/cocv:2007027.

[6]

N. DinhB. S. Mordukhovich and T. T. A. Nghia, Qualification and optimality conditions for DC programs with infinite constraints, Acta Mathematica Vietnamica, 34 (2009), 125-155. 

[7]

D. H. Fang, Some relationships among the constraint qualifications for Lagrangian dualities in DC infinite optimization problems, Journal of Inequalities and Applications, 2015 (2015), 41-55.  doi: 10.1186/s13660-015-0561-3.

[8]

D. H. Fang and X. Gong, Extended Farkas lemma and strong duality for composite optimization problems with DC functions, Optimization, 66 (2017), 179-196.  doi: 10.1080/02331934.2016.1266628.

[9]

D. H. FangG. M. LeeC. Li and J. C. Yao, Extended Farkas's lemma and strong Lagrange dualities for DC infinite programming, Journal of Nonlinear and Convex Analysis, 14 (2013), 747-767. 

[10]

D. H. FangC. Li and K. F. Ng, Constraint qualifications for extended Farkas' lemmas and Lagrangian dualities in convex infinite programming, SIAM Journal on Optimization, 20 (2009), 1311-1332.  doi: 10.1137/080739124.

[11]

D. H. FangM. D. Wang and X. P. Zhao, The strong duality for DC optimization problems with composite convex functions, Journal of Nonlinear Convex Analysis, 16 (2015), 1337-1352. 

[12]

M. A. GobernaV. Jeyakumar and M. A. López, Necessary and sufficient conditions for solvability of systems of infinite convex inequalities, Nonlinear Analysis, 68 (2008), 1184-1194.  doi: 10.1016/j.na.2006.12.014.

[13]

M. A. Goberna and M. A. López, Linear Semi-Infinite Optimization, Wiley, Chichester, 1998.

[14]

V. JeyakumarA. RubinovB. M. Glover and Y. Ishizuka, Inequality systems and global optimization, Journal of Mathematical Analysis and Applications, 202 (1996), 900-919.  doi: 10.1006/jmaa.1996.0353.

[15]

C. LiD. H. FangG. López and M. A. López, Stable and total Fenchel duality for convex optimization problems in locally convex spaces, SIAM Journal on Optimization, 20 (2009), 1032-1051.  doi: 10.1137/080734352.

[16]

C. Li and K. F. Ng, On constrint qualification for infinite system of convex inequalities in a Banach space, SIAM Journal on Optimization, 15 (2005), 488-512.  doi: 10.1137/S1052623403434693.

[17]

C. LiK. F. Ng and T. K. Pong, Constrint qualifications for convex inequality systems with applications in constrained optimization, SIAM Journal on Optimization, 19 (2008), 163-187.  doi: 10.1137/060676982.

[18]

G. LiX. Q. Yang and Y. Y. Zhou, Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces, Journal of Industrial and Management Optimization, 9 (2013), 671-687.  doi: 10.3934/jimo.2013.9.669.

[19]

G. LiL. P. Zhang and Z. Liu, The stable duality of DC programs for composite convex functions, Journal of Industrial and Management Optimization, 13 (2017), 63-79.  doi: 10.3934/jimo.2016004.

[20]

W. LiC. Nahak and I. Singer, Constrint qualifications for semi-infinite systems of convex inequalities, SIAM Journal on Optimization, 11 (2000), 31-52.  doi: 10.1137/S1052623499355247.

[21]

J.-E. Martínez-Legaz and M. Volle, Duality in D.C. programming: The case of several D.C. constraints, Journal of Mathematical Analysis and Applications, 237 (1999), 657-671.  doi: 10.1006/jmaa.1999.6496.

[22]

X.-K. Sun, Regularity conditions characterizing Fenchel-Lagrange duality and Farkas-type results for DC infinite programming, Journal of Mathematical Analysis and Applications, 414 (2014), 590-611.  doi: 10.1016/j.jmaa.2014.01.033.

[23]

X.-K. Sun and H.-Y. Fu, A note on optimality conditions for DC programs involving composite functions, Abstract and Applied Analysis, 2014 (2014), 203467, 6 pp. doi: 10.1155/2014/203467.

[24]

X.-K. Sun, H.-Y. Fu and J. Zeng, Robust approximate optimality conditions for uncertain nonsmooth optimization with infinite number of constraints, Mathematics, 7 (2019). doi: 10.3390/math7010012.

[25]

X.-K. SunX.-L. Guo and Y. Zhang, Fenchel-Lagrange duality for DC programs with composite functions, Journal of Nonlinear Convex Analysis, 16 (2015), 1607-1618. 

[26]

X.-K. SunS.-J. Li and D. Zhao, Duality and Farkas-type results for DC infinite programming with inequality constraints, Taiwanese Journal of Mathematics, 17 (2013), 1227-1244.  doi: 10.11650/tjm.17.2013.2675.

[27]

X.-K. SunX.-J. Long and J. Zeng, Constraint qualifications characterizing Fenchel duality in composed convex optimization, Journal of Nonlinear Convex Analysis, 17 (2016), 325-347. 

[28]

C. Zǎlinescu, Convex Analysis in General Vector Space, World Sciencetific Publishing, Singapore, 2002. doi: 10.1142/9789812777096.

show all references

References:
[1]

R. I. BoţS.-M. Grad and G. Wanka, A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces, Mathematische Nachrichten, 281 (2008), 1088-1107.  doi: 10.1002/mana.200510662.

[2]

R. I. BoţS.-M. Grad and G. Wanka, Generalized Moreau-Rockafellar results for composed convex functions, Optimization, 58 (2009), 917-933.  doi: 10.1080/02331930902945082.

[3]

R. I. BoţS.-M. Grad and G. Wanka, On strong and total Lagrange duality for convex optimization problems, Journal of Mathematical Analysis and Applications, 337 (2008), 1315-1325.  doi: 10.1016/j.jmaa.2007.04.071.

[4]

R. I. BoţI. B. Hodrea and G. Wanka, Some new Farkas-type results for inequality system with DC functions, Journal of Global Optimization, 39 (2007), 595-608.  doi: 10.1007/s10898-007-9159-8.

[5]

N. DinhM. A. GobernaM. A. López and T. Q. Song, New Farkas-type constraint qualifications in convex infinite programming, ESAIM: Control Optimisation and Calculus of Variations, 13 (2007), 580-597.  doi: 10.1051/cocv:2007027.

[6]

N. DinhB. S. Mordukhovich and T. T. A. Nghia, Qualification and optimality conditions for DC programs with infinite constraints, Acta Mathematica Vietnamica, 34 (2009), 125-155. 

[7]

D. H. Fang, Some relationships among the constraint qualifications for Lagrangian dualities in DC infinite optimization problems, Journal of Inequalities and Applications, 2015 (2015), 41-55.  doi: 10.1186/s13660-015-0561-3.

[8]

D. H. Fang and X. Gong, Extended Farkas lemma and strong duality for composite optimization problems with DC functions, Optimization, 66 (2017), 179-196.  doi: 10.1080/02331934.2016.1266628.

[9]

D. H. FangG. M. LeeC. Li and J. C. Yao, Extended Farkas's lemma and strong Lagrange dualities for DC infinite programming, Journal of Nonlinear and Convex Analysis, 14 (2013), 747-767. 

[10]

D. H. FangC. Li and K. F. Ng, Constraint qualifications for extended Farkas' lemmas and Lagrangian dualities in convex infinite programming, SIAM Journal on Optimization, 20 (2009), 1311-1332.  doi: 10.1137/080739124.

[11]

D. H. FangM. D. Wang and X. P. Zhao, The strong duality for DC optimization problems with composite convex functions, Journal of Nonlinear Convex Analysis, 16 (2015), 1337-1352. 

[12]

M. A. GobernaV. Jeyakumar and M. A. López, Necessary and sufficient conditions for solvability of systems of infinite convex inequalities, Nonlinear Analysis, 68 (2008), 1184-1194.  doi: 10.1016/j.na.2006.12.014.

[13]

M. A. Goberna and M. A. López, Linear Semi-Infinite Optimization, Wiley, Chichester, 1998.

[14]

V. JeyakumarA. RubinovB. M. Glover and Y. Ishizuka, Inequality systems and global optimization, Journal of Mathematical Analysis and Applications, 202 (1996), 900-919.  doi: 10.1006/jmaa.1996.0353.

[15]

C. LiD. H. FangG. López and M. A. López, Stable and total Fenchel duality for convex optimization problems in locally convex spaces, SIAM Journal on Optimization, 20 (2009), 1032-1051.  doi: 10.1137/080734352.

[16]

C. Li and K. F. Ng, On constrint qualification for infinite system of convex inequalities in a Banach space, SIAM Journal on Optimization, 15 (2005), 488-512.  doi: 10.1137/S1052623403434693.

[17]

C. LiK. F. Ng and T. K. Pong, Constrint qualifications for convex inequality systems with applications in constrained optimization, SIAM Journal on Optimization, 19 (2008), 163-187.  doi: 10.1137/060676982.

[18]

G. LiX. Q. Yang and Y. Y. Zhou, Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces, Journal of Industrial and Management Optimization, 9 (2013), 671-687.  doi: 10.3934/jimo.2013.9.669.

[19]

G. LiL. P. Zhang and Z. Liu, The stable duality of DC programs for composite convex functions, Journal of Industrial and Management Optimization, 13 (2017), 63-79.  doi: 10.3934/jimo.2016004.

[20]

W. LiC. Nahak and I. Singer, Constrint qualifications for semi-infinite systems of convex inequalities, SIAM Journal on Optimization, 11 (2000), 31-52.  doi: 10.1137/S1052623499355247.

[21]

J.-E. Martínez-Legaz and M. Volle, Duality in D.C. programming: The case of several D.C. constraints, Journal of Mathematical Analysis and Applications, 237 (1999), 657-671.  doi: 10.1006/jmaa.1999.6496.

[22]

X.-K. Sun, Regularity conditions characterizing Fenchel-Lagrange duality and Farkas-type results for DC infinite programming, Journal of Mathematical Analysis and Applications, 414 (2014), 590-611.  doi: 10.1016/j.jmaa.2014.01.033.

[23]

X.-K. Sun and H.-Y. Fu, A note on optimality conditions for DC programs involving composite functions, Abstract and Applied Analysis, 2014 (2014), 203467, 6 pp. doi: 10.1155/2014/203467.

[24]

X.-K. Sun, H.-Y. Fu and J. Zeng, Robust approximate optimality conditions for uncertain nonsmooth optimization with infinite number of constraints, Mathematics, 7 (2019). doi: 10.3390/math7010012.

[25]

X.-K. SunX.-L. Guo and Y. Zhang, Fenchel-Lagrange duality for DC programs with composite functions, Journal of Nonlinear Convex Analysis, 16 (2015), 1607-1618. 

[26]

X.-K. SunS.-J. Li and D. Zhao, Duality and Farkas-type results for DC infinite programming with inequality constraints, Taiwanese Journal of Mathematics, 17 (2013), 1227-1244.  doi: 10.11650/tjm.17.2013.2675.

[27]

X.-K. SunX.-J. Long and J. Zeng, Constraint qualifications characterizing Fenchel duality in composed convex optimization, Journal of Nonlinear Convex Analysis, 17 (2016), 325-347. 

[28]

C. Zǎlinescu, Convex Analysis in General Vector Space, World Sciencetific Publishing, Singapore, 2002. doi: 10.1142/9789812777096.

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