Optimality conditions based on the Fréchet second-order subdifferential (Q2231323)

This paper studies second-order optimality conditions for constrained minimization problems. The authors consider problems in a general real Banach space. When the objective function is twice differentiable (i.e., we are in the classical setting), the authors show that strengthened second-order necessary optimality conditions hold if the constraint set is assumed to be generalized polyhedral convex (i.e., when the constraint set is the intersection of a polyhedral set with a closed affine subspace). When the objective function is just assumed to be \(C^1\)-smooth and the constraint set is generalized polyhedral convex, the authors 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. The authors apply their results to quadratic programming problems. They also show with specific examples that the hypotheses they impose cannot be relaxed.