Theoretical Smoothing Frameworks for Nonsmooth Simple Bilevel Problems

Bilevel programming has recently received a great deal of attention because of its abundant applications in many areas. We study a class of bilevel problems in which the lower-level feasible set is independent of the upper-level variables. The optimal value function approach provides a useful reformulation of the bilevel problem, but its utility is often limited because of the nonsmoothness of the value function even in cases when the associated lower-level function is smooth. In this paper, we present two smoothing strategies for the value function associated with lower-level functions that are not necessarily smooth but are Lipschitz continuous. The first method employs quadratic regularization for partially convex lower-level functions, whereas the second utilizes entropic regularization for general lower-level objective functions. Meanwhile, the property known as gradient consistency is crucial in ensuring that a designed smoothing algorithm is globally subsequentially convergent to stationary points of the value function reformulation. With this motivation, we prove that the proposed smooth approximations satisfy the gradient consistent property under certain conditions on the lower-level function. Funding: This work was supported by Core Research for Evolutionary Science and Technology (JST CREST) [Grant JPMJCR24Q2] and the Japan Society for the Promotion of Science (JSPS KAKENHI) [Grant 23H03351].