# Optimality conditions based on the Fr'echet second-order subdifferential

arXiv: https://arxiv.org/abs/2007.14593 · HTML: https://mlsanity.com/paper/2007.14593/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jul 2020.

## Built on by (most cited first, 1 of 1)

- [Similarity-based fuzzy clustering scientific articles: potentials and challenges from mathematical and computational perspectives](https://mlsanity.com/paper/2506.04045/) (2025)

## Abstract (arXiv)

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 C2-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 C1-smooth and the constraint set is generalized polyhedral convex, we establish sharp second-order necessary optimality conditions based on the Fr'echet 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.