Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization

TLDR

The nuclear norm is widely used to promote low‑rank solutions, and recent work shows that the augmented Lagrangian and alternating direction methods are highly efficient for convex problems when subproblems admit closed‑form solutions. This study investigates applying the augmented Lagrangian and alternating direction methods to nuclear‑norm minimization problems. When subproblems lack closed‑form solutions, the authors linearize them to obtain tractable closed‑form updates and demonstrate the methods’ effectiveness through numerical experiments. Global convergence of the linearized augmented Lagrangian and alternating direction methods is proven under standard assumptions.

Abstract

The nuclear norm is widely used to induce low-rank solutions for many optimization problems with matrix variables. Recently, it has been shown that the augmented Lagrangian method (ALM) and the alternating direction method (ADM) are very efficient for many convex programming problems arising from various applications, provided that the resulting subproblems are sufficiently simple to have closed-form solutions. In this paper, we are interested in the application of the ALM and the ADM for some nuclear norm involved minimization problems. When the resulting subproblems do not have closed-form solutions, we propose to linearize these subproblems such that closed-form solutions of these linearized subproblems can be easily derived. Global convergence results of these linearized ALM and ADM are established under standard assumptions. Finally, we verify the effectiveness and efficiency of these new methods by some numerical experiments.