Simulated Annealing

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Simulated Annealing

• General Idea
• Start with an initial solution
• Generate a neighboring solution.
• If the neighbor is better, move to it.
• If the neighbor is not better move to it with
  some probability, else stay put.
• Repeat step 2 for n iterations, however
  continually reduce the probability of moving to a
  poorer neighbor over time.
• The continuous reduction in the probability of
  moving to a poorer neighbor over time effectively
  reduces the annealing process to a local
  improvement in the end.

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Simulated Annealing
Issue 1 What to use as the initial solution
x? Options a) Heuristic b) Randomly
generated Issue 2 What defines a neighbor or
neighborhood? Same as local improvement. Issue
3 Search strategy. Randomly generate a neighbor
or simple programmatic approach.
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Simulated Annealing
Issue 4 Evaluation Function How do you know
the adjacent (or neighboring) solution is better
or not? Speed need efficient mechanism to
evaluate solutions.
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Simulated Annealing
Issue 5 Probability Function - What probability
function to use for determining whether to move
to a poorer solution or not? Let V be the value
of the current solution and V the value of the
neighboring solution. Then Johnson et al
proposed using the following for minimization
problem, let D V V if D lt 0, then
downhill move, so take it. if D gt 0, then uphill
move, move with probability e-D/T, where T is
some value referred to as the temperature.
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Simulated Annealing
"e(-d/T) Temp Temp Temp Temp Temp Temp Temp Temp Temp
delta 100.0 75.0 56.3 42.2 31.6 23.7 17.8 13.3 10.0
0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
5 0.95 0.94 0.91 0.89 0.85 0.81 0.76 0.69 0.61
10 0.90 0.88 0.84 0.79 0.73 0.66 0.57 0.47 0.37
15 0.86 0.82 0.77 0.70 0.62 0.53 0.43 0.33 0.22
20 0.82 0.77 0.70 0.62 0.53 0.43 0.33 0.22 0.14
25 0.78 0.72 0.64 0.55 0.45 0.35 0.25 0.15 0.08
30 0.74 0.67 0.59 0.49 0.39 0.28 0.19 0.11 0.05
35 0.70 0.63 0.54 0.44 0.33 0.23 0.14 0.07 0.03
40 0.67 0.59 0.49 0.39 0.28 0.19 0.11 0.05 0.02
45 0.64 0.55 0.45 0.34 0.24 0.15 0.08 0.03 0.01
50 0.61 0.51 0.41 0.31 0.21 0.12 0.06 0.02 0.01
55 0.58 0.48 0.38 0.27 0.18 0.10 0.05 0.02 0.00
60 0.55 0.45 0.34 0.24 0.15 0.08 0.03 0.01 0.00
65 0.52 0.42 0.31 0.21 0.13 0.06 0.03 0.01 0.00
70 0.50 0.39 0.29 0.19 0.11 0.05 0.02 0.01 0.00
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Simulated Annealing
Issue 6 Cooling Function How to reduce the
temperature as the annealing process
runs. After n iterations, set T rT, where r lt
1. r is typically set somewhere between .95
and .8
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Simulated Annealing
The classic Johnson, et al. simulated annealing
algorithm 1. Get an initial solution S. 2. Get
an initial temperature T gt 0. 3. While not yet
frozen do the following 3.1 Perform the
following loop l time. 3.1.1 Pick a random
neighbor S of S. 3.1.2 Let D cost(S)
cost(S) 3.1.3 If D lt 0 (downhill move), Set S
S. 3.1.4 If D gt 0 (uphill move), Set S S
with probability e-D/T. 3.2 Set T rT
(reduce temperature). 4. Return S.
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Simulated Annealing
Issue 7 What should the initial temperature T
be? Large enough to make some uphill moves.
This becomes problem instance specific. Could be
found programmatically by finding comparing the
value of S to the value of several neighboring
solutions and then ensuring that where X is
some value say 0.8 or 0.9
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Simulated Annealing
Issue 8 How do you perform the probability test
to see if an uphill move is to be made? Generate
a random number R between 0 and 1. If,
then set S S, move uphill. else,
do not move. Sample C code
x (rand( )1000)/1000.0 if(x lt
exp(-float(result - finalreport-gtBestLmax)/Temp))
Accepted newjoblist
true else
rev_seq_switch(Mach_1,Seq_1,CenterList,offset,n)
NotAccepted
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Simulated Annealing

• Issue 9 How do you know if the process if
  frozen?
• Rules of thumb
• - If loop 3 completes 3 times without any move
  being accepted (uphill or downhill).
• - No move being accepted after M iterations.
• Minimum temperature reached.
• Issue 10 How many iterations per temperature, or
  what is the value of l?
• Use experimentation to determine good value.
  Possible numbers are 50,000 or 100,000, or
  200,000, etc

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Simulated Annealing

• Extensions to Johnson et al. algorithm
• Alternately heat up and cool down the process.
• Use a re-centering approach. For example when
  the process has become frozen, replace the
  current solution with the best solution found so
  far, S S, then restart the annealing process
  by setting the T back to the original
  temperature.
• Use a different probability function.
• Vary the number of iterations l as a function of
  temperature T.
• Others?