Second-Order Optimality Conditions for Nonsmooth Constrained Optimization with Applications to Bilevel Programming
Xiang Liu, Mengwei Xu, Liwei Zhang
TL;DR
The paper develops a no-gap second-order framework for constrained nonsmooth optimization by introducing second-order gph-regularity and leveraging outer second-order regularity via the parabolic curve method under MSCQ. It then applies these ideas to bilevel programming, establishing second-order conditions for bi-local solutions based on lower-level regularity assumptions (MFCQ, SSOSC, CRCQ) and showing that under LICQ these conditions can be written purely in terms of the problem data. The contributions provide a robust, implementable set of criteria for optimality verification and algorithmic guidance in nonsmooth constrained and bilevel settings, without needing convexity of the constraint set or lower-level multiplier uniqueness. Together, the results unify primal and dual viewpoints and give explicit SOSC under standard qualifications, enhancing tractability and interpretability in complex hierarchical optimization tasks.
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.