Linear Programming & Games Uni Notes Part II

mom
equation




Maximise 3x t Iz


S t X t 6 2 23 Ia Z 3
7 2 I 4 5
I t Iz t 23 1

3 I c
E Z



x unresistricted

12,313,514 Z O

Maximise 3x.tt 3 taz
t
X ai 6 2 Sts 1 Ii E 3
7 2 Ic 5
sci y Sez t x 3 I

23 24 E 2

, it sci Iz X3 I4 Z O


to make standard equation
form change inequalities
into equations with slave variables


Maximise 3x it t 3 i t xz
t
x ai 6 2 Iz t I 4 t s 3
23 14 t Sz 2
7 2 I4 5
x T y t x z t x 3 I

X f ai Iz X3 I4 S1 S2 Z O



maximise I

5t A b
a ZQ


Theorem

linear program 1 in standard equation form has an optimal
Ig a

solution then it has an optimal solution then it has an optimal
solution that is an extreme point solution

, Extreme point




Claim an optimal solution I is not an extreme
If point then
we can find a feasible solution with Ix z EE and
has one more zero entry than x



Proof

If I is not an extreme point then we can write it as a

convex combination


I Xy t l l X E

where X E lo il and E Z And E feasible
y y
Since is optimal


EE Z Ey and E I EE

I j either ETE s Ey or Eto et E

, Then

II I lay l H Z
Exy t et Ct X I
X of I H Et E
X of I a Etz
Ete

SO ETI ETF and ETI Etz

Note since and Z are we have
y feasible


A be AZ L

Pick 0 in IR set II t O ly Z




yon x

or
Z




Consider AI