Regularity conditions and Farkas-type results for systems with fractional functions
RAIRO - Operations Research - Recherche Opérationnelle, Tome 54 (2020) no. 5, pp. 1369-1384.

This paper deals with some new versions of Farkas-type results for a system involving cone convex constraint, a geometrical constraint as well as a fractional function. We first introduce some new notions of regularity conditions in terms of the epigraphs of the conjugate functions. By using these regularity conditions, we obtain some new Farkas-type results for this system using an approach based on the theory of conjugate duality for convex or DC optimization problems. Moreover, we also show that some recently obtained results in the literature can be rediscovered as special cases of our main results.

DOI : 10.1051/ro/2019070
Classification : 90C26, 90C32, 90C46
Mots-clés : Regularity conditions, Farkas-type results, fractional functions 
@article{RO_2020__54_5_1369_0,
     author = {Sun, Xiangkai and Long, Xian-Jun and Tang, Liping},
     title = {Regularity conditions and {Farkas-type} results for systems with fractional functions},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {1369--1384},
     publisher = {EDP-Sciences},
     volume = {54},
     number = {5},
     year = {2020},
     doi = {10.1051/ro/2019070},
     mrnumber = {4127958},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ro/2019070/}
}
TY  - JOUR
AU  - Sun, Xiangkai
AU  - Long, Xian-Jun
AU  - Tang, Liping
TI  - Regularity conditions and Farkas-type results for systems with fractional functions
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2020
SP  - 1369
EP  - 1384
VL  - 54
IS  - 5
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ro/2019070/
DO  - 10.1051/ro/2019070
LA  - en
ID  - RO_2020__54_5_1369_0
ER  - 
%0 Journal Article
%A Sun, Xiangkai
%A Long, Xian-Jun
%A Tang, Liping
%T Regularity conditions and Farkas-type results for systems with fractional functions
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2020
%P 1369-1384
%V 54
%N 5
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ro/2019070/
%R 10.1051/ro/2019070
%G en
%F RO_2020__54_5_1369_0
Sun, Xiangkai; Long, Xian-Jun; Tang, Liping. Regularity conditions and Farkas-type results for systems with fractional functions. RAIRO - Operations Research - Recherche Opérationnelle, Tome 54 (2020) no. 5, pp. 1369-1384. doi : 10.1051/ro/2019070. http://archive.numdam.org/articles/10.1051/ro/2019070/

[1] R.I. Boţ, Conjugate Duality in Convex Optimization. Springer, Berlin (2010). | DOI | MR | Zbl

[2] 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 | MR | Zbl

[3] R.I. Boţ, I.B. Hodrea and G. Wanka, Farkas-type results for fractional programming problems. Nonlinear Anal. 67 (2007) 1690–1703. | DOI | MR | Zbl

[4] R.S. Burachik and V. Jeyakumar, A new geometric condition for Fenchels duality in infinite dimensional spaces. Math. Program. Ser. B 104 (2005) 229–233. | DOI | MR | Zbl

[5] N. Dinh and V. Jeyakumar, Farkas’s lemma: three decades of generalizations for mathematical optimization. Top 22 (2014) 1–22. | DOI | MR | Zbl

[6] N. Dinh, G. Vallet and T.T.A. Nghia, Farkas-typeresuts and dualityfor DC programs with convex constraints. J. Convex Anal. 15 (2008) 235–262. | MR | Zbl

[7] 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 | MR | Zbl

[8] N. Dinh, G. Vallet and M. Volle, Functional inequalities and theorems of the alternative involving composite functions. J. Glob. Optim. 59 (2014) 837–863. | DOI | MR | Zbl

[9] J. Farkas, Theorie der einfachen Ungleichungen. J. Reine Angew. Math. 124 (1901) 1–27. | JFM | MR

[10] 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 | MR | Zbl

[11] D.H. Fang and X.Y. Wang, Stable and total Fenchel duality for composed convex optimization problems. Acta Math. Appl. Sin. Engl. Ser. 34 (2018) 813–827. | DOI | MR | Zbl

[12] D.H. Fang and Y. Zhang, Extended Farkas’s lemmas and strong dualities for conic programming involving composite functions. J. Optim. Theory Appl. 176 (2018) 351–376. | DOI | MR

[13] D.H. Fang, C. 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 | MR | Zbl

[14] 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 | MR | Zbl

[15] D.H. Fang, C. Li and X.Q. Yang, Asymptotic closure condition and Fenchel duality for DC optimization problems in locally convex spaces. Nonlinear Anal. 75 (2012) 3672–3681. | DOI | MR | Zbl

[16] B.L. Gorissen, Robust fractional programming. J. Optim. Theory Appl. 166 (2015) 508–528. | DOI | MR

[17] J. Gwinner, Results of Farkas-type. Numer. Funct. Anal. Optim. 9 (1987) 471–520. | DOI | MR | Zbl

[18] V. Jeyakumar and G.M. Lee, Complete characterizations of stable Farkas lemma and cone-convex programming duality. Math. Program. Ser. A 114 (2008) 335–347. | DOI | MR | Zbl

[19] X.J. Long, N.J. Huang and Z.B. Liu, Optimality conditions, duality and saddle points for nondifferentiable multiobjective fractional programs. J. Ind. Manag. Optim. 4 (2008) 287–298. | DOI | MR | Zbl

[20] X.J. Long, X.K. Sun and Z.Y. Peng, Approximate optimality conditions for composite convex optimization problems. J. Oper. Res. Soc. China 5 (2017) 469–485. | DOI | MR

[21] 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 | MR | Zbl

[22] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton, NJ (1970). | DOI | MR | Zbl

[23] S. Schaible, Duality in fractional programming. Oper. Res. 24 (1976) 452–461. | DOI | MR | Zbl

[24] S. Schaible and T. Ibaraki, Fractional programming. Eur. J. Oper. Res. 12 (1983) 325–338. | DOI | MR | Zbl

[25] I.M. Stancu-Minasian, A eighth bibliography of fractional programming. Optimization 66 (2017) 439–470. | DOI | MR

[26] 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 | MR

[27] X.K. Sun, S.J. Li and D. Zhao, Duality and Farkas-type results for DC infinite programming with inequality constraints. Taiwan. J. Math. 17 (2013) 1227–1244. | MR | Zbl

[28] X.K. Sun, Y. Chai and J. Zeng, Farkas-type results for constrained fractional programming with DC functions. Optim. Lett. 8 (2014) 2299–2313. | DOI | MR | Zbl

[29] X.K. Sun, X.J. Long and M.H. Li, Some characterizations of duality for DC optimization with composite functions. Optimization 66 (2017) 1425–1443. | DOI | MR | Zbl

[30] X.K. Sun, X.J. Long, H.Y. Fu and X.B. Li, Some characterizations of robust optimal solutions for uncertain fractional optimization and applications. J. Ind. Manag. Optim. 13 (2017) 803–824. | DOI | MR | Zbl

[31] X.K. Sun, L.P. Tang, X.J. Long and M.H. Li, Some dual characterizations of Farkas-type results for fractional programming problems. Optim. Lett. 12 (2018) 1403–1420. | DOI | MR | Zbl

[32] 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) 12. | DOI

[33] H.J. Wang and C.Z. Cheng, Duality and Farkas-type results for DC fractional programming with DC constraints. Math. Comput. Model. 53 (2011) 1026–1034. | DOI | MR | Zbl

[34] X.M. Yang, K.L. Teo and X.Q. Yang, Symmetric duality for a class of nonlinear fractional programming problems. J. Math. Anal. Appl. 271 (2002) 7–15. | DOI | MR | Zbl

[35] X.M. Yang, X.Q. Yang and K.L. Teo, Duality and saddle-point type optimality for generalized nonlinear fractional programming. J. Math. Anal. Appl. 289 (2004) 100–109. | DOI | MR | Zbl

[36] C. Zălinescu, Convex Analysis in General Vector Spaces. World Scientific, London (2002). | DOI | MR | Zbl

[37] X.H. Zhang and C.Z. Cheng, Some Farkas-type results for fractional programming with DC functions. Nonlinear Anal. Real World Appl. 10 (2009) 1679–1690. | DOI | MR | Zbl

Cité par Sources :