Adaptive Simulated
Annealing
Final Project Report
ECE 556: Design Automation of Digital
Systems
Professor Yu Hen Hu
Greg Knowles
12/16/04
Introduction (Problem statement, Motivation)
Simulated Annealing (SA) is a heuristic for minimizing hard combinatorial
optimization problems. Dating back to 1953 when Metropolis proposed
(SA) to Monte Carlo was modelled after a thermal equilibrium equation
representing the metal annealing process. The process was then developed
by Kirkpatrick, Gelatt and Vecchi, in “Optimization by Simulated
Annealing”, IBM Research Report RC 9355 (1992) and the Metropolis
heuristic became what we know today as Simulated Annealing. The
technique has proven its capability to minimize complicated n-dimensional
cost functions very efficiently.
Simulated Annealing is not a replacement for a heuristic designed
specifically for the problem at hand. Each problem is different and a method
tuned to minimize a specific function, such as Newton Minimization for a
parabola will almost always work better. SA is a randomized heuristic with
rules to maximize the quality of its guesses, and methods of escaping local
minima “traps” to continue exploring the rest of the solution set. SA is also
excellent at solving real time continuous optimization problems. Real Time
data can be fed into the system and analyzed to find the ideal adjustment
parameters to keep a system stable.
SA operates on two basic principles; all moves that improve the
solution are always accepted, and moves that increase the solution cost are
accepted with low probability according to the Metropolis criterion. This
criterion is based on the Boltzmann average [e-(Ei-Ej)/kT]. This average is
compared to a random number, R, between 0.0 and 1.0
 If R ≤ e-(Ei-Ej)/kT, the new solution is accepted
 If R > e-(Ei-Ej)/kT, the new state is rejected
This probability of accepting worse solutions is known as “hillclimbing” and
it allows for the escape from local minima in hopes of finding a global
minimum.
Simulated Annealing has a number of very important parameters that
change with every problem. When these parameters are carefully chosen,
SA can produce excellent solutions with great efficiency. However, these
parameters are hard to guess if you don’t know much about the system to
begin with. This paper introduces some techniques that were employed into
adaptive software that dynamically changes the critical parameters based on
the data samples, as they are collected in an attempt to reach a global
minimum value as quickly as possible.
Method, algorithm
There are five key parameters to focus on:
 Ti (Initial Temperature)
 Tf (Final Temperature)
 Alpha (Cooling Rate)
 Iterations (Solutions generated at each Temperature)
 Stop conditions (“Freeze” Detection)
Both the initial and final temperatures have closed form solutions, but still
require additional information. The closed form solutions are dependent on
selecting an acceptance probability for any given increase in cost. How
should one go about determining what the ideal probability is to accept a
given worse solution?
kTi = -dEi/ln(Pi)
kTf = -dEf/ln(Pf)
Intuition tells us that we want to accept bad moves more frequently at the
beginning and start accepting them less at the end. We still lack prerequisite
knowledge of the range of our system, and the only way to find that out is by
testing a few solutions. Once we get a feel for how big is big and how little
is little, then we will know what probabilities to choose.
The next two parameters are intimately related. Alpha is the rate at
which the system cools, and the iteration counter keeps track of how many
solutions to generate at each temperature. On closer review, one will notice
that these two parameters do essentially the same thing in different ways. A
steep temperature slope will send you down to zero quickly, so you want to
take a lot of samples at each temperature plateau. If one were to use a mild
temperature slope, it wouldn’t be necessary to take as many measurements at
each spot. The advantage to the latter scenario is a much more contoured
graph that is better suited for curve fitting. Rapid temperature drops result in
block like charts with no smooth lines.
The number of iterations taken at high temperatures is less important
then taking several iterations on the breaking point of the solution set. The
breaking point is the sweet spot of every function that produces a 40-60%
ratio of accepted moves. This is the period of time when the “best” found
solution moves the quickest. Once the solution set has been adequately
sampled, the temperature should continue to decrease until the system
becomes frozen. Once there are essentially no moves being accepted, it is
up to your stopping criteria to decide when to cut the cord.
Stopping criteria are very important because the majority of the
iterative calculations performed are at low temperatures. Lots of
computation time will be wasted if the algorithm is let to run to zero. Then
again, the system could be hiding a very isolated solution that has far better
costs, and would only be detected with an extremely thorough search of that
local minimum.
It should be obvious now that there is no step – by – step best way to
perform these calculations and estimates. Professor Lester Ingber of
CalTech has spent over twenty years researching ASA and has published
several papers on the topic as well as releasing source code to optimize
almost any problem that can be modelled with mathematical constraints.
Martial Arts for instance is one of his focus areas. Big Blue will really have
to step up to the challenge this time around.
Program structure and highlight
The attached code is based on optimizing a two
dimensional cost function. The perturb function is
simply a random point generator that is more likely to
produce points near the current solution then further
away. The ideal perturb function is capable of
reaching every solution in the set from every other
point. These criteria do not hold true here, as only
one dimension can be changed at a time, however any
given point can be reached in exactly two moves from
every other point with the encoded perturb function.
This implementation of ASA generates a completely
random initial solution. Research has proven that
given a sufficiently high initial temperature the
outcome is uncorrelated with the initial solution. The
algorithm then performs a search for the ideal
temperature based on the acceptance rate. The
algorithm will even increase the temperature if the
starting one was too low. Once the ideal temperature
has been reached, the only decision left to be made is
how low your temperature should go before stopping.
This algorithm picks a stopping point proportional to
the optimal temperature found.
This code highlights a Comma Separated Value (CSV)
output. The output file can be read by any modern
spreadsheet analysis software package. This allows for
quick graphical interpretation of your results, as well
as a report on how well the adaptive components are
functioning.
Results: Performance
Since these test points are completely random in
nature, it is not uncommon to get “unlucky” and have
the adaptive system perform to different levels of
effectiveness on every successive run. This particular
data run displays near ideal performance based on the
goals that were set forth in this paper.
Temp
400000
350000
300000
250000
200000
150000
100000
50000
0
1
10
19
28
37
46
55
64
73
82
91
100 109 118 127 136 145 154 163 172
Alpha
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1
9
17
25
33
41
49
57
65
73
81
89
97 105 113 121 129 137 145 153 161 169
The two charts on the previous page show the temperature and the cooling
rate parameters. Notice the temperature drops every time the cooling rate,
alpha migrates from zero. Also notice the initial temperature was decided to
be too low, and was rapidly raised to an appropriate level. The temperature
was not directly changed. It was only affected by modifying the cooling rate
parameter, alpha. The value for alpha was decided with trigger levels on the
acceptance ratio. The temperature is raised until 60% of solutions are
accepted, and then slowly lowered while in the 60-40% range, before being
more noticeably decremented at acceptance rates lower then 40%.
AcceptRatio
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
9
17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145 153 161 169
What may be more interesting to some is the quality of the solutions
being generated and accepted. Again at higher temperatures, bad solutions
are accepted more frequently, and at lower temperatures, a local minima of
solutions will have been located, and each new solution selected will be a
good one. The graph shown happens to start out with very good solutions,
and this is due to luck in picking a good local minimum when the
temperature was at a low value. Typically the graph would start at a very
high value and work its way down following the behavior observed in the
remainder of the graph. The shape of this graph directly shows a linear
representation of the path chosen throughout the solution set on this run. A
few million runs like this could potentially create a complete graph of the
solution space, however this is often impractical in real world applications
involving much more then two dimensions of flexibility.
AvgCost
300000
250000
200000
150000
100000
50000
0
1
10
19
28
37
46
55
64
73
82
91 100 109 118 127 136 145 154 163 172
Iterations
1200000
1000000
800000
600000
iter
400000
200000
0
1
11
21
31
41
51
61
71
81
91
101 111 121 131 141 151 161 171
The number of iterations chart and best-found solution chart look like
complete opposites from each other in this example. As the complexity and
resolution of the solution space increases, both of these graphs will move
inward, requiring longer to converge to a global minimum. The iteration
count may need to start getting turned up higher at an earlier temperature if
the solution is not stabilizing.
Best Cost
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
1
10
19
28
37
46
55
64
73
82
91 100 109 118 127 136 145 154 163 172
References
S. Nahar, S. Sahni, and E. Shragowitz. Experiments
with simulated annealing. Proceedings of 22nd
Design Automation Conference(1985) pp. 748-752.
S. Kirkpatrick, C.D. Gelatt and M.P. Vecchi, Science
220 (1983) 671-680.
N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth. A.H.
Teller and E. Teller, "Equation of State
Calculations by Fast Computing Machines," J. Chem.
Phys. 21 (1953) 1087-1092.
Cerny, V., "Thermodynamical Approach to the Traveling
Salesman Problem: An Efficient Simulation Algorithm",
J. Opt. Theory Appl., 45, 1, 41-51, 1985
Kirkpatrick, S., C. D. Gelatt Jr., M. P. Vecchi,
"Optimization by Simulated Annealing",Science, 220,
4598, 671-680, 1983.
Press, Wm. H., B. Flannery, S. Teukolsky, Wm.
Vettering, Numerical Recipes, 326-334, Cambridge
University Press, New York, NY, 1986.