Local Convergence Analysis of ADMM for Nonconvex Composite Optimization

In this paper, we study the local convergence of the standard ADMM scheme for a class of nonconvex composite optimization problems motivated by applications in signal processing and machine learning.

The problems are constrained by a closed convex set, while their objective is the sum of a continuously differentiable, possibly nonconvex, smooth term and a polyhedral convex nonsmooth term composed with a linear mapping.

Motivated by recent works of Rockafellar, we first provide an elementary proof of a local strong convexity property of the Moreau envelope of polyhedral convex functions on the orthogonal complement of an appropriate subspace.

Building on this property, we establish the strong variational sufficiency of the reduced augmented Lagrangian under an appropriate second-order condition.

We then derive a descent inequality for the ADMM iterates that is analogous to the classical descent inequality for convex ADMM.

For a sufficiently large penalty parameter, and under suitable initialization and local trajectory conditions, we prove that the ADMM sequence converges to a stationary primal-dual point.

When the constraint set is polyhedral convex, we further show that the weighted distance of the primal-dual sequence to the local solution set converges Q-linearly, while the primal sequence converges R-linearly.

Finally, we present three illustrative examples together with an application-oriented verification for a class of possibly nonconvex quadratic programs, illustrating the role of the second-order condition, the local nature of the convergence theory, and its applicability.