# A second order primal-dual method for nonsmooth convex composite optimization

arXiv: https://arxiv.org/abs/1709.01610 · HTML: https://mlsanity.com/paper/1709.01610/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Sep 2017.

## Built on by (most cited first, 1 of 1)

- [Stability of Primal-Dual Gradient Flow Dynamics for Multi-Block Convex Optimization Problems](https://mlsanity.com/paper/2408.15969/) (2024)

## Abstract (arXiv)

We develop a second order primal-dual method for optimization problems in which the objective function is given by the sum of a strongly convex twice differentiable term and a possibly nondifferentiable convex regularizer. After introducing an auxiliary variable, we utilize the proximal operator of the nonsmooth regularizer to transform the associated augmented Lagrangian into a function that is once, but not twice, continuously differentiable. The saddle point of this function corresponds to the solution of the original optimization problem. We employ a generalization of the Hessian to define second order updates on this function and prove global exponential stability of the corresponding differential inclusion. Furthermore, we develop a globally convergent customized algorithm that utilizes the primal-dual augmented Lagrangian as a merit function. We show that the search direction can be computed efficiently and prove quadratic/superlinear asymptotic convergence. We use the ℓ1-regularized model predictive control problem and the problem of designing a distributed controller for a spatially-invariant system to demonstrate the merits and the effectiveness of our method.