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

Bilevel optimizaiton serves as a powerful tool for many machine learning applications. Perturbed pessimistic bilevel problem PBP$\epsilon$, with $\epsilon$ 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 PBP$\epsilon$ with a nonlinear lower level problem are lacking. To fill the gap, we consider in the paper the problem PBP$\epsilon$ 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 PBP$\epsilon$, which has not been given before, and we show that PVFIM can converge to a stationary point of PBP$\epsilon$. Finally, experiments are presented to verify the theoretical results and to show the application of the algorithm to GAN.