  We investigate a class of nonconvex optimization problems characterized by a
feasible set consisting of level-bounded nonconvex regularizers, with a
continuously differentiable objective. We propose a novel hybrid approach to
tackle such structured problems within a first-order algorithmic framework by
combining the Frank-Wolfe method and the gradient projection method. The
Frank-Wolfe step is amenable to a closed-form solution, while the gradient
projection step can be efficiently performed in a reduced subspace. A notable
characteristic of our approach lies in its independence from introducing
smoothing parameters, enabling efficient solutions to the original nonsmooth
problems. We establish the global convergence of the proposed algorithm and
show the $O(1/\sqrt{k})$ convergence rate in terms of the optimality error for
nonconvex objectives under reasonable assumptions. Numerical experiments
underscore the practicality and efficiency of our proposed algorithm compared
to existing cutting-edge methods. Furthermore, we highlight how the proposed
algorithm contributes to the advancement of nonconvex regularizer-constrained
optimization.
