  In this paper, an inexact proximal-point penalty method is studied for
constrained optimization problems, where the objective function is non-convex,
and the constraint functions can also be non-convex. The proposed method
approximately solves a sequence of subproblems, each of which is formed by
adding to the original objective function a proximal term and quadratic penalty
terms associated to the constraint functions. Under a weak-convexity
assumption, each subproblem is made strongly convex and can be solved
effectively to a required accuracy by an optimal gradient-based method. The
computational complexity of the proposed method is analyzed separately for the
cases of convex constraint and non-convex constraint. For both cases, the
complexity results are established in terms of the number of proximal gradient
steps needed to find an $\varepsilon$-stationary point. When the constraint
functions are convex, we show a complexity result of $\tilde
O(\varepsilon^{-5/2})$ to produce an $\varepsilon$-stationary point under the
Slater's condition. When the constraint functions are non-convex, the
complexity becomes $\tilde O(\varepsilon^{-3})$ if a non-singularity condition
holds on constraints and otherwise $\tilde O(\varepsilon^{-4})$ if a feasible
initial solution is available.
