Question $2$

For primal linear program

$$

$=$

min

$$

$$

$$

$$

$,$

s

u

b

j

e

c

t

$$

t

o

$$

$$

$$

$>=$

$$

P$=$min

x

$$

c

T

x$,$subjecttoAx$>=$b$,$ with

$$

$=$

$[$

$1$

$,$

$$

$1$

$]$

$$

$,$

$$

$=$

$[$

$$

$1$

$,$

$$

$2$

$]$

$$

$,$

$$

$=$

$[$

$1$

$$

$1$

$2$

$1$

$]$

$,$

$$

$=$

$[$

$1$

$,$

$1$

$]$

$$

c$=[1,1]$

T

$,$x$=[$x

$1$

$$

$,$x

$2$

$$

$]$

T

$,$A$=[$

$1$

$2$

$$

$1$

$1$

$$

$],$b$=[1,1]$

T

$.$ Suppose the variable in dual program is

$$

$=$

$[$

$$

$1$

$,$

$$

$2$

$]$

$$

u$=[$u

$1$

$$

$,$u

$2$

$$

$]$

T

$.$ From the constrains, you can solve for

$$

$1$

u

$1$

$$

and

$$

$2$

u

$2$

$$

$.$ Suppose strong duality holds, what is the optimal value of the primal linear program

$$

P$?$