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Bilevel optimizaiton serves as a powerful tool for many machine learningapplications. Perturbed pessimistic bilevel problem PBP$\epsilon$, with$\epsilon$ being an arbitrary positive number, is a variant of the bilevelproblem to deal with the case where there are multiple solutions in the lowerlevel problem. However, the provably convergent algorithms for PBP$\epsilon$with a nonlinear lower level problem are lacking. To fill the gap, we considerin the paper the problem PBP$\epsilon$ with a nonlinear lower level problem. Byintroducing a log-barrier function to replace the inequality constraintassociated with the value function of the lower level problem, andapproximating this value function, an algorithm named PerturbedValue-Function-based Interior-point Method(PVFIM) is proposed. We present astationary condition for PBP$\epsilon$, which has not been given before, and weshow that PVFIM can converge to a stationary point of PBP$\epsilon$. Finally,experiments are presented to verify the theoretical results and to show theapplication of the algorithm to GAN.

A Perturbed Value-Function-Based Interior-Point Method for Perturbed Pessimistic Bilevel Problems