An inexact perturbed path-following method for Lagrangian decomposition in large-scale separable convex optimization

TL;DR

This paper introduces an inexact perturbed path-following algorithm for large-scale convex optimization, allowing inexact primal solutions and approximate Hessians, with proven convergence and complexity analysis.

Contribution

It presents a novel inexact perturbed path-following method within Lagrangian dual decomposition, extending existing exact algorithms to practical inexact computations.

Findings

Convergence of the proposed algorithm is rigorously analyzed.

Worst-case complexity estimates are provided for both algorithm phases.

Numerical results demonstrate the method's effectiveness and practical applicability.

Abstract

In this paper, we propose an inexact perturbed path-following algorithm in the framework of Lagrangian dual decomposition for solving large-scale structured convex optimization problems. Unlike the exact versions considered in literature, we allow one to solve the primal problem inexactly up to a given accuracy. The inexact perturbed algorithm allows to use both approximate Hessian matrices and approximate gradient vectors to compute Newton-type directions for the dual problem. The algorithm is divided into two phases. The first phase computes an initial point which makes use of inexact perturbed damped Newton-type iterations, while the second one performs the path-following algorithm with inexact perturbed full-step Newton-type iterations. We analyze the convergence of both phases and estimate the worst-case complexity. As a special case, an exact path- following algorithm for Lagrangian relaxation is derived and its worst-case complexity is estimated. This variant possesses some differences compared to the previously known methods. Implementation details are discussed and numerical results are reported.