Neural Networks, CAP 6615 Spring 2011
Homework 7, Due 18 April 2011
1.
(Haykin Problem 11.3) Consider the Markov chain depicted in Figure P11.3, which is reducible.
Identify the classes of states contained in this state transition diagram.
Figure P11.3
The classes of this Markov chain are {x
1
} and {x
2
, x
3
}
2.
(Haykin Problem 11.4) Calculate the steadystate probabilities of the Markov chain shown in
Fig. P11.4
Figure P11.4
The stochastic matrix for this Markov chain is
P
=
[
3
/
4
1
/
4
0
0
2
/
3
1
/
3
1
/
4
0
3
/
4
]
yielding equations
π
i
=π
i
(
3
/
4
)+π
2
(
0
)+π
3
(
1
/
4
)
π
2
=π
1
(
1
/
4
)+π
2
(
2
/
3
)+π
3
(
0
)
π
3
=π
1
(
0
)+π
2
(
1
/
3
)+π
3
(
3
/
4
)
thus,
π
1
=π
3
,
π
2
=π
1
(
3
/
4
)
. We also have
π
1
+π
2
+π
3
=
1
by definition.
Thus,
π
1
+π
1
(
3
/
4
)+π
1
=
1
so
π
1
=π
3
=
4
/
11
and
π
2
=
3
/
11
.
1/3
2/3
3/4
1/4
1/2
1/2
x
1
x
2
x
3
1/4
1/4
1/3
3/4
3/4
2/3
x
1
x
2
x
3
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View Full DocumentNeural Networks, CAP 6615 Spring 2011
Homework 7, Due 18 April 2011
3.
(Haykin Problem 11.7) In this problem, we consider the use of simulated annealing for solving
the
travelingsalesman problem
(TSP). You are given the following:
◦
N
cities
◦
the distance between each pair of cities,
d
◦
a tour represented by a closed path visiting each city once, and only once.
The objective is to find a tour (i.e., permutation of the order in which the cities are visited) that
is of minimal total length
L
. In this problem, the different possible tours are the configurations,
and the total length of a tour is the cost function to be minimized.
(a) Devise an iterative method of generating valid configurations.
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 Thermodynamics, Energy, Entropy, Neural Networks

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