Optimality conditions based on the Fréchet second-order subdifferential

This paper focuses on second-order necessary optimality conditions for constrained optimization problems Banach spaces. For in the classical setting, where objective function is $$C^2$$ -smooth, we show that strengthened are valid if constraint set generalized polyhedral convex. a new just assumed to be $$C^1$$ -smooth and convex, establish sharp based Fréchet subdifferential of tangent set. Three examples given used hypotheses essential theorems. Our refine extend several existing results.