Complexity of a Quadratic Penalty Accelerated Inexact Proximal Point Method for Solving Linearly Constrained Nonconvex Composite Programs

Abstract

This paper analyzes the iteration complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs. More specifically, the objective function is of the form $f+h$, where $f$ is a differentiable function whose gradient is Lipschitz continuous and $h$ is a closed convex function with possibly unbounded domain. The method, basically, consists of applying an accelerated inexact proximal point method for approximately solving a sequence of quadratic penalized subproblems associated with the linearly constrained problem. Each subproblem of the proximal point method is in turn approximately solved by an accelerated composite gradient (ACG) method. It is shown that the proposed scheme generates a $\rho$-approximate stationary point in at most $\mathcal{O}(\rho^{-3})$ ACG iterations. Finally, numerical results showing the efficiency of the proposed method are also given.