A primal-dual majorization-minimization method for large-scale linear programs

TL;DR

This paper introduces a primal-dual majorization-minimization method for large-scale linear programs, leveraging a smooth barrier augmented Lagrangian function to achieve global linear convergence without requiring feasibility assumptions.

Contribution

It develops a novel SBAL function-based approach that simplifies computations and guarantees convergence for large-scale linear programs, independent of step size tuning.

Findings

Method depends only on matrix factorization, no step size computation

Achieves global linear convergence under regular conditions

Provides a new iteration complexity bound for large-scale LPs

Abstract

We present a primal-dual majorization-minimization method for solving large-scale linear programs. A smooth barrier augmented Lagrangian (SBAL) function with strict convexity for the dual linear program is derived. The majorization-minimization approach is naturally introduced to develop the smoothness and convexity of the SBAL function. Our method only depends on a factorization of the constant matrix independent of iterations and does not need any computation on step sizes, thus can be expected to be particularly appropriate for large-scale linear programs. The method shares some similar properties to the first-order methods for linear programs, but its convergence analysis is established on the differentiability and convexity of our SBAL function. The global convergence is analyzed without prior requiring either the primal or dual linear program to be feasible. Under the regular conditions, our method is proved to be globally linearly convergent, and a new iteration complexity result is given.