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Fenchel-Lagrange duality for DC infinite programs with inequality constraints. (English) Zbl 1481.90264

A composite DC optimization problem with constraints consisting in possibly infinitely many DC inequalities is considered. New closedness type constraint qualifications for completely characterizing the weak/zero/strong/stable Fenchel-Lagrange duality for such problems are provided. Several relevant special cases are given as applications. Some examples illustrate the theoretical achievements.

MSC:

90C26 Nonconvex programming, global optimization
90C30 Nonlinear programming
90C46 Optimality conditions and duality in mathematical programming
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[1] Keller, A. A., Convex underestimating relaxation techniques for nonconvex polynomial programming problems: Computational overview, J. Mech. Behav. Mater., 24, 3-4, 129-143 (2015)
[2] Sun, X. K.; Teo, K. L.; Tang, L. P., Dual approaches to characterize robust optimal solution sets for a class of uncertain optimization problems, J. Optim. Theory Appl., 182, 3, 984-1000 (2019) · Zbl 1422.49028
[3] Sun, X. K.; Fu, H. Y.; Zeng, J., Robust approximate optimality conditions for uncertain nonsmooth optimization with infinite number of constraints, Mathematics, 7, 12 (2019)
[4] Fang, D. H., Some relationships among the constraint qualifications for Lagrangian dualities in DC infinite optimization problems, J. Inequal. Appl., 2015, 1, 41-55 (2015) · Zbl 1308.90133
[5] Sun, X. K., Regularity conditions characterizing Fenchel-Lagrange duality and Farkas-type results for DC infinite programming, J. Math. Anal. Appl., 414, 590-611 (2014) · Zbl 1312.49047
[6] Long, X. J.; Sun, X. K.; Peng, Z. Y., Approximate optimality conditions for composite convex optimization problems, J. Oper. Res. Soc. China, 5, 469-485 (2017) · Zbl 1386.90106
[7] Boţ, R. I.; Hodrea, I. B.; Wanka, G., Farkas-type results for inequality systems with composed convex functions via conjugate duality, J. Math. Anal. Appl., 322, 316-328 (2006) · Zbl 1104.90054
[8] Boţ, R. I.; Hodrea, I. B.; Wanka, G., Some new Farkas-type results for inequality system with DC functions, J. Global Optim., 39, 595-608 (2007) · Zbl 1182.90071
[9] Boţ, R. I.; Grad, S. M.; Wanka, G., A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces, Math. Nachr., 281, 8, 1088-1107 (2008) · Zbl 1155.49019
[10] Combari, C.; Laghdir, M.; Thibault, L., A note on subdifferentials of convex composite functionals, Arch. Math. (Basel), 67, 239-252 (1996) · Zbl 0854.49017
[11] Dinh, N.; Nghia, T. T.A.; Vallet, G., A closedness condition and its applications to DC programs with convex constraints, Optimization, 59, 4, 541-560 (2010) · Zbl 1218.90155
[12] Dinh, N.; Vallet, G.; Nghia, T. T.A., Farkas-type results and duality for DC programs with convex constraint, J. Convex Anal., 15, 235-262 (2008) · Zbl 1145.49016
[13] Fang, D. H.; Ansari, Q. H.; Zhao, X. P., Constraint qualifications and zero duality gap properties in conical programming involving composite functions, J. Nonlinear Convex Anal., 19, 1, 53-69 (2018) · Zbl 1390.49043
[14] Li, C.; Fang, D. H.; López, G.; López, M. A., Stable and total Fenchel duality for convex optimization problem in locally convex spaces, SIAM J. Optim., 20, 2, 1032-1051 (2009) · Zbl 1189.49051
[15] Jeyakumar, V.; Rubinov, A.; Glover, B. M.; Ishizuka, Y., Inequality systems and global optimization, J. Math. Anal. Appl., 202, 900-919 (1996) · Zbl 0856.90128
[16] Li, G.; Yang, X. Q.; Zhou, Y. Y., Stable strong and total parametrized dualities for DC optimization problems in locally convex spaces, J. Ind. Manag. Optim., 9, 671-687 (2013) · Zbl 1281.90035
[17] Li, G.; Zhou, Y. Y., The stable Farkas lemma for composite convex functions in infinite dimensional spaces, Acta Math. Appl. Sin-E., 31, 677-692 (2015) · Zbl 1359.90104
[18] Martínez-Legaz, J. E.; Volle, M., Duality in DC programming: the case of several DC constraints, J. Math. Anal. Appl., 237, 657-671 (1999) · Zbl 0946.90064
[19] Long, X. J.; Huang, N. J.; O’Regan, D., Farkas-type results for general composed convex optimization problems with inequality constraints, Math. Inequal. Appl., 13, 1, 135-143 (2010) · Zbl 1206.90119
[20] Shi, L. Y.; Ansari, Q. H.; Yao, J. C., Incremental gradient projection algorithm for constrained composite minimization problems, J. Nonlinear Var. Anal., 1, 253-264 (2017) · Zbl 1398.90205
[21] Zhou, Y. Y.; Li, G., The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions, Numer. Algebra Control Optim., 4, 1, 9-23 (2014) · Zbl 1292.90241
[22] Dinh, N.; Mordukhovich, B. S.; Nghia, T. T.A., Qualification and optimality conditions for DC programs with infinite constraints, Acta Math. Vietnam., 34, 1, 125-155 (2009) · Zbl 1190.90264
[23] Sun, X. K.; Li, S. J.; Zhao, D., Duality and Farkas-type results for DC infinite programming with inequality constraints, Taiwanese J. Math., 4, 1227-1244 (2013) · Zbl 1276.49028
[24] Fang, D. H.; Gong, X., Extended Farkas lemma and strong duality for composite optimization problems with DC functions, Optimization, 66, 2, 179-196 (2017) · Zbl 1406.90128
[25] Fang, D. H.; Wang, M. D.; Zhao, X. P., The strong duality for DC optimization problems with composite convex functions, J. Nonlinear Convex Anal., 16, 7, 1337-1352 (2015) · Zbl 1332.90206
[26] Sun, X. K.; Fu, H. Y., A note on optimality conditions for DC programs involving composite functions, Abstr. Appl. Anal., 2014, 1-6 (2014) · Zbl 1469.90155
[27] Sun, X. K.; Guo, X. L.; Zhang, Y., Fenchel-Lagrange duality for DC programs with composite functions, J. Nonlinear Convex Anal., 16, 8, 1607-1618 (2015) · Zbl 1327.90223
[28] Li, G.; Zhang, L. P.; Liu, Z., The stable duality of DC programs for composite convex functions, J. Ind. Manag. Optim., 13, 1, 63-79 (2017) · Zbl 1365.90218
[29] Sun, X. K.; Long, X. J.; Li, M. H., Some characterizations of duality for DC optimization with composite functions, Optimization, 66, 9, 1425-1443 (2017) · Zbl 1434.90157
[30] Boţ, R. I.; Grad, S. M.; Wanka, G., On strong and total Lagrange duality for convex optimization problems, J. Math. Anal. Appl., 337, 1315-1325 (2008) · Zbl 1160.90004
[31] Zălinescu, C., Convex Analysis in General Vector Space (2002), World Sciencetific: World Sciencetific River Edge, New Jersey · Zbl 1023.46003
[32] Fang, D. H.; Li, C.; Ng, K. F., Constraint qualifications for extended Farkas’ lemmas and Lagrangian dualities in convex infinite programming, SIAM J. Optim., 20, 6, 1311-1332 (2009) · Zbl 1206.90198
[33] Boţ, R. I.; Hodrea, I. B.; Wanka, G., Farkas-type results for fractional programming problems, Nonlinear Anal., 67, 1690-1703 (2007) · Zbl 1278.90395
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