×

A note on optimality conditions for DC programs involving composite functions. (English) Zbl 1469.90155

Summary: By using the formula of the \(\epsilon\)-subdifferential for the sum of a convex function with a composition of convex functions, some necessary and sufficient optimality conditions for a DC programming problem involving a composite function are obtained. As applications, a composed convex optimization problem, a DC optimization problem, and a convex optimization problem with a linear operator are examined at the end of this paper.

MSC:

90C46 Optimality conditions and duality in mathematical programming
49J52 Nonsmooth analysis
90C26 Nonconvex programming, global optimization
90C48 Programming in abstract spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Horst, R.; Thoai, N. V., DC programming: overview, Journal of Optimization Theory and Applications, 103, 1, 1-43 (1999) · Zbl 1073.90537
[2] Laghdir, M., Optimality conditions and Toland’s duality for a nonconvex minimization problem, Matematichki Vesnik, 55, 1-2, 21-30 (2003) · Zbl 1051.49020
[3] Boţ, R. I.; Grad, S.-M.; Wanka, G., Generalized Moreau-Rockafellar results for composed convex functions, Optimization, 58, 7, 917-933 (2009) · Zbl 1201.90154
[4] Fang, D. H.; Li, C.; Ng, K. F., Constraint qualifications for extended Farkas’s lemmas and Lagrangian dualities in convex infinite programming, SIAM Journal on Optimization, 20, 3, 1311-1332 (2009) · Zbl 1206.90198
[5] Boţ, R. I., Conjugate Duality in Convex Optimization (2010), Berlin, Germany: Springer, Berlin, Germany · Zbl 1190.90002
[6] Dinh, N.; Nghia, T. T. A.; Vallet, G., A closedness condition and its applications to DC programs with convex constraints, Optimization, 59, 3-4, 541-560 (2010) · Zbl 1218.90155
[7] Fang, D. H.; Li, C.; Yang, X. Q., Stable and total Fenchel duality for DC optimization problems in locally convex spaces, SIAM Journal on Optimization, 21, 3, 730-760 (2011) · Zbl 1236.90140
[8] Fang, D. H.; Li, C.; Yang, X. Q., Asymptotic closure condition and Fenchel duality for DC optimization problems in locally convex spaces, Nonlinear Analysis: Theory, Methods & Applications, 75, 8, 3672-3681 (2012) · Zbl 1266.90147
[9] Fang, D. H.; Lee, G. M.; Li, C.; Yao, J. C., Extended Farkas’s lemmas and strong Lagrange dualities for DC infinite programming, Journal of Nonlinear and Convex Analysis, 14, 4, 747-767 (2013) · Zbl 1311.90107
[10] Sun, X. K.; Li, S. J., Duality and Farkas-type results for extended Ky Fan inequalities with DC functions, Optimization Letters, 7, 3, 499-510 (2013) · Zbl 1287.90069
[11] Sun, X.-K.; Li, S.-J.; Zhao, D., Duality and Farkas-type results for DC infinite programming with inequality constraints, Taiwanese Journal of Mathematics, 17, 4, 1227-1244 (2013) · Zbl 1276.49028
[12] Sun, X.-K., Regularity conditions characterizing Fenchel-Lagrange duality and Farkas-type results in DC infinite programming, Journal of Mathematical Analysis and Applications, 414, 2, 590-611 (2014) · Zbl 1312.49047
[13] Thibault, L., Sequential convex subdifferential calculus and sequential Lagrange multipliers, SIAM Journal on Control and Optimization, 35, 4, 1434-1444 (1997) · Zbl 0891.90138
[14] Jeyakumar, V.; Lee, G. M.; Dinh, N., New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs, SIAM Journal on Optimization, 14, 2, 534-547 (2003) · Zbl 1046.90059
[15] Jeyakumar, V.; Wu, Z. Y.; Lee, G. M.; Dinh, N., Liberating the subgradient optimality conditions from constraint qualifications, Journal of Global Optimization, 36, 1, 127-137 (2006) · Zbl 1131.90069
[16] Boţ, R. I.; Csetnek, E. R.; Wanka, G., Sequential optimality conditions for composed convex optimization problems, Journal of Mathematical Analysis and Applications, 342, 2, 1015-1025 (2008) · Zbl 1220.90087
[17] Bai, F. S.; Wu, Z. Y.; Zhu, D. L., Sequential Lagrange multiplier condition for \(\epsilon \)-optimal solution in convex programming, Optimization, 57, 5, 669-680 (2008) · Zbl 1152.90558
[18] Boţ, R. I.; Hodrea, I. B.; Wanka, G., \( \epsilon \)-optimality conditions for composed convex optimization problems, Journal of Approximation Theory, 153, 1, 108-121 (2008) · Zbl 1158.46029
[19] Dinh, N.; Mordukhovich, B. S.; Nghia, T. T. A., Qualification and optimality conditions for DC programs with infinite constraints, Acta Mathematica Vietnamica, 34, 1, 123-153 (2009) · Zbl 1190.90264
[20] Dinh, N.; Strodiot, J. J.; Nguyen, V. H., Duality and optimality conditions for generalized equilibrium problems involving DC functions, Journal of Global Optimization, 48, 2, 183-208 (2010) · Zbl 1228.90078
[21] Kim, G. S.; Lee, G. M., On \(\epsilon \)-optimality theorems for convex vector optimization problems, Journal of Nonlinear and Convex Analysis, 12, 3, 473-482 (2011) · Zbl 1235.90138
[22] Lee, G. M.; Kim, G. S.; Dinh, N.; Ansari, Q. H.; Yao, J. C., Optimality conditions for approximate solutions of convex semi-infinite vector optimization problems, Recent Developments in Vector Optimization, Vector Optimization, 1, 275-295 (2012), Berlin, Germany: Springer, Berlin, Germany · Zbl 1247.90238
[23] Guo, X. L.; Li, S. J.; Teo, K. L., Subdifferential and optimality conditions for the difference of set-valued mappings, Positivity, 16, 2, 321-337 (2012) · Zbl 1255.49029
[24] Guo, X. L.; Li, S. J., Optimality conditions for vector optimization problems with difference of convex maps, Journal of Optimization Theory and Applications (2013) · Zbl 1307.90160
[25] Fang, D. H.; Zhao, X. P., Local and global optimality conditions for DC infinite optimization problems, Taiwanese Journal of Mathematics (2013) · Zbl 1357.90161
[26] Hiriart-Urruty, J. B.; Gilbert, R. P.; Panagiotopoulos, P. D.; Pardalos, P. M., From convex optimization to nonconvex optimization necessary and sufficient conditions for global optimality, From Convexity to Nonconvexity, 219-239 (2001), London, UK: Kluwer, London, UK
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.