  We consider a class of constrained optimization problems with a possibly
nonconvex non-Lipschitz objective and a convex feasible set being the
intersection of a polyhedron and a possibly degenerate ellipsoid. Such problems
have a wide range of applications in data science, where the objective is used
for inducing sparsity in the solutions while the constraint set models the
noise tolerance and incorporates other prior information for data fitting. To
solve this class of constrained optimization problems, a common approach is the
penalty method. However, there is little theory on exact penalization for
problems with nonconvex and non-Lipschitz objective functions. In this paper,
we study the existence of exact penalty parameters regarding local minimizers,
stationary points and $\epsilon$-minimizers under suitable assumptions.
Moreover, we discuss a penalty method whose subproblems are solved via a
nonmonotone proximal gradient method with a suitable update scheme for the
penalty parameters, and prove the convergence of the algorithm to a KKT point
of the constrained problem. Preliminary numerical results demonstrate the
efficiency of the penalty method for finding sparse solutions of
underdetermined linear systems.
