3. 30/01/15 3
Formal definition
Simulated annealing – is a technique of
optimization based on the analogy between
the way the metal cools and freezes in a
minimum energy of the crystalline structure
(the annealing process) and the search for a
minimum in a more general system.
Netreba Kirill, SPbSPU
Simulated AnnealingIntroduction
4. 30/01/15 4
Natural motivation
Properties of structure depend on cooling factor after the substance was
heated to melting point. Slow cooling – large crystals are formed, that is useful
for a substance structure. Spasmodic cooling– the weak structure is formed.
«Agitation» at a heat is accompanied by high molecular activity in physical
system.
Disturbance
Disturbance
Netreba Kirill, SPbSPU
Introduction Simulated Annealing
5. 30/01/15 5
SA algorithm
The initial solution
For the majority of problems the
initial solution is casual.
Solution estimation
The solution estimation consists
of decoding of the current
solution and performance of the
necessary act, allowing to
fathom its expediency for the
solution of the given problem.
Casual search of the solution
Solution search begins with
copying of the current solution
in the working solution which is
any way inoculated further.
Create the initial
solution
Evaluate the solution
Change the solution
in a random way
Evaluate the new
solution
Criterion of the
admission
Reduce temperature
The current solution
The working solution
The best solution
Netreba Kirill, SPbSPU
Simulated Annealing
6. 30/01/15 6
Criterion of the admission
At this stage of algorithm two solutions are available.
First - the current solution, second - the working
solution. Certain energy (E) is connected with each
solution and represents its efficiency.
The working solution is accepted as the current
solution if :
In the beginning of search the temperature has the
greatest value and ξ is close to 1. Therefore the
sampling probability of the solution increasing value
of energy is great. Taking of such solutions
corresponds to movement to saddle point B, instead
of to minimum A. As approaching a global minimum
the temperature decreases and probability of
increase in energy drops.
Create the initial
solution
Evaluate the solution
Change the solution
in a random way
Evaluate the new
solution
Criterion of the
admission
Reduce temperature
The current solution
The working solution
The current solution
ð ò
/T
E E E 0
0 & r, e ,r [0,1]−∆
∆ = − ≤
∆ > ξ > ξ = ∈
Netreba Kirill, SPbSPU
SA algorithm Simulated Annealing
7. 30/01/15 7
Temperature decrease
After a number of iterations on algorithm
at the given temperature we reduce it.
There are a lot of alternatives of
decrease in temperature. Simple
function T=αT, 0<α<1 is usually used.
Other strategy of decrease in
temperature, including linear and
nonlinear functions are also possible.
Iteration
Several iterations are carried out at one
temperature. After iteration is finished
temperature reduceed. The process
continues until the temperature will not attain
null..
Create the initial
solution
Evaluate the solution
Change the solution
in a random way
Evaluate the new
solution
Criterion of the
admission
Reduce temperature
The current solution
The working solution
The current solution
Netreba Kirill, SPbSPU
SA algorithm Simulated Annealing
8. 30/01/15 8
The N queens puzzle is the problem of placing N chess queens on an N×N
chessboard so that none of them is able to capture any other using the
standard chess queen's moves.:
Netreba Kirill, SPbSPU
One of 92 solutions of 8 queens puzzle
Example Simulated Annealing
9. 30/01/15 9
Energy
Energy of the solution is defined as quantity of conflicts which appear in the coding. The
problem consists in finding the coding at which energy is equal to null (that is on a board
there are no conflicts).
Temperature
For the given problem solution search began with temperature 100° and gradually decreased
it to null, using formula T=αT. Thus value α = 0,98. Apparently from the schedule the
temperature shows at first sweeping decrease, and then a slow convergence to final
temperature - to null.
At each change of temperature we will execute 100 iterations. It will allow
algorithm to carry out some operations of search at each level.
Netreba Kirill, SPbSPU
Example of SA's realization
for a problem with 40 queens
100
80
60
40
20
0
0 50 100 150 200 250 300
Accepted
Energy
Temperature
Example Simulated Annealing
10. 30/01/15 10
Example of solution of 40 queens puzzle
Netreba Kirill, SPbSPU
Example Simulated Annealing
11. 30/01/15 11
Temperature
The initial temperature should be enough high to make possible
sampling of other areas of a range of solutions. If the maximum
distance between the next solutions is known it is easy to count
initial temperature:
The initial temperature also can be changed dynamically. If the
statistics on criterion of the admission of the worst solutions and a
finding of new best solutions is set, it is possible to raise
temperature until the necessary quantity of admission (opening of
new solutions) will be attained. This process is analogous to heating
of substance to its transition in the liquid form then already there is
no sense to raise temperature.
Final temperature. Though the zero is convenient final
temperature, geometrical function which is used in an instance,
shows, that the algorithm will work much longer, than it is really
necessary. Therefore the final temperature usually is accepted
hardly more null (for example, 0.5)
Netreba Kirill, SPbSPU
/T
e r (r [0,1], 0)−∆
ξ = > ∈ ∆ >
Настройка алгоритмаSimulated Annealing
12. 30/01/15 12
Advantages of annealing
absence of restrictions of the form of the minimizing function;
search of a global minimum;
efficiency in a solving of the various classes of problems
demanding optimization.
Annealing deficiencies
the demand of infinitely slow cooling, in practice meaning
slow work of algorithm;
complexity of tuning
Netreba Kirill, SPbSPU
Conclusion Simulated Annealing
13. 30/01/15 13
Ranges of application
way creation
image reconstruction
assignment routine and planning
network placement
global routing
detection and recognition of visual targets
design of special digital filters
Netreba Kirill, SPbSPU
Conclusion Simulated Annealing
