Simulated Annealing Simulated annealing is a probabilistic search - - PDF document

Simulated Annealing

Simulated Annealing

• Simulated annealing is a probabilistic search

algorithm.

• The terminology is borrowed from the physics

literature.

• For the simulated annealing algorithm we need

– a scoring function, – a move set, – a selection probability, – an acceptance function.

1

Simulated Annealing

The Traveling Salesman Problem

• S

Which is the shortest path that connects all cities?

2

Simulated Annealing

The Traveling Salesman Problem (cont.) · · · 43 44 35 36 37 47 46 56 55 45 34 24 14 13 · · ·

• · · · 43 44 45 55 56 46 47 37 36 35 34 24 14 13 · · ·

↑ ↑

• 3

Simulated Annealing

Properties of our Simulated Annealing run

• We use the acceptance function

α(ǫold, ǫnew, t) = min {1, exp ([ǫold − ǫnew] /t)}

• We run homogeneous Markov chains at constant

temperatures.

• We constructed the move set to be irreducible

and aperiodic, therefore each homogeneous Markov chain has a limiting distribution πt(S).

• The limit (as t → 0) of those distributions assigns

probability 1 to the optimal scoring states.

• With limited resources, we cannot guarantee to

find an optimal scoring state.

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Simulated Annealing

The Traveling Salesman Problem (cont.)

log10(temperature) score 100 200 300 400 500 3 2 1

• 1
• 2
• 5

Simulated Annealing

The Traveling Salesman Problem (cont.)

• The initial path.
• The tour after 60 steps.
• The tour after 90 steps.
• The tour after 125 steps.

6

Simulated Annealing

The Traveling Salesman Problem (cont.)

score 100 200 300 400 500 600 0.0 0.05 0.10 0.15 1 55 60 65 70 75 80 85 90

Anything noteworthy in the score distributions?

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Simulated Annealing

The Traveling Salesman Problem (cont.)

mean [scores] 100 200 300 400 500

• log10(temperature)

deviation [scores] 5 10 15 20 3 2 1

• 1
• 2
• The mean and the standard deviation of the scores.

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Simulated Annealing

The Traveling Salesman Problem (cont.)

mean [scores] 100 200 300 400 500

• log10(temperature)

deviation [scores] 5 10 15 20 3 2 1

• 1
• 2
• Normal scores with mean µ(t) = µ∞ − σ2

∞

t

and standard deviation σ(t) = σ∞.

9

Simulated Annealing

The Traveling Salesman Problem (cont.)

mean [scores] 100 200 300 400 500

• log10(temperature)

deviation [scores] 5 10 15 20 3 2 1

• 1
• 2
• Gamma scores with mean µ(t) = µ∞ − σ2

∞

T

• 2 − t

T

• and standard deviation σ(t) = σ∞ t

T added.

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Simulated Annealing

The Traveling Salesman Problem (cont.) There is a region of weak and a region of strong control! Let µ∞ be the mean and σ∞ be the standard deviation at temperature t = ∞. t ≥ T te < t ≤ T µ(t) µ∞ − σ2

∞

t

µ∞ − σ2

∞

T

• 2 − t

T

• σ(t)

σ∞ σ∞ t

T

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Simulated Annealing

The Traveling Salesman Problem (cont.)

• iterations

proportions 0.0 0.2 0.4 0.6 0.8 1.0 1000 10000 100000

• Iterations versus success rate on a slimmed down

version of the algorithm.

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