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doi: 10.3934/jimo.2022064
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Optimality conditions for composite DC infinite programming problems

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: Gang Li

Received  July 2021 Revised  March 2022 Early access April 2022

Fund Project: The first author is supported by the Scientific Research Fund of Zhejiang Provincial Education Department (19060042-F) and the Science Foundation of Zhejiang Sci-Tech University (19062150-Y). The second author is supported by the Zhejiang Provincial Natural Science Foundation of China (LY18A010030)

This paper is concerned with the optimality conditions for the composite DC optimization problems with infinite inequality constraints. By virtue of the epigraph of the conjugate functions and the $ \varepsilon $-subdifferential of convex functions, we introduce some new regularity conditions. Under these new regularity conditions, we derive some necessary and/or sufficient conditions for the $ \varepsilon $-optimal solutions, the global optimal solutions and the local optimal solutions. Applications to the conical programs with composite DC functions and the DC programs with infinite inequality constraints are also given. The results obtained in this paper extend or cover many results in the literature.

Citation: Gang Li, Yinghong Xu, Zhenhua Qin. Optimality conditions for composite DC infinite programming problems. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022064
References:
[1]

F. BaiZ. Wu and D. Zhu, Sequential Lagrange multiplier condition for $\epsilon$-optimal solution in convex programming, Optimization, 57 (2008), 669-680.  doi: 10.1080/02331930802355234.

[2]

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367-426.  doi: 10.1137/S0036144593251710.

[3]

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

[4]

R. I. BoƫI. B. Hodrea and G. Wanka, $\varepsilon$-optimality conditions for composed convex optimization problems, J. Approx. Theory, 153 (2008), 108-121.  doi: 10.1016/j.jat.2008.03.002.

[5]

N. DinhT. 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.

[6]

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.

[7]

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

[8]

D. H. FangC. 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.

[9]

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

[10]

D. H. Fang and Y. Zhang, Optimality conditions and total dualities for conic programming involving composite function, Optimization, 69 (2020), 305-327.  doi: 10.1080/02331934.2018.1561695.

[11]

D. H. Fang and X. P. Zhao, Local and global optimization conditions for DC infinite optimization problems, Taiwan J. Math., 18 (2014), 817-834.  doi: 10.11650/tjm.18.2014.3888.

[12]

Y. GaoS. H. Hou and X. M. Yang, Existence and optimality conditions for approximate solutions to vector optimization problems, J. Optim. Theory Appl., 152 (2012), 97-120.  doi: 10.1007/s10957-011-9891-6.

[13]

X. L. Guo and S. J. Li, Optimality conditions for vector optimization problems with difference of convex maps, J. Optim. Theory Appl., 162 (2014), 821-844.  doi: 10.1007/s10957-013-0327-3.

[14]

J. B. Hiriart-Urruty, $\epsilon$-Subdifferential, in Convex Analysis and Optimization (eds. J. P. Aubin and R. Vinter), Pitman Advanced Pub. Program, (1982), 43–92.

[15]

Y. H. HuC. Li and X. Q. Yang, On convergence rates of linearized proximal algorithms for convex composite optimization with applications, SIAM J. Optim., 26 (2016), 1207-1235.  doi: 10.1137/140993090.

[16]

Y. H. HuX. Q. Yang and C.-K. Sim, Inexact subgradient methods for quasi-convex optimization problems, Eur. J. Oper. Res., 240 (2015), 315-327.  doi: 10.1016/j.ejor.2014.05.017.

[17]

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

[18]

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

[19]

G. LiY. H. Xu and Z. H. Qin, Fenchel-Lagrange duality for DC infinite programs with inequality constraints, J. Comput. Appl. Math., 391 (2021), 113426.  doi: 10.1016/j.cam.2021.113426.

[20]

X. J. LongX. B. Li and J. Zeng, Lagrangian conditions for approximate solutions on nonconvex setvalued optimization problems, Optim. Lett., 7 (2013), 1847-1856.  doi: 10.1007/s11590-012-0527-z.

[21]

X. J. LongX. K. Sun and Z. Y. Peng, Approximate optimality conditions for composite convex optimization problems, J. Oper. Res. Soc. China, 5 (2017), 469-485.  doi: 10.1007/s40305-016-0140-4.

[22]

X. K. Sun, Regularity conditions characterizing Fenchel-Lagrange duality and Farkas-type results in DC infinite programming, J. Math. Anal. Appl., 414 (2014), 590-611.  doi: 10.1016/j.jmaa.2014.01.033.

[23]

X. K. SunX. L. Guo and J. Zeng, Necessary optimality conditions for DC infinite programs with inequality constraints, J. Nonlinear Sci. Appl., 9 (2016), 617-626.  doi: 10.22436/jnsa.009.02.25.

[24]

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

[25]

X. K. SunX. L. Long and M. H. Li, Some characterizations of duality for DC optimization with composite functions, Optimization, 66 (2017), 1425-1443.  doi: 10.1080/02331934.2017.1338289.

[26]

X. K. SunK. L. Teo and X. J. Long, Some characterizations of approximate solutions for robust semi-infinite optimization problems, J. Optim. Theory Appl., 191 (2021), 281-310.  doi: 10.1007/s10957-021-01938-4.

[27]

X. K. SunK. L. Teo and L. P. Tang, Dual approaches to characterize robust optimal solution sets for a class of uncertain optimization problems, J. Optim. Theory Appl., 182 (2019), 984-1000.  doi: 10.1007/s10957-019-01496-w.

[28]

C. Z$\breve{a}$linescu, Convex Analysis in General Vector Spaces, World Scientific, New Jersey (NJ), 2002. doi: 10.1142/9789812777096.

show all references

References:
[1]

F. BaiZ. Wu and D. Zhu, Sequential Lagrange multiplier condition for $\epsilon$-optimal solution in convex programming, Optimization, 57 (2008), 669-680.  doi: 10.1080/02331930802355234.

[2]

H. H. Bauschke and J. M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev., 38 (1996), 367-426.  doi: 10.1137/S0036144593251710.

[3]

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

[4]

R. I. BoƫI. B. Hodrea and G. Wanka, $\varepsilon$-optimality conditions for composed convex optimization problems, J. Approx. Theory, 153 (2008), 108-121.  doi: 10.1016/j.jat.2008.03.002.

[5]

N. DinhT. 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.

[6]

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.

[7]

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

[8]

D. H. FangC. 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.

[9]

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

[10]

D. H. Fang and Y. Zhang, Optimality conditions and total dualities for conic programming involving composite function, Optimization, 69 (2020), 305-327.  doi: 10.1080/02331934.2018.1561695.

[11]

D. H. Fang and X. P. Zhao, Local and global optimization conditions for DC infinite optimization problems, Taiwan J. Math., 18 (2014), 817-834.  doi: 10.11650/tjm.18.2014.3888.

[12]

Y. GaoS. H. Hou and X. M. Yang, Existence and optimality conditions for approximate solutions to vector optimization problems, J. Optim. Theory Appl., 152 (2012), 97-120.  doi: 10.1007/s10957-011-9891-6.

[13]

X. L. Guo and S. J. Li, Optimality conditions for vector optimization problems with difference of convex maps, J. Optim. Theory Appl., 162 (2014), 821-844.  doi: 10.1007/s10957-013-0327-3.

[14]

J. B. Hiriart-Urruty, $\epsilon$-Subdifferential, in Convex Analysis and Optimization (eds. J. P. Aubin and R. Vinter), Pitman Advanced Pub. Program, (1982), 43–92.

[15]

Y. H. HuC. Li and X. Q. Yang, On convergence rates of linearized proximal algorithms for convex composite optimization with applications, SIAM J. Optim., 26 (2016), 1207-1235.  doi: 10.1137/140993090.

[16]

Y. H. HuX. Q. Yang and C.-K. Sim, Inexact subgradient methods for quasi-convex optimization problems, Eur. J. Oper. Res., 240 (2015), 315-327.  doi: 10.1016/j.ejor.2014.05.017.

[17]

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

[18]

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

[19]

G. LiY. H. Xu and Z. H. Qin, Fenchel-Lagrange duality for DC infinite programs with inequality constraints, J. Comput. Appl. Math., 391 (2021), 113426.  doi: 10.1016/j.cam.2021.113426.

[20]

X. J. LongX. B. Li and J. Zeng, Lagrangian conditions for approximate solutions on nonconvex setvalued optimization problems, Optim. Lett., 7 (2013), 1847-1856.  doi: 10.1007/s11590-012-0527-z.

[21]

X. J. LongX. K. Sun and Z. Y. Peng, Approximate optimality conditions for composite convex optimization problems, J. Oper. Res. Soc. China, 5 (2017), 469-485.  doi: 10.1007/s40305-016-0140-4.

[22]

X. K. Sun, Regularity conditions characterizing Fenchel-Lagrange duality and Farkas-type results in DC infinite programming, J. Math. Anal. Appl., 414 (2014), 590-611.  doi: 10.1016/j.jmaa.2014.01.033.

[23]

X. K. SunX. L. Guo and J. Zeng, Necessary optimality conditions for DC infinite programs with inequality constraints, J. Nonlinear Sci. Appl., 9 (2016), 617-626.  doi: 10.22436/jnsa.009.02.25.

[24]

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

[25]

X. K. SunX. L. Long and M. H. Li, Some characterizations of duality for DC optimization with composite functions, Optimization, 66 (2017), 1425-1443.  doi: 10.1080/02331934.2017.1338289.

[26]

X. K. SunK. L. Teo and X. J. Long, Some characterizations of approximate solutions for robust semi-infinite optimization problems, J. Optim. Theory Appl., 191 (2021), 281-310.  doi: 10.1007/s10957-021-01938-4.

[27]

X. K. SunK. L. Teo and L. P. Tang, Dual approaches to characterize robust optimal solution sets for a class of uncertain optimization problems, J. Optim. Theory Appl., 182 (2019), 984-1000.  doi: 10.1007/s10957-019-01496-w.

[28]

C. Z$\breve{a}$linescu, Convex Analysis in General Vector Spaces, World Scientific, New Jersey (NJ), 2002. doi: 10.1142/9789812777096.

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