Simulated Annealing

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Simulated Annealing
http//www.math.iastate.edu/wu/math597.html
http//www.math.iastate.edu/wu/Math597HW0000/index
.htm

• Math/BCB/ComS597
• Zhijun Wu
• Department of Mathematics

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Simulated Annealing
Algorithm input initial x0 y0 f (x0) set x
x0 y y0 for T T0, T1, , Tm
(decreasing) for k 1, , n x1
perturb (x0) y1 f (x1)
dy y1 y0 e exp (- dy / T)
if (rand y1 end update x, y
end end
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Annealing Strategies

• Cooling Schedule
• Cooling Steps at Each Temperature
• Step Sizes
• Starting State at Each Temperature
• Local Minimization

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Simple Version
annealing2simple.m function x, y
annealing2simple (f, x0) y0 feval (f, x0) x
x0 y y0 alpha 0.9 T 2 s 2 for k 1
20 T alpha T accept 0 for l
1 100 x1 x0 (0.5 rand (size (x0)))
s y1 feval (f, x1) dy y1
y0 if rand x1 y0 y1 accept accept 1 if y0
accept 75, s
2 s end disp ( accept steps ,
num2str(accept)) disp (current energy ,
num2str(y)) end
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function f, g ljfun (x) n max (size (x)) /
3 f 0 g zeros(3n, 1) for i 1 n for
j i1 n i1 3(i-1) 1 i2 i1 1
i3 i2 1 j1 3(j-1) 1 j2 j1 1
j3 j2 1 r1 x(i1)-x(j1) r2
x(i2)-x(j2) r3 x(i3)-x(j3) rr r1 r1
r2 r2 r3 r3 r sqrt (rr)
r6 1 / rr / rr / rr f f (r6 - 2)
r6 dr - 12 (r6 - 1) r6 / r
g(i1) g(i1) dr r1 / r g(i2) g(i2)
dr r2 / r g(i3) g(i3) dr r3 /
r g(j1) g(j1) - dr r1 / r g(j2)
g(j2) - dr r2 / r g(j3) g(j3) - dr
r3 / r end end
Lennard-Jones Function
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Minimizing LJ Potential
singlerun.m rand (state, 0) x0 rand
(36, 1) x1, y1 annealing2simple (ljfun,
x0) x1 y1
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Decreasing the Step Size
annealing2simple.m function x, y
annealing2simple (f, x0) y0 feval (f, x0) x
x0 y y0 alpha 0.9 T 2 s 0.2 for k
1 20 T alpha T accept 0 for l
1 100 x1 x0 (0.5 rand (size
(x0))) s y1 feval (f, x1) dy
y1 y0 if rand x0 x1 y0 y1 accept accept 1
if y0 if accept
75, s 2 s end disp ( accept steps ,
num2str(accept)) disp (current energy ,
num2str(y)) end
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Reset Starting States
annealing2simple.m function x, y
annealing2simple (f, x0) y0 feval (f, x0) x
x0 y y0 alpha 0.9 T 2 s 0.2 for k
1 20 T alpha T accept 0 for l
1 100 x1 x0 (0.5 rand (size
(x0))) s y1 feval (f, x1) dy
y1 y0 if rand x0 x1 y0 y1 accept accept 1
if y0 if accept
75, s 2 s end disp ( accept steps ,
num2str(accept)) disp (current energy ,
num2str(y)) x0 x y0 y end
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Increasing Cooling Steps
annealing2simple.m function x, y
annealing2simple (f, x0) y0 feval (f, x0) x
x0 y y0 alpha 0.9 T 2 s 0.2 for k
1 20 T alpha T for j 1 10
accept 0 for l 1 100 x1 x0
(0.5 rand (size (x0))) s y1 feval
(f, x1) dy y1 y0 if rand (- dy / T) x0 x1 y0 y1 accept
accept 1 if y0 end end end if accept 2 end if accept 75, s 2 s end disp
( of accepted steps , num2str(accept))
disp (current energy level , num2str(y))
x0 x y0 y end end
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Multiple Runs
multiruns.m rand (state, 0) for i 1
5 x0 rand (36, 1) x1, y1
annealing2simple (ljfun, x0) options
optimset (GradObj, on) x2, y2 fminunc
(ljfun, x1, options) x2 y2 end
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Computer Project 3 General Policy

• Make your programs and related running results
  accessible from your homework page. The programs
  should be written in a relatively formal fashion,
  be easy to read, and contain minimal comments for
  major computing steps.
• The programs should run correctly on test data.
  If not before turned in, please explain a little
  bit what you have done and the problem you have
  not resolved.
• Discussion with classmates is encouraged but you
  are required to finish and turn in your own
  project separately.

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Computer Project 3 Due 6pm, Monday, December 12th

• Develop a global optimization code for the
  minimization of the Lennard-Jones potential using
  multistart, simulated annealing, or a method of
  your choice.
• Use your code to find the minimum energy
  configurations of the Lennard-Jones clusters with
  five to twenty-four atoms. Record your solutions
  and display them with RasMol.
• If you cannot find the minimum configuration of a
  specific cluster, you may change your code
  specifically to obtain the solution for the
  cluster. However, you need to describe the
  specific procedure.
• There are twenty clusters. You need to find the
  minimum energy configurations of all the clusters
  to earn the total twenty points for the project.
  Only global energy minima are counted.

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RasMol Input Format
COMPNDARG12 REMARK ATOM 1 O 1
0.186 -0.322 0.674 TER ATOM 2 O
1 -0.136 0.150 1.505 TER ATOM
3 O 1 0.896 0.251
0.244 TER ATOM 4 O 1 0.375
1.016 1.589 TER ATOM 5 O 1
1.160 -0.559 0.782 TER ATOM 6 O
1 0.570 -0.623 1.581 TER ATOM 7
O 1 1.012 1.079
0.810 TER ATOM 8 O 1 0.790
0.220 1.211 TER ATOM 9 O 1
0.627 0.225 2.161 TER ATOM 10 O
1 0.102 0.676 0.686 TER ATOM 11
O 1 1.683 0.329
0.869 TER ATOM 12 O 1 1.354
0.813 1.720 TER
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Global Energy Minima of LJ-Cluster 4 to 24
Cluster Size Global Energy
Minimum 3

-3.0000e00 4

-6.0000e00 5

-9.1040e00 6

-1.2712e01 7

-1.6505e01 8

-1.9822e01 9

-2.4113e01 10

-2.8420e01 11

-3.2765e01 12

-3.7967e01 13

-4.4327e01 14

-4.7845e01 15

-5.2323e01 16

-5.6816e01 17

-6.1318e01 18

-6.6531e01 19

-7.2660e01 20

-7.7117e01 21

-8.1685e01 22

-8.6810e01 23

-9.2844e01 24

-9.7349e01