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Some remarks on Assignment 4
Question 1
1. The statement
“
Ax
≤
b
Ax
≥
d
x
≥
0
is infeasible”
does not imply
“
Ax
≤
b
x
≥
0
is in
feasible”
. (In general, the logical statement “
not (
A
and
B
and
C
)
” does not
imply “
not (
A
and
B
)
”.)
2. The dual of
max 0
T
x
s.t.
Ax
≤
b, Ax
≥
d,x
≥
0
is given by
min
y ,z
b
T
y

d
T
z
s.t.
A
T
y

A
T
z
≥
0
, y
≥
0
, z
≥
0
(not min
b
T
y

d
T
z
s.t.
A
T
y
≥
0
, A
T
z
≥
0
, y
≥
0
, z
≥
0. See Pg. 3537 of
the printed course notes for the method of constructing duals.)
3.
y
T
b
≥
z
T
b
does not imply
y
≥
z
(even if
b >
0).
4.
a < b
does not imply
a <
0
< b
.
Question 7
Majority of the class can answer this question correctly. One common mistake in
this question is to pick the pairs (
x
2
,x
1
) and (
x
5
,x
1
) in
(i)
and the pair (
x
5
,x
1
)
in
(ii)
. They are not valid pairs for the respective questions because they fail the
minimumratio test (which is
necessary
for choosing the leaving variable; see Pg.
77, Algorithm Step 5 on the printed courses notes). This is not to be confused
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 Fall '07
 S.Furino,B.Guenin

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