Complexity of an inexact proximal-point penalty method for constrained smooth non-convex optimization

Abstract

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. This 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 this approach 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 ε-stationary point. When the constraint functions are convex, we show a complexity result of O~ (ε- 5 / 2) to produce an ε-stationary point under the Slater’s condition. When the constraint functions are non-convex, the complexity becomes O~ (ε- 3) if a non-singularity condition holds on constraints and otherwise O~ (ε- 4) if a feasible initial solution is available.