# Abstract:

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.

# Keywords:

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

# References:

1. T. Antczak: An $\eta$-approximation approach to nonlinear mathematical programming involving invex functions. Numer. Funct. Anal. Optim. 25 (2004), 5-6, 423-438.   CrossRef
2. T. Antczak: Saddle points criteria in an $\eta$-approximation approach for nonlinear mathematical programming involving invex functions. J. Optim. Theory Appl. {\mi 132} (2007), 1, 71-87.   CrossRef
3. T. Antczak: A modified objective function method in mathematical programming with second order invexity. Numer. Funct. Anal. Optim. 28 (2007), 1-2, 1-13.   CrossRef
4. T. Antczak: A second order $\eta$-approximation method for constrained optimization problems involving second order invex functions. Appl. Math. 54 (2009), 433-445.   CrossRef
5. M. S. Bazaraa, H. D. Sherali and C. M. Shetty: Nonlinear Programming: Theory and Algorithms. John Wiley and Sons, New York 1991.   CrossRef
6. C. R. Bector and B. K. Bector: (Generalized)-bonvex functions and second order duality for a nonlinear programming problem. Congr. Numer. 52 (1985), 37-52.   CrossRef
7. C. R. Bector and B. K. Bector: On various duality theorems for second order duality in nonlinear programming. Cahiers Centre Études Rech. Opér. 28 (1986), 283-292.   CrossRef
8. C. R. Bector and S. Chandra: Generalized Bonvex Functions and Second Order Duality in Mathematical Programming. Research Report No. 85-2, Department of Actuarial and Management Sciences, University of Manitoba, Winnepeg, Manitoba 1985.   CrossRef
9. A. Ben-Tal: Second-order and related extremality conditions in nonlinear programming. J. Optim. Theory Appl. {\mi 31} (1980), 2, 143-165.   CrossRef
10. B. D. Craven: Invex functions and constrained local minima. Bull. Austral. Math. Soc. 24 (1981), 357-366.   CrossRef
11. M. A. Hanson: On sufficiency of the Kuhn-Tucker conditions. J. Math. Anal. Appl. 80 (1981) 545-550.   CrossRef
12. O. L. Mangasarian: Nonlinear Programming. McGraw-Hill, New York 1969.   CrossRef
13. R. T. Rockafellar: Convex Analysis. Princeton University Press, 1970.   CrossRef