Math 482Lecture 7

Announcements

Awk 2 due Friday

Evening officehours this week thµ8pm_

on class zoom link not OH link

Rearrest supplementary problems if taking

the 4 unit version

ReadingsLect 7 GM 5.5 5,6 5 so

PS 2.5 2.62 I 2.2

handoutBT chap 3 I

Lect 8 DegeneraciesGm 5,7 5,8PS 2,3 2.8chap 3

BJ 24 4

today An unbounded examplematrix notation

Anunbound edexampleMax 2 X 2 Xz

i

as

In standard form

Max 2 X 2 2

Subj X Xz 1 3 I

I Is Elise

Xi 42,43 4 30

Tableau Xi Xz Xs XL solve

I I is

I I 0 O lE not auite solved

Does this work

Ra

EE'd to keep entries in this firstpushescolumn non negative for feasibility notwork

Instead try putting in basis

EERIEI II 0 O I

pivot here

i H 7Bi 17isfENext

try to pivot againMax LP pivot column must be XyBut no restriction on how big X4 grows

Claim this means LP is unbounded

Let Xz o whereDictionary

4 M M isZ X 2 1 4 1 as large

qX XIII the Miz asHE

X M 11

ButI 2 Mtl

MatrixNotationKEIRNconsider the LP

q I o aTyberm

Let Aj be the jth column of A

Given basis 93 let BCD Bc2 Bang

to denote basic indices

Basic variables XBas XBc XBCm

XB X Bcn XBes n XBcm

CB CBcis CBas CBCm

AB mxm sub matrix of A

BCD Bcm

Claim If X is a bfs

AB is invertible by def of bfs

CTX CTzXB since xjjop.frp XB AI't

2 x E'AI b follows fromprevious

2Cx CTXI'BXB

P fpEBAfb

Doing Row operations on A

to turn The columns ABCD ABCm

to the identity matrix Im

is conivalent to multiplying on left

by Ats

EEE KITENE ENAE AB

Row operations

must on left by Ats

Since Apj'AXB Inxs XB

qB

AI Axa'Eb

a

MainpiotingstepSuppose your tableau has form

a fit

pivoting can be thought of as left multiplication

by seauence of elementarymatrices

T Xk XzX T

What is form for X

T dTI dX

Uo U8 Cmas xConti

x if.ECEi aEaI

Suppose XT s solved for basis B

Then UA Im

so U AIWe also know that Iida D

So CBT dTAB

d cigars

xt

ffn.IN bifaaaJTableau solved for basis B

Xss Ats'bFirst column tells us values ofbasic variables

relatist'sIT CT CBTAB'A u reducedcosts

Top

DI Ej.is osEfXj

Cg is cost per unit increase in

variable Xj

CBTABTAj is cost of compensating changein basic variables necessitated

by constraint Ax b

Them If X is a bfs for a

minimization LP then

if c 20 then x is optimal

maximization LP ESO n