MAE 552 – Heuristic Optimization
Lecture 6
February 6, 2002
http://www.stockheim.net/cosa.html
Simulated Annealing
A psuedo-code of this algorithm might look like this.
T=current temperature
Do i=1,k
Generate a random displacement for a particle.
Calculate the change in energy, E = E’-E
If (E0) then
it’s a downhill move to lower energy so accept and update configuration
else
it’s an uphill move so generate random number P’[0,1]
compare with Pr(E)=exp(- E/KBT)
if (P’<Pr(E) then
accept move and update configuration
else
reject move – keep original configuration
endif
endif
enddo
The SA Algorithm
• SA is a application of the Metropolis Algorithm to
function optimization.
• It assumes a similarity between the physical annealing
of a solid and the global optimization of a function by
the following:
1. The value of an objective function can be viewed as
the energy of a solid.
2. The values of the Design Variables can be viewed
as the configuration of the particles of a solid.
• So, optimizing a function is analogous to finding the
ground state of a solid.
The SA Algorithm
•A parameter, T, called the control parameter is used in
place of the temperature as the Metropolis algorithm is
used for function optimization.
•In physical annealing, T has a true physical meaning the temperature of the material undergoing the annealing
process.
•In function optimization, the parameter T, is simply an
artificial control parameter that governs both the jumps
that move out of local minima and the search for the
global optimum.
•SA can be considered as a sequence of Metropolis
algorithms evaluate for a decreasing sequence of the
control parameter, T.
The SA Algorithm
1. For a high value of T, the objective is ‘melted’ and thus most
uphill moves are accepted which allows a large-scale random
search to be performed.
2. As the value of T decreases, fewer uphill moves are accepted.
At this stage, searches are confined to a smaller region of the
design space and the hill-jumping behavior is somewhat limited.
However some local optima can still be avoided.
3. As the control parameter, T, approaches zero, almost no uphill
moves are accepted and the solution almost ‘frozen’ to its final
form. At this stage, SA acts like a traditional downhill only
technique.
The SA Algorithm
•At each value of the control parameter, SA accepts or
rejects a new configuration by using the Metropolis
algorithm.
•The difference between the values of the evaluation
function at two configurations is:
f = f (X’) - f (X)
X is the latest accepted solution.
X’ is the trial configuration
The SA Algorithm
• If f  0 - Accept the new configuration and use as a
starting point for your next move.
• If f > 0:
Generate a random number P’=U[0,1]
Calculate the probability of acceptance of the move
Pr( f )  e
(  f / Tk )
Where Tk is the kth value of the control parameter after the
starting value of the control parameter.
• If P’ < Pr(f ), the new configuration is accepted,
otherwise it is rejected.
The SA Algorithm
•To achieve ‘thermal equilibrium’ at each value of the
control parameter, the SA process must go through
sufficiently many iterations for the objective function to
reach a steady state.
•Then as the control parameter approaches zero, the
algorithm converges asymptotically to the global
optimum.
The SA Algorithm
T0:
m10, m20, m30, m40, …………………………………mm0
T1:
m11, m21, m31, m41, …………………………………mm1
T2:
m12, m22, m32, m42, …………………………………mm2
T3:
m13, m23, m33, m43, …………………………………mm3
T4:
m14, m24, m34, m44, …………………………………mm4
T5:
m15, m25, m35, m45, …………………………………mm5
…..
Tn:
m1n, m2n, m3n, m4n, …………………………………mmn
n=number of levels in cooling schedule
m=number of transitions in each Markov chain
The SA Algorithm
The following musty be specified in implementing SA:
1. An unambiguous description for the evaluation function
(analogous to energy) and possible constraints.
2. A clear representation of the design vector (analogous to
the configuration of a solid) over which an optimum is
sought.
3. A ‘cooling schedule’ – this includes the starting value of
the control parameter, To, and rules to determine when
the current value of the control parameter should be
reduced and by how much (‘the decrement rule’) and a
stopping criterion to determine when the optimization
process should be terminated.
The SA Algorithm
4. A ‘move set generator’ which generates candidate
points.
5. An ‘acceptance criterion; which decides whether or
not a new move is accepted.
• Steps 4 and 5 together are called a ‘transition
mechanism’ which results in the transformation of a
current state into a subsequent one.
The SA Algorithm
4. A ‘move set generator’ which generates candidate
points.
5. An ‘acceptance criterion; which decides whether or
not a new move is accepted.
• Steps 4 and 5 together are called a ‘transition
mechanism’ which results in the transformation of a
current state into a subsequent one.
The SA Algorithm
•The SA algorithm is outlined as follows:
Step 1
Input the starting value of the control parameter
(temperature) Tk and set k=0.
Step 2
Choose a starting point (initial configuration) X0 and
calculate the value of the objective function (energy) at
X0, f(X0). Then set X=X0 and f(X)=f(X0).
The SA Algorithm
Step 3
Use the transition mechanism to generate a random point X’ and compute
f(X’).
Evaluate f = f(X’) – f(X).
If f 0:
accept X’,
set X=X’,
set f(X)=f(X’);
else
generate a random number P’ from [0,1],
compare P’ with Pr(f )=exp(- f /Tk),
if P’<Pr(f ):
accept X’,
set X = X’,
set f(X)=f(X’);
else: reject X’ and keep the original point;
endif;
endif;
The SA Algorithm
Step 4
Use the cooling schedule to decide if the steady state
(thermal equilibrium) of the objective
Function has been reached at the current value of the
control parameter.
If it is true:
reduce the control parameter by the decrement rule,
set k = k+1
else:
go to Step 3
endif.
The SA Algorithm
Step 5
Use the stopping criterion to decide if the simulated
annealing algorithm has to be terminated.
If it is true:
stop;
else:
go to Step 3;
endif.
The SA Algorithm
Common Stopping Criteria
1. If Xbest does not change for successive Markov
chains, then stop.
2. Fixed length cooling schedule – algorithms
automatically stops when T reaches a certain level
3. Maximum number of function evaluations
The SA Algorithm
• 2 Loops in the SA Algorithm
•There is an inner loop that generates a sequence of
trial points unit “thermal equilibrium” is reached at
that value of the control parameter.
•There is also an outer loop that constantly decreases
the control parameter and checks to see if the
optimization process should be terminated.
A
B
C
• Start with a ball at point A. Shake it up and it might jump out of A and
into B.
•Give it another shake (adding energy) and it might go to C.
•This is the general idea behind SAs.
The SA Algorithm – Convergence Issues
• Since the convergence of an stochastic algorithm is
asymptotic SA asymptotically obtain a global optimum
with probability of 1.
• The larger the number of samples, the higher the
probability of the algorithm finding the global
optimum.
• In general an infinite number of moves is required to
obtain the exact global optimum.
• In practical implementations, this is not realizable and
the asymptotic convergence must be approximated.
This is done using a proper cooling mechanism which
will be discussed next.
Cooling Schedules
•
A cooling schedule is used to achieve convergence to a global
optimum in function optimization.
Cooling schedule describes how control parameter T changes
during optimization process.
First let us look at the concept of acceptance ratio, X(Tk).
X(Tk) = (# of Accepted Moves / # of Attempted Moves)
•
•
•
If T is large almost all moves are accepted
–
•
As T decreases:
–
•
X(Tk)->1
X(Tk)->1
For maximum efficiency, it is important to set the proper
value of To.
Simulated Annealing – Cooling Schedule
Steps in a cooling schedule:
1. Choose the starting value of the control parameter, T0.
• It should be large enough to melt the objective function,
to leap over all peaks.
• This is accomplished by ensuring that the initial X(T0) is
close to 1.0 (most random moves are accepted).
2. Start the SA Algorithm
• At some T0 and execute for some number of transitions
and check X(T0).
• If not close to 1.0 multiply Tk by a factor greater than 1.0
and execute again.
• Repeat until X(T0) close to 1.0.