A Lagrange function approach to study second-order optimality conditions for infinite-dimensional optimization problems

In this paper, we focus on the second-order optimality conditions for infinite-dimensional optimization problems constrained by generalized polyhedral convex sets. Our aim is to further explore the role of the generalized polyhedral convex property, which is inspired by the findings of other authors. To this end, we employ the concept of Fréchet second-order subdifferential, a tool in variational analysis, to establish optimality conditions. Furthermore, by applying this concept to the Lagrangian function associated with the problem, we are able to derive refined optimality conditions that surpass existing results. The unique properties of generalized polyhedral convex sets play a crucial role in enabling these improvements.