On the Convergence of Alternating Direction Lagrangian Methods for Nonconvex Structured Optimization Problems

Nonconvex and structured optimization problems arise in many engineering applications that demand scalable and distributed solution methods. The study of the convergence properties of these methods is, in general, difficult due to the nonconvexity of the problem. In this paper, two distributed solution methods that combine the fast convergence properties of augmented Lagrangian-based methods with the separability properties of alternating optimization are investigated. The first method is adapted from the classic quadratic penalty function method and is called the alternating direction penalty method (ADPM). Unlike the original quadratic penalty function method, where single-step optimizations are adopted, ADPM uses an alternating optimization which, in turn, makes it scalable. The second method is the well-known alternating direction method of multipliers (ADMM). It is shown that ADPM for nonconvex problems asymptotically converges to a primal feasible point under mild conditions and an additional condition ensuring that it asymptotically reaches the standard first-order necessary conditions for local optimality is introduced. In the case of the ADMM, novel sufficient conditions under which the algorithm asymptotically reaches the standard first-order necessary conditions are established. Based on this, complete convergence of the ADMM for a class of low-dimensional problems is characterized. Finally, the results are illustrated by applying ADPM and ADMM to a nonconvex localization problem in wireless-sensor networks.