On second order duality for nondifferentiable minimax fractional programming problems involving type-I functions

We introduce second order $(C,\alpha ,\rho ,d)$ type-I functions and formulate a second order dual model for a nondifferentiable minimax fractional programming problem. The usual duality relations are established under second order $(F,\alpha ,\rho ,d)/(C,\alpha ,\rho ,d)$ type-I assumptions. By citing a nontrivial example, it is shown that a second order $(C,\alpha ,\rho ,d)$ type-I function need not be $(F,\alpha ,\rho ,d)$ type-I. Several known results are obtained as special cases. References Ahmad, I., Husain, Z., Optimality conditions and duality in nondifferentiable minimax fractional programming with generalized convexity. J. Optimiz. Theory Appl. 129:255–275, 2006. doi:10.1007/s10957-006-9057-0 Ahmad, I., Husain, Z., Sharma, S., Second-order duality in nondifferentiable minmax programming involving type-I functions. J. Comput. Appl. Math. 215:91–102, 2008. doi:10.1016/j.cam.2007.03.022 Antczak, T., Generalized fractional minimax programming with $B$-$(p, r)$-invexity. Comput. Math. Appl. 56:1505–1525, 2008. doi:10.1016/j.camwa.2008.02.039 Chinchuluun, A., Yuan, D. H., Pardalos, P. M., Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity. Ann. Oper. Res. 154:133–147, 2007. doi:10.1007/s10479-007-0180-6 Du, D.-Z., Pardalos, P. M., Minimax and applications, Kluwer Academic Publishers, Dordrecht, 1995. http://vlsicad.eecs.umich.edu/BK/Slots/cache/www.wkap.nl/prod/b/0-7923-3615-1 Hachimi, M., Aghezzaf, B., Second order duality in multiobjective programming involving generalized type I functions. Numer. Funct. Anal. Optimiz. 25:725–736, 2005. doi:10.1081/NFA-200045804 Husain, Z., Ahmad, I., Sharma, S., Second order duality for minmax fractional programming. Optimiz. Lett. 3:277–286, 2009. doi:10.1007/s11590-008-0107-4 Hu, Q., Yang, G., Jian, J., On second order duality for minimax fractional programming. Nonlinear Anal. 12:3509–3514, 2011. doi:10.1016/j.nonrwa.2011.06.011 Lai, H. C., Lee, J. C., On duality theorems for a nondifferentiable minimax fractional programming. J. Comput. Appl. Math. 146:115–126, 2002. doi:10.1016/S0377-0427(02)00422-3 Lai, H. C., Liu, J. C., Tanaka, K., Necessary and sufficient conditions for minimax fractional programming. J. Math. Anal. Appl. 230:311–328, 1999. doi:10.1006/jmaa.1998.6204 Liu, J. C., Wu, C. S., On minimax fractional optimality conditions with invexity. J. Math. Anal. Appl. 219:21–35, 1998. doi:10.1006/jmaa.1997.5786 Long, X. J., Optimality conditions and duality for nondifferentiable multiobjective fractional programming problems with $(C,\alpha ,\rho ,d)$-convexity. J. Optimiz. Theory Appl. 148:197–208, 2011. doi:10.1007/s10957-010-9740-z Schmitendorf, W. E., Necessary conditions and sufficient conditions for static minmax problems. J. Math. Anal. Appl. 57:683–693, 1977. doi:10.1016/0022-247X(77)90255-4 Sharma, S., Gulati, T. R., Second order duality in minmax fractional programming with generalized univexity. J. Glob. Optimiz. 52:161–169, 2012. doi:10.1007/s10898-011-9694-1 Yuan, D. H., Liu, X. L., Chinchuluun, A., Pardalos, P. M., Nondifferentiable minimax fractional programming problems with $(C,\alpha , \rho , d)$-convexity. J. Optimiz. Theory Appl. 129:185–199, 2006. doi:10.1007/s10957-006-9052-

Prevalence and prognostic value of cardiac troponin in elderly patients hospitalized for COVID-19

BACKGROUND Increases in cardiac troponin (cTn) in coronavirus disease 2019 (COVID-19) have been associated with worse prognosis. Nonetheless, data about the significance of cTn in elderly subjects with COVID-19 are lacking. METHODS From a registry of consecutive patients with COVID-19 admitted to a hub hospital in Italy from 25/02/2020 to 03/07/2020, we selected those 65 60 year-old and with cTnI measured within three days from the molecular diagnosis of SARSCoV-2 infection. When available, a second cTnI value within 48 h was also extracted. The relationship between increased cTnI and all-cause in-hospital mortality was evaluated by a Cox regression model and restricted cubic spline functions with three knots. RESULTS Of 343 included patients (median age: 75.0 (68.0 1283.0) years, 34.7% men), 88 (25.7%) had cTnI above the upper-reference limit (0.046 \u3bcg/L). Patients with increased cTnI had more comorbidities, greater impaired respiratory exchange and higher inflammatory markers on admission than those with normal cTnI. Furthermore, they died more (73.9% vs. 37.3%, P < 0.001) over 15 (6 1225) days of hospitalization. The association of elevated cTnI with mortality was confirmed by the adjusted Cox regression model (HR = 1.61, 95%CI: 1.06 122.52, P = 0.039) and was linear until 0.3 \u3bcg/L, with a subsequent plateau. Of 191 (55.7%) patients with a second cTnI measurement, 49 (25.7%) had an increasing trend, which was not associated with mortality (univariate HR = 1.39, 95%CI: 0.87 122.22, P = 0.265). CONCLUSIONS In elderly COVID-19 patients, an initial increase in cTn is common and predicts a higher risk of death. Serial cTn testing may not confer additional prognostic information