CS621: Artificial Intelligence
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Transcript of CS621: Artificial Intelligence
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CS621: Artificial Intelligence
Pushpak BhattacharyyaCSE Dept., IIT Bombay
Lecture 7: Traveling Salesman Problem as search; Simulated Annealing; Comparison
with GA
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4-city TSP
1
34
2
d12
d23
d34
d31
d14
d23
dij not necessarilyEqual to dji
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TSP: State Representation
10004
00103
01002
00011
4321Position
(α)City (i) `i’ varies over cities
`α’ varies over positions
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Objective Functions
Minimize F = F1 + F2
F1 = k1 ∑i ((∑α xiα) – 1)2 + k2 ∑β ((∑j xjβ) – 1)2
F2 = k3 ∑i ∑j ∑α dij (xiα xi,α+1 + xiα xi,α-1)
1(a) 1(b)
2
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Travelling Salesperson problem through Simulated Annealing, State representation
1 2 3 4
1 1 0 0 0
2 0 0 1 0
3 0 0 0 1
4 0 1 0 0
PositionCity
43
1
2
State
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1 2 3 4
1 1 0 0 0
2 0 0 1 0
3 0 0 0 1
4 0 1 0 0
1 2 3 4
1 1 0 0 0
2 0 1 0 0
3 0 0 0 1
4 0 0 1 0
1 2 3 4
1 1 0 0 0
2 0 0 0 1
3 0 0 1 0
4 0 1 0 0
State/Node expansion
1
43
2
1
4
3
2
1
23
4
3 2 4 2
Parent State
Children States
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State-Energy diagram
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Metropolis Algorithm
1) Initialize: Start with a random state matrix S. Compute the objective function value at S. Call this the energy of the state E(S).
2) The states are transformed by the application of an operator (for TSP, inversion of adjacent cities)
3) Compute change the energy ΔE=Enew-Eold
4) if ΔE <=0, accept the new state Snew
5) Else, accept Snew with probability
(‘T’ is the “temperature” and KB, the Boltzmann constant)
TK
stateE
Be)(
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Metropolis Algorithm (contd)
6) Continue 2-5 until there is no appreciable change in energy
7) The current state may be one of the local minima
8) Increase the temperature and continue 2-7 until the global minimum is reached
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How to probabilistically accept a state? Suppose the probability =p
Generate a random number from a uniform distribution [0,1]
Number generated is in the range [0-p]: Accept the new state, else continue search from the old state itself
TK
stateE
Be)(
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Why? The significance of p (= ) is that
if the states are generated infinite number of times then a proportion p of them will be the concerned new state
(0,0) (0,1)
(1,1)(0,0)
Uniform distributionOf [0,1]
(0,p)
TK
stateE
Be)(
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Why? (contd) If numbers in the range [0,1] are
generated randomly, p% of them will be in the range [0,p]. Hence this process can simulate the state generation process
(0,0) (0,1)
(1,1)(0,0)
Uniform distributionOf [0,1]
(0,p)
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Chromosome Fitness % of total1 6.82 312 1.11 53 8.48 384 2.57 125 3.08 14Total 22.0 100
Acknowledgement: http://www.edc.ncl.ac.uk/highlight/rhjanuary2007g02.php/
Compare with Roulette Wheel Algorithm for Selection
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Roulette Wheel SelectionRoulette Wheel SelectionLet i = 1, where i denotes chromosome index;Calculate P(xi) using proportional selection;
sum = P(xi);
choose r ~ U(0,1);
while sum < r doi = i + 1; i.e. next chromosomesum = sum + P(xi);
endreturn xi as one of the selected parent;
repeat until all parents are selected
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Significance of “temperature” We have a pseudo temperature T As T increases so does T is a parameter in the algorithm When stuck in the local minima, we
increase the temperature The probability of going to a higher
energy state increases Shaken out of local minima Similar to annealing of metal
TK
stateE
Be)(
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Annealing of Metal
The metal should have a stable crystal structure so that it is not brittle
For this it is repeatedly heated and then cooled slowly