  Bilevel programming has recently received a great deal of attention due to
its abundant applications in many areas. The optimal value function approach
provides a useful reformulation of the bilevel problem, but its utility is
often limited due to 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, while 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.
