Convergence rates for an inexact linearized ADMM for nonsmooth nonconvex optimization with nonlinear equality constraints
Lahcen El Bourkhissi, Ion Necoara
TL;DR
This work advances constrained nonsmooth nonconvex optimization by introducing iL-ADMM, an inexact linearized ADMM that linearizes the smooth parts of both the objective and the nonlinear equality constraints inside the augmented Lagrangian and augments with a Gauss-Newton-inspired regularization. The algorithm updates the x-block inexactly, the y-block via projection, and the dual variable in the standard ADMM manner, yielding simple subproblems while preserving convergence guarantees. The authors establish global convergence to stationary points and derive $\mathcal{O}(\epsilon^{-2})$ iteration complexity for attaining $\epsilon$-KKT points, with improved rates under the KL property. Numerical experiments on nonlinear model predictive control and nonnegative orthogonal matrix factorization validate the theory and demonstrate competitive performance against state-of-the-art solvers, highlighting robustness to penalty parameter choices and practical applicability.
Abstract
In this paper, we consider nonconvex optimization problems with nonsmooth nonconvex objective function and nonlinear equality constraints. We assume that both the objective function and the functional constraints can be separated into 2 blocks. To solve this problem, we introduce a new inexact linearized alternating direction method of multipliers (ADMM) algorithm. Specifically, at each iteration, we linearize the smooth part of the objective function and the nonlinear part of the functional constraints within the augmented Lagrangian and add a dynamic quadratic regularization. We then compute the new iterate of the block associated with nonlinear constraints inexactly. This strategy yields subproblems that are easily solvable and their (inexact) solutions become the next iterates. Using Lyapunov arguments, we establish convergence guarantees for the iterates of our method toward an $ε$-first-order solution within $\mathcal{O}(ε^{-2})$ iterations. Moreover, we demonstrate that in cases where the problem data exhibit e.g., semi-algebraic properties or more general the KL condition, the entire sequence generated by our algorithm converges, and we provide convergence rates. To validate both the theory and the performance of our algorithm, we conduct numerical simulations for several nonlinear model predictive control and matrix factorization problems.