Second-Order Optimality Conditions for Nonsmooth Constrained Optimization with Applications to Bilevel Programming

TL;DR

This paper develops comprehensive second-order optimality conditions for nonsmooth constrained optimization problems, especially applied to bilevel programming, without convexity assumptions and using a new regularity concept.

Contribution

It introduces the second-order gph-regularity concept and derives no-gap necessary and sufficient conditions for nonsmooth problems, including bilevel optimization, under various constraint qualifications.

Findings

Established second-order necessary and sufficient conditions for nonsmooth problems.

Extended these conditions to bilevel optimization without requiring lower-level solution uniqueness.

Provided conditions expressed solely in second derivatives when LICQ holds.

Abstract

Second-order optimality conditions are essential for nonsmooth optimization, where both the objective and constraint functions are Lipschitz continuous and second-order directionally differentiable. This paper provides no-gap second-order necessary and sufficient optimality conditions for such problems without requiring convexity assumptions on the constraint set. We introduce the concept of second-order gph-regularity for constraint functions, which ensures the outer second-order regularity of the feasible region and enables the formulation of comprehensive optimality conditions through the parabolic curve approach. An important application of our results is bilevel optimization, where we derive second-order necessary and sufficient optimality conditions for bi-local optimal solutions, which are based on the local solutions of the lower-level problem. By leveraging the Mangasarian-Fromovitz constraint qualification (MFCQ), strong second-order sufficient condition (SSOSC) and constant rank constraint qualification (CRCQ) of lower-level problem, these second-order conditions are derived without requiring the uniqueness of the lower-level multipliers. In addition, if the linear independence constraint qualification (LICQ) holds, these conditions are expressed solely in terms of the second-order derivatives of the functions defining the bilevel problem, without relying on the second-order information from the solution mapping, which would introduce implicit complexities.