Spring 2010
CS530 – Analysis of Algorithms
S
ELF

CHECK QUESTIONS
2
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Problem 1.
Let
A
be a positive semidefinite matrix.
(a) Show that
a
ii
0 for all
i
.
Solution.
Consider a vector
x
with
x
i
=
1 and
x
j
=
0 for all
j
=
i
. Now,
a
ii
=
x
T
Ax
0.
(b) Show that if
a
ii
=
0 then
a
i j
=
a
ji
=
0 for all
j
.
Solution.
Consider a vector
x
with
x
i
=
1 and
x
k
=
0 for
k
=
j
. Then
0
x
T
Ax
=
a
ii
+
2
a
i j
x
j
+
a
j j
x
2
j
=
2
a
i j
x
j
+
a
j j
x
2
j
.
The minimum of the righthand side is
0 only if
a
i j
=
0.
Problem 2.
How do you recognize it in the simplex algorithm that an optimal
solution has been reached?
Solution.
Each nonbasic variable enters the objective function with a non
positive coefficient.
Problem 3.
Prove that if a linear program has a nonbasic optimal solution
then it has infinitely many optimal solutions.
Solution.
Let
x
be a nonbasic optimal solution.
For this solution, let us
choose the basis in which the number of nonzero nonbasic variables is as
small as possible. Let
x
j
be a nonbasic variable whose value is not 0. Then
any basic variable
x
i
for which
a
i j
=
0 in the current slack form also a
nonzero value, otherwise a pivot operation would decrease the number of
nonzero basic variables.
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 Spring '09
 Algorithms, Optimization, Analysis of algorithms, 2m, linear programming problem, 1 bits

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