Abstract
In this paper, we introduce the various types of generalized invexities, i.e., alpha(f)-invex/ alpha(f)-pseudoinvex and (G ,alpha(f) )-bonvex/ (G ,alpha(f) )-pseudobonvexfunctions.furthermore, we construct nontrivial numerical examples of (G ,alpha(f) )-bonvexity/ (G ,alpha(f) )-pseudobonvexity, which is neither alpha(f)-bonvex/ alpha(f)-pseudobonvex nor alpha(f)-invex/ alpha(f)-pseudoinvex with the same eta.further, weformulate a pair of second-order non-differentiable symmetric dual models and prove the duality relations under alpha(f)-invex/ alpha(f)-pseudoinvex and (G ,alpha(f) )-bonvex/ (G ,alpha(f) )-pseudobonvex assumptions.finally, we construct a nontrivial numerical example justifying the weak duality result presented in the paper.