A Perturbed Value-Function-Based Interior-Point Method for Perturbed Pessimistic Bilevel Problems
arxiv(2024)
摘要
Bilevel optimizaiton serves as a powerful tool for many machine learning
applications. Perturbed pessimistic bilevel problem PBPϵ, with
ϵ being an arbitrary positive number, is a variant of the bilevel
problem to deal with the case where there are multiple solutions in the lower
level problem. However, the provably convergent algorithms for PBPϵ
with a nonlinear lower level problem are lacking. To fill the gap, we consider
in the paper the problem PBPϵ with a nonlinear lower level problem. By
introducing a log-barrier function to replace the inequality constraint
associated with the value function of the lower level problem, and
approximating this value function, an algorithm named Perturbed
Value-Function-based Interior-point Method(PVFIM) is proposed. We present a
stationary condition for PBPϵ, which has not been given before, and we
show that PVFIM can converge to a stationary point of PBPϵ. Finally,
experiments are presented to verify the theoretical results and to show the
application of the algorithm to GAN.
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