Second-order optimality conditions for bilevel programs

TL;DR

This paper introduces bi-local solutions for bilevel programs, enabling the derivation of second-order optimality conditions that depend only on the problem's defining functions, and demonstrates their application to convergence analysis.

Contribution

It proposes the bi-local solution concept and establishes new second-order optimality conditions that avoid complex derivatives of value functions.

Findings

Bi-local solutions are local minimizers under Jacobian conditions.

New second-order optimality conditions involve only defining functions.

Second-order sufficient conditions ensure Q-linear convergence of augmented Lagrangian methods.

Abstract

Second-order optimality conditions of the bilevel programming problems are dependent on the second-order directional derivatives of the value functions or the solution mappings of the lower level problems under some regular conditions, which can not be calculated or evaluated. To overcome this difficulty, we propose the notion of the bi-local solution. Under the Jacobian uniqueness conditions for the lower level problem, we prove that the bi-local solution is a local minimizer of some one-level minimization problem. Basing on this property, the first-order necessary optimality conditions and second-order necessary and sufficient optimality conditions for the bi-local optimal solution of a given bilevel program are established. The second-order optimality conditions proposed here only involve second-order derivatives of the defining functions of the bilevel problem. The second-order sufficient optimality conditions are used to derive the Q-linear convergence rate of the classical augmented Lagrangian method.