In this paper, by using the second order $\eta $-approximation method introduced by Antczak \cite{antczak3}, new saddle point results are obtained for a nonlinear mathematical programming problem involving second order invex functions with respect to the same function $\eta $. Moreover, a second order $\eta $-saddle point and a second order $\eta $-Lagrange function are defined for the so-called second order $\eta $-approximated optimization problem constructed in this method. Then, the equivalence between an optimal solution in the original mathematical programming problem and a second order $\eta $-saddle point of the second order $\eta $ -Lagrangian in the associated second order $\eta $-approximated optimization problem is established. Finally, some example of using this approach to characterize of solvability of some O.R. problem is given.
second order optimality conditions, second order $\eta $-approximated optimization problem, second order $\eta $-saddle point, second order $\eta $-Lagrange function, second order invex function with respect to $\eta $
90C26, 90C46