# An Inexact Interior-Point Lagrangian Decomposition Algorithm with Inexact Oracles

## TL;DR

This paper introduces an inexact interior-point Lagrangian decomposition algorithm for convex optimization, combining dual decomposition, barrier smoothing, and proximal-Newton methods, with proven polynomial-time complexity and practical numerical validation.

## Contribution

It presents a novel inexact interior-point method that handles inexact primal and dual solutions, extending the applicability and efficiency of convex optimization algorithms.

## Key findings

- Algorithm achieves polynomial-time complexity.
- Effective primal recovery strategy provided.
- Numerical experiments demonstrate competitive performance.

## Abstract

We develop a new inexact interior-point Lagrangian decomposition method to solve a wide range class of constrained composite convex optimization problems. Our method relies on four techniques: Lagrangian dual decomposition, self-concordant barrier smoothing, path-following, and proximal-Newton technique. It also allows one to approximately compute the solution of the primal subproblems (called the slave problems), which leads to inexact oracles (i.e., inexact gradients and Hessians) of the smoothed dual problem (called the master problem). The smoothed dual problem is nonsmooth, we propose to use an inexact proximal-Newton method to solve it. By appropriately controlling the inexact computation at both levels: the slave and master problems, we still estimate a polynomial-time iteration-complexity of our algorithm as in standard short-step interior-point methods. We also provide a strategy to recover primal solutions and establish complexity to achieve an approximate primal solution. We illustrate our method through two numerical examples on well-known models with both synthetic and real data and compare it with some existing state-of-the-art methods.