CO350 Assignment 6
–
Wing Man Fok
20300650
,
June 25
2010
:

Instructor
Mathieu Guay Paquet
1a
Suppose there is a tie for the choice of leaving variable in an iteration of the
.
simplex method
, =
=
∈
, , ∉
≠ .
That is
t
biaik
bjajk for some k
N i j
B with i
j
Assume WLOG that the new
=
∪ ∖ .
, ,
basis is B'
B
k i
When pivoting k i
we subtract ajkaik times the xi row from
.
=

= .
,
the xj row
So bj'
bj ajkaikbi
0
Hence
the new basic variable has value zero in
,
.

the new solution
and B' is degenerate
Since every perturbed LP is non
,
degenerate
there is never a tie for the choice of leaving variable in the
.
,
,
perturbation method
Otherwise
the new basis is degenerate
which is
.
impossible
2a
Suppose there is an iteration of the simplex method which starts with basis B
.
>
=

.
=
and ends with basis B'
We can find t 0 and Ak such that xB'
xB tAk
Let d
.
,
=
(
)=

=
(  )= .
Ak
Since B and B' are both feasible
Ad
1tA td
1tAxB' AxB
1t b b
0
Also
note that
=
=

>
cTd
1tcTtd
1tcTxB' cTxB 0
=

=
 >
dk
1txB'k xBk
1txB'k 0 0
=

=

<
dr
1txB'r xBr
1t0 xBr 0
=
=
∉
∪
di
aik
0 for i
B
k
2b
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 Fall '07
 S.Furino,B.Guenin
 Linear Programming, Optimization, Duality, Wing Man Fok

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