Computational performance of a projection and rescaling algorithm

TL;DR

This paper presents a MATLAB implementation and computational analysis of a projection and rescaling algorithm for solving specific feasibility problems involving linear subspaces and nonnegative constraints, demonstrating promising results.

Contribution

It provides a detailed MATLAB implementation of a recent projection and rescaling algorithm and evaluates its performance on synthetic problems.

Findings

Algorithm shows promising computational performance

Implementation is straightforward and adaptable to other environments

Numerical experiments validate effectiveness on synthetic instances

Abstract

This paper documents a computational implementation of a {\em projection and rescaling algorithm} for finding most interior solutions to the pair of feasibility problems \[ \text{find} \; x\in L\cap\mathbb{R}^n_{+} \;\;\;\; \text{ and } \; \;\;\;\; \text{find} \; \hat x\in L^\perp\cap\mathbb{R}^n_{+}, \] where $L$ denotes a linear subspace in $\mathbb{R}^n$ and $L^\perp$ denotes its orthogonal complement. The projection and rescaling algorithm is a recently developed method that combines a {\em basic procedure} involving only low-cost operations with a periodic {\em rescaling step.} We give a full description of a MATLAB implementation of this algorithm and present multiple sets of numerical experiments on synthetic problem instances with varied levels of conditioning. Our computational experiments provide promising evidence of the effectiveness of the projection and rescaling algorithm. Our MATLAB code is publicly available. Furthermore, the simplicity of the algorithm makes a computational implementation in other environments completely straightforward.