  In this paper, in terms of three types of generalized second-order
derivatives of a nonsmooth function, we mainly study the corresponding
second-order optimality conditions in a Hilbert space and prove the equivalence
among these optimality conditions for paraconcave functions. As applications,
we use these second-order optimality conditions to study strict local
minimizers of order two and provide sufficient and/or necessary conditions for
ensuring the local minimizer. This work extends and generalizes the study on
second-order optimality conditions from the finite-dimensional space to the
Hilbert space.
