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

本文关注Banach空间上约束优化问题的二阶必要最优性条件。对于经典环境中的目标函数为\（C ^ 2 \）- smooth的问题，我们表明，如果约束集为广义多面体凸，则增强的二阶必要最优性条件是有效的。对于新设置中的问题，假设目标函数只是\（C ^ 1 \）-平滑且约束集为广义多面凸，我们基于目标函数的弗雷谢二阶次微分和约束集的二阶切线，建立了尖锐的二阶必要最优性条件。给出了三个例子，表明所使用的假设对于新定理是必不可少的。我们的二阶必要最优性条件完善并扩展了多个现有结果。