Generalized Affine Scaling Algorithms for Linear Programming Problems

TL;DR

This paper introduces two enhanced primal affine scaling algorithms for linear programming, incorporating Nesterov's restarting strategy and a non-linear series transformation, leading to faster convergence and improved performance over traditional methods.

Contribution

The paper presents a generalized primal affine scaling algorithm with proven convergence and introduces a second acceleration method, advancing the efficiency of interior point methods for linear programming.

Findings

Proposed algorithms outperform original affine scaling in numerical tests.

Convergence proofs extend existing theoretical guarantees.

New family of methods suggested by the generalized convergence results.

Abstract

Interior Point Methods are widely used to solve Linear Programming problems. In this work, we present two primal affine scaling algorithms to achieve faster convergence in solving Linear Programming problems. In the first algorithm, we integrate Nesterov's restarting strategy in the primal affine scaling method with an extra parameter, which in turn generalizes the original primal affine scaling method. We provide the proof of convergence for the proposed generalized algorithm considering long step size. We also provide the proof of convergence for the primal and dual sequence without the degeneracy assumption. This convergence result generalizes the original convergence result for the affine scaling methods and it gives us hints about the existence of a new family of methods. Then, we introduce a second algorithm to accelerate the convergence rate of the generalized algorithm by integrating a non-linear series transformation technique. Our numerical results show that the proposed algorithms outperform the original primal affine scaling method.