Immunity to Increasing Condition Numbers of Linear Superiorization versus Linear Programming

TL;DR

This paper compares the robustness of Linear Programming and Linear Superiorization methods against increasing condition numbers in linear systems, highlighting their respective sensitivities to data perturbations.

Contribution

It provides an experimental analysis of how LP and LinSup perform under ill-conditioned data, focusing on their sensitivity to condition numbers, which was not previously studied.

Findings

LinSup shows greater resilience to high condition numbers than LP.

LP's performance deteriorates significantly with increasing condition numbers.

LinSup maintains feasible solutions better under data perturbations.

Abstract

Given a family of linear constraints and a linear objective function one can consider whether to apply a Linear Programming (LP) algorithm or use a Linear Superiorization (LinSup) algorithm on this data. In the LP methodology one aims at finding an optimal point, i.e., a point that fulfills the constraints and has the minimal value of the objective function over these constraints. The Linear Superiorization approach considers the same data as linear programming problems but instead of attempting to solve those with linear programming methods it employs perturbation resilient feasibility-seeking algorithms and steers them toward a feasible point with reduced (not necessarily minimal) objective function value. This aim of the superiorization method (SM) is less demanding than aiming to reach full-fledged constrained optimality and it places more importance on reaching feasibility than on reaching optimality. Previous studies (e. g. [12]) compared LP and LinSup in terms of their respective outputs and the resources they use. This paper is a follow-up analysis of [12], where we investigate classical LP approaches and LinSup in terms of their sensitivity to condition numbers of the system of linear constraints. Condition numbers are a measure for the impact of deviations in the input data on the output of a problem and, in particular, they describe the factor of error propagation when given wrong or erroneous data. Therefore, the ability of LP and LinSup to cope with increased condition numbers, thus with ill-posed problems, is an important matter to consider which was not studied until now. We investigate experimentally the advantages and disadvantages of both LP and LinSup on exemplary problems of linear programming with multiple condition numbers and different problem dimensions.