  Bilevel optimization (BO) has recently gained prominence in many machine
learning applications due to its ability to capture the nested structure
inherent in these problems. Recently, many hypergradient methods have been
proposed as effective solutions for solving large-scale problems. However,
current hypergradient methods for the lower-level constrained bilevel
optimization (LCBO) problems need very restrictive assumptions, namely, where
optimality conditions satisfy the differentiability and invertibility
conditions and lack a solid analysis of the convergence rate. What's worse,
existing methods require either double-loop updates, which are sometimes less
efficient. To solve this problem, in this paper, we propose a new hypergradient
of LCBO leveraging the theory of nonsmooth implicit function theorem instead of
using the restrive assumptions. In addition, we propose a \textit{single-loop
single-timescale} algorithm based on the double-momentum method and adaptive
step size method and prove it can return a $(\delta, \epsilon)$-stationary
point with $\tilde{\mathcal{O}}(d_2^2\epsilon^{-4})$ iterations. Experiments on
two applications demonstrate the effectiveness of our proposed method.
