Optimization problems involving equilibrium constraints capture diverse optimization settings such as bi-level optimization, min-max problems and games, and the minimization over non-linear constraints.
This paper introduces an Augmented Lagrangian approach with Hessian-vector product approximation to address an equilibrium constrained nonconvex nonsmooth optimization problem.
The underlying model in particular captures various settings of bi-level optimization problems, including those in which the inner problem may have a non-singleton set of optimal solutions.
The proposed method attains approximated critical points and enjoys a standard rate of convergence after stabilization. It does not require double-loops, nested procedures, nor any Hessian computation, and subsequently bypasses any matrix storage requirements.
We complement the theoretical results with numerical illustrations demonstrating the implementation of our method in a bi-level application.
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