Unformatted text preview: Points possible: 10
1. Consider the problem given in the figure below.
Let N(Si) = Si plus all nodes connected by an edge; therefore N(S1) ={ S1, S2, S3}.
Your objective is to minimize the cost, which is given inside the circles for each S j.
For each of the following heuristics
i) draw the configuration graph and give the probabilities (using the same convention
for locating probabilities near the head of the arrow in the corresponding direction
and
ii) find the transition probability matrix Θ =P A.
iii) Also find the convergence probability vector π. Check your answer by seeing if
π = π Θ . You can solve this by making a good guess or by solving the equation π =
π Θ for π. (This is small enough that you can do it by hand if you wish.) You a re not
being asked to compute PM or ΘM for large M.. For each case explain your intuition
for the reason behind the answer , i.e. why some πi are zero or why some πi equal
each other.
a. Random Search ( Walk) (5 points)
(i)
Configuration graph (1 point): 1/3 10
S1 1/2 1/2
1/3
30
S3
1/2 (ii) 1/3
20
S2
1/2 (2 points) Note: students do not have to have expressions for pij and Aij , only
Θ. 1 Thus the transition matrix, Θ=PʘA where ʘ means element byelement multiplication: (iii)
Guess (2 points) This makes sense intuitively since S1 has 3 neighbors, and S2 and S3 both have 2
neighbors, and you have an equal probability of starting at S 1, S2 and S3.
Check that Therefore you have verified your
b. Greedy Search (5 points) 2 (i) (1 point) Configuration graph: 1 10
S1 1/2 1/2
0
30
S3
1/2 (ii) 0
20
S2
1/2 (2 points) Note (students do not have to have expressions for p ij and Aij , only
Θ). Thus the transition matrix, Θ=PʘA where ʘ means element byelement multiplication: (iii) (2 points) Guess that This makes intuitive sense because the optimal solution (lowest cost solution) is S 1.
Check that
3 2. (Not graded) Assume you have the following values of the cost function
S
1
COST(S) 15 2
20 3
10 4
13 5
11 6
14 7
16 Also assume that the neighborhood N (S ) S 1, S 2 (mod 7 , so 7, 1, and 2
are neighbors of each other and 6,7, and 1 are neighbors of each other..). The
relationships between different Si are shown below. S1
15
S2
20 S3
10 S4
13 S5
11 S6
14 S7
16 Note that each node should have four neighbors. Also, S is not a neighbor of itself.
Assume you want to use simulated annealing for this optimization problem. Give the
transition matrix for this problem when T=2. You can express the answers in
terms of exponentials and numbers. Can you guess what * is (without solving any
matrix equation or doing repeated matrix multiplication? Why do you think * has
that form? 4 ...
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 Fall '11
 Shoemaker
 aij, configuration graph

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