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

TL;DR

This paper develops refined second-order necessary optimality conditions for constrained optimization in Banach spaces, utilizing the Fréchet second-order subdifferential, especially under generalized polyhedral convex constraints.

Contribution

It introduces sharp second-order necessary conditions based on the Fréchet second-order subdifferential for problems with less smooth objectives and generalized polyhedral convex constraints.

Findings

Strengthened second-order conditions for $C^2$-smooth objectives.

New sharp second-order conditions for $C^1$-smooth objectives.

Examples demonstrating the necessity of hypotheses.

Abstract

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 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.