The purpose of this paper is to establish a fairly large number of sets of second-order parameter-free sufficient optimality conditions for a discrete minmax fractional programming problem. Our effort to accomplish this goal is by utilizing various new classes of generalized second-order $(\phi,\eta,\rho,\theta,m)$-invex functions, which generalize most of the concepts available in the literature.
"Higher-order parameter-free sufficient optimality conditions in discrete minmax fractional programming." Tbilisi Math. J. 10 (2) 211 - 233, Feb 2017. https://doi.org/10.1515/tmj-2017-0038