Second-order Optimality Conditions by Generalized Derivatives and Applications in Hilbert Spaces

TL;DR

This paper extends second-order optimality conditions using generalized derivatives to Hilbert spaces, providing theoretical foundations and applications for nonsmooth optimization problems.

Contribution

It introduces new second-order optimality conditions in Hilbert spaces and proves their equivalence for paraconcave functions, extending finite-dimensional results.

Findings

Equivalence of second-order optimality conditions for paraconcave functions

Characterization of strict local minimizers of order two

Conditions for ensuring local minimality in Hilbert spaces

Abstract

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.