Centered Solutions for Uncertain Linear Equations (revision of CentER DP 2015-044)

Jianzhe Zhen, Dick den Hertog

Research output: Working paperDiscussion paperOther research output

178 Downloads (Pure)

Abstract

Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we show that the intersection of the set of possible solutions and any orthant is convex.We derive a convex representation of this intersection. Secondly, to obtain centered solutions for systems of uncertain linear equations, we compute the maximum size inscribed convex body (MCB) of the set of all possible solutions. The obtained MCB is an inner approximation of the solution set, and its center is a potential solution to the system. We compare our method both theoretically and numerically with an existing method that minimizes the worst-case violation. Applications to the input-output model, Colley's Matrix Rankings and Article Influence Scores demonstrate the advantages of the new method.
Original languageEnglish
Place of PublicationTilburg
PublisherCentER, Center for Economic Research
Number of pages26
Volume2016-048
Publication statusPublished - 22 Dec 2016

Publication series

NameCentER Discussion Paper
Volume2016-048

Fingerprint

Linear equations
Uncertainty

Keywords

  • interval linear systems
  • uncertain linear equations
  • Maximum volume inscribed ellipsoid

Cite this

Zhen, J., & den Hertog, D. (2016). Centered Solutions for Uncertain Linear Equations (revision of CentER DP 2015-044). (CentER Discussion Paper; Vol. 2016-048). Tilburg: CentER, Center for Economic Research.
Zhen, Jianzhe ; den Hertog, Dick. / Centered Solutions for Uncertain Linear Equations (revision of CentER DP 2015-044). Tilburg : CentER, Center for Economic Research, 2016. (CentER Discussion Paper).
@techreport{297fa3b1529048b5bbc000766c1288f8,
title = "Centered Solutions for Uncertain Linear Equations (revision of CentER DP 2015-044)",
abstract = "Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we show that the intersection of the set of possible solutions and any orthant is convex.We derive a convex representation of this intersection. Secondly, to obtain centered solutions for systems of uncertain linear equations, we compute the maximum size inscribed convex body (MCB) of the set of all possible solutions. The obtained MCB is an inner approximation of the solution set, and its center is a potential solution to the system. We compare our method both theoretically and numerically with an existing method that minimizes the worst-case violation. Applications to the input-output model, Colley's Matrix Rankings and Article Influence Scores demonstrate the advantages of the new method.",
keywords = "interval linear systems, uncertain linear equations, Maximum volume inscribed ellipsoid",
author = "Jianzhe Zhen and {den Hertog}, Dick",
year = "2016",
month = "12",
day = "22",
language = "English",
volume = "2016-048",
series = "CentER Discussion Paper",
publisher = "CentER, Center for Economic Research",
type = "WorkingPaper",
institution = "CentER, Center for Economic Research",

}

Zhen, J & den Hertog, D 2016 'Centered Solutions for Uncertain Linear Equations (revision of CentER DP 2015-044)' CentER Discussion Paper, vol. 2016-048, CentER, Center for Economic Research, Tilburg.

Centered Solutions for Uncertain Linear Equations (revision of CentER DP 2015-044). / Zhen, Jianzhe; den Hertog, Dick.

Tilburg : CentER, Center for Economic Research, 2016. (CentER Discussion Paper; Vol. 2016-048).

Research output: Working paperDiscussion paperOther research output

TY - UNPB

T1 - Centered Solutions for Uncertain Linear Equations (revision of CentER DP 2015-044)

AU - Zhen, Jianzhe

AU - den Hertog, Dick

PY - 2016/12/22

Y1 - 2016/12/22

N2 - Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we show that the intersection of the set of possible solutions and any orthant is convex.We derive a convex representation of this intersection. Secondly, to obtain centered solutions for systems of uncertain linear equations, we compute the maximum size inscribed convex body (MCB) of the set of all possible solutions. The obtained MCB is an inner approximation of the solution set, and its center is a potential solution to the system. We compare our method both theoretically and numerically with an existing method that minimizes the worst-case violation. Applications to the input-output model, Colley's Matrix Rankings and Article Influence Scores demonstrate the advantages of the new method.

AB - Our contribution is twofold. Firstly, for a system of uncertain linear equations where the uncertainties are column-wise and reside in general convex sets, we show that the intersection of the set of possible solutions and any orthant is convex.We derive a convex representation of this intersection. Secondly, to obtain centered solutions for systems of uncertain linear equations, we compute the maximum size inscribed convex body (MCB) of the set of all possible solutions. The obtained MCB is an inner approximation of the solution set, and its center is a potential solution to the system. We compare our method both theoretically and numerically with an existing method that minimizes the worst-case violation. Applications to the input-output model, Colley's Matrix Rankings and Article Influence Scores demonstrate the advantages of the new method.

KW - interval linear systems

KW - uncertain linear equations

KW - Maximum volume inscribed ellipsoid

M3 - Discussion paper

VL - 2016-048

T3 - CentER Discussion Paper

BT - Centered Solutions for Uncertain Linear Equations (revision of CentER DP 2015-044)

PB - CentER, Center for Economic Research

CY - Tilburg

ER -

Zhen J, den Hertog D. Centered Solutions for Uncertain Linear Equations (revision of CentER DP 2015-044). Tilburg: CentER, Center for Economic Research. 2016 Dec 22. (CentER Discussion Paper).