## Strong and total Lagrange dualities for quasiconvex programming.(English)Zbl 1442.90142

Summary: We consider the strong and total Lagrange dualities for infinite quasiconvex optimization problems. By using the epigraphs of the $$z$$-quasi-conjugates and the Greenberg-Pierskalla subdifferential of these functions, we introduce some new constraint qualifications. Under the new constraint qualifications, we provide some necessary and sufficient conditions for infinite quasiconvex optimization problems to have the strong and total Lagrange dualities.

### MSC:

 90C25 Convex programming 90C34 Semi-infinite programming 90C46 Optimality conditions and duality in mathematical programming
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### References:

 [1] Boţ, R. I.; Wanka, G., Farkas-type results with conjugate functions, SIAM Journal on Optimization, 15, 2, 540-554 (2005) · Zbl 1114.90147 [2] Boţ, R. I.; Grad, S. M.; Wanka, G., On strong and total Lagrange duality for convex optimization problems, Journal of Mathematical Analysis and Applications, 337, 2, 1315-1325 (2008) · Zbl 1160.90004 [3] Dinh, N.; Goberna, M. A.; López, M. A., From linear to convex systems: consistency, Farkas’ lemma and applications, Journal of Convex Analysis, 13, 1, 279-290 (2006) · Zbl 1137.90684 [4] Dinh, N.; Goberna, M. A.; López, M. A.; Son, T. Q., New Farkas-type constraint qualifications in convex infinite programming, ESAIM Control, Optimisation and Calculus of Variations, 13, 3, 580-597 (2007) · Zbl 1126.90059 [5] 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 [6] Fang, D. H.; Li, C.; Ng, K. F., Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming, Nonlinear Analysis: Theory, Methods & Applications, 73, 5, 1143-1159 (2010) · Zbl 1218.90200 [7] Goberna, M. A.; Jeyakumar, V.; López, M. A., Necessary and sufficient constraint qualifications for solvability of systems of infinite convex inequalities, Nonlinear Analysis: Theory, Methods & Applications, 68, 5, 1184-1194 (2008) · Zbl 1145.90051 [8] Goberna, M. A.; López, M. A., Linear Semi-Infinite Optimization, 2 (1998), Chichester, UK: John Wiley & Sons, Chichester, UK · Zbl 0909.90257 [9] Jeyakumar, V., Constraint qualifications characterizing Lagrangian duality in convex optimization, Journal of Optimization Theory and Applications, 136, 1, 31-41 (2008) · Zbl 1194.90069 [10] Jeyakumar, V.; Mohebi, H., Limiting ϵ-subgradient characterizations of constrained best approximation, Journal of Approximation Theory, 135, 2, 145-159 (2005) · Zbl 1138.41304 [11] Li, C.; Ng, K. F.; Pong, T. K., Constraint qualifications for convex inequality systems with applications in constrained optimization, SIAM Journal on Optimization, 19, 1, 163-187 (2008) · Zbl 1170.90009 [12] Li, W.; Nahak, C.; Singer, I., Constraint qualifications for semi-infinite systems of convex inequalities, SIAM Journal on Optimization, 11, 1, 31-52 (2000) · Zbl 0999.90045 [13] Greenberg, H. J.; Pierskalla, W. P., Quasi-conjugate functions and surrogate duality, Cahiers du Centre d’Études de Recherche Opérationnelle, 15, 437-448 (1973) · Zbl 0276.90051 [14] Luenberger, D. G., Quasi-convex programming, SIAM Journal on Applied Mathematics, 16, 1090-1095 (1968) · Zbl 0212.23905 [15] Penot, J. P.; Volle, M., On quasi-convex duality, Mathematics of Operations Research, 15, 4, 597-625 (1990) · Zbl 0717.90058 [16] Suzuki, S.; Kuroiwa, D., Set containment characterization for quasiconvex programming, Journal of Global Optimization, 45, 4, 551-563 (2009) · Zbl 1185.26021 [17] Suzuki, S.; Kuroiwa, D., Optimality conditions and the basic constraint qualification for quasiconvex programming, Nonlinear Analysis: Theory, Methods & Applications, 74, 4, 1279-1285 (2011) · Zbl 1236.90142 [18] Suzuki, S.; Kuroiwa, D., On set containment characterization and constraint qualification for quasiconvex programming, Journal of Optimization Theory and Applications, 149, 3, 554-563 (2011) · Zbl 1229.90208 [19] Zălinescu, C., Convex Analysis in General Vector Spaces (2002), River Edge, NJ, USA: World Scientific Publishing, River Edge, NJ, USA · Zbl 1023.46003 [20] Thach, P. T., Diewert-Crouzeix conjugation for general quasiconvex duality and applications, Journal of Optimization Theory and Applications, 86, 3, 719-743 (1995) · Zbl 0839.90114 [21] Martínez-Legaz, J. E.; Sach, P. H., A new subdifferential in quasiconvex analysis, Journal of Convex Analysis, 6, 1, 1-11 (1999) · Zbl 0942.49020
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