Decomposing Linearly Constrained Nonconvex Problems by a Proximal Primal Dual Approach: Algorithms, Convergence, and Applications

TL;DR

This paper introduces the Proximal Primal-Dual Algorithm (Prox-PDA) for nonconvex linearly constrained problems, providing convergence analysis, and demonstrating its application to distributed nonconvex optimization and matrix factorization.

Contribution

The paper develops a new primal-dual decomposition method for nonconvex problems, proving its convergence and linking it to existing algorithms like EXTRA in nonconvex settings.

Findings

Prox-PDA converges globally to stationary solutions under certain conditions.

The algorithm can decompose variables to simplify subproblems.

Application to distributed nonconvex optimization aligns with the EXTRA algorithm.

Abstract

In this paper, we propose a new decomposition approach named the proximal primal dual algorithm (Prox-PDA) for smooth nonconvex linearly constrained optimization problems. The proposed approach is primal-dual based, where the primal step minimizes certain approximation of the augmented Lagrangian of the problem, and the dual step performs an approximate dual ascent. The approximation used in the primal step is able to decompose the variable blocks, making it possible to obtain simple subproblems by leveraging the problem structures. Theoretically, we show that whenever the penalty parameter in the augmented Lagrangian is larger than a given threshold, the Prox-PDA converges to the set of stationary solutions, globally and in a sublinear manner (i.e., certain measure of stationarity decreases in the rate of $\mathcal{O}(1/r)$, where $r$ is the iteration counter). Interestingly, when applying a variant of the Prox-PDA to the problem of distributed nonconvex optimization (over a connected undirected graph), the resulting algorithm coincides with the popular EXTRA algorithm [Shi et al 2014], which is only known to work in convex cases. Our analysis implies that EXTRA and its variants converge globally sublinearly to stationary solutions of certain nonconvex distributed optimization problem. There are many possible extensions of the Prox-PDA, and we present one particular extension to certain nonconvex distributed matrix factorization problem.