Optimality and Saddle Point for Vector Optimization Under Semilocally B-preinvex

In this paper, a nonlinear programming problem is considered where the functions involved are η-semi-differentiable. An equivalent η-approximated vector optimization problem is constructed by a modification of the objective and the constraint functions in the original multi-objective programming problem. The connection between (weak) efficient points in the original multi-objective programming problem and its equivalent η-approximated vector optimization is proved. Furthermore, a modified Lagrange function is introduced for a constructed vector optimization problem. By the help of the modified Lagrange function, saddle point results are presented for the original multi-objective programming problem.