1. Simulated Annealing

2. Iterative Improvement 1
General method to solve combinatorial optimization problems
Principle:
Start with initial configuration
Repeatedly search neighborhood and select a neighbor as candidate
Evaluate some cost function (or fitness function) and accept candidate if "better"; if not, select another neighbor
Stop if quality is sufficiently high, if no improvement can be found or after some fixed time

3. Iterative Improvement 2
Needed are:
A method to generate initial configuration
A transition or generation function to find a neighbor as next candidate
A cost function
An Evaluation Criterion
A Stop Criterion

4. Iterative Improvement 3
Simple Iterative Improvement or Hill Climbing:
Candidate is always and only accepted if cost is lower (or fitness is higher) than current configuration
Stop when no neighbor with lower cost (higher fitness) can be found
Disadvantages:
Local optimum as best result
Local optimum depends on initial configuration
Generally no upper bound on iteration length

5. Hill climbing

6. Convergence of simulated annealing

7. How to cope with disadvantages
Repeat algorithm many times with different initial configurations
Use information gathered in previous runs
Use a more complex Generation Function to jump out of local optimum
Use a more complex Evaluation Criterion that accepts sometimes (randomly) also solutions away from the (local) optimum

8. Simulated Annealing Use a more complex Evaluation Function:
Do sometimes accept candidates with higher cost to escape from local optimum
Adapt the parameters of this Evaluation Function during execution
Based upon the analogy with the simulation of the annealing of solids

9. Other Names
Monte Carlo Annealing
Statistical Cooling
Probabilistic Hill Climbing
Stochastic Relaxation
Probabilistic Exchange Algorithm

10. Analogy
Slowly cool down a heated solid, so that all particles arrange in the ground energy state
At each temperature wait until the solid reaches its thermal equilibrium
Probability of being in a state with energy E :
Pr { E = E } = 1/Z (T ) . exp (-E / kB.T )
E Energy
T Temperature
kB Boltzmann constant
Z(T) Normalization factor (temperature dependant)

11. Simulation of cooling (Metropolis 1953)
At a fixed temperature T :
Perturb (randomly) the current state to a new state
E is the difference in energy between current and new state
If E < 0 (new state is lower), accept new state as current state
If E  0 , accept new state with probability
Pr (accepted) = exp (- E / kB.T)
Eventually the systems evolves into thermal equilibrium at temperature T ; then the formula mentioned before holds
When equilibrium is reached, temperature T can be lowered and the process can be repeated

12. Performance
SA is a general solution method that is easily applicable to a large number of problems
"Tuning" of the parameters (initial c, decrement of c, stop criterion) is relatively easy
Generally the quality of the results of SA is good, although it can take a lot of time
Results are generally not reproducible: another run can give a different result
SA can leave an optimal solution and not find it again (so try to remember the best solution found so far)
Proven to find the optimum under certain conditions; one of these conditions is that you must run forever

13. Theoretical Results
If the temperature T is kept constant,
then the transition matrix,Pij , is independent of the iteration number.
Correspond to a homogeneous Markov chain
The stationary distribution q(i) , corresponds to Boltzmann distribution.
The limit as T→0 of this stationary distribution is a uniform distribution over set of optimal solutions.
The SA algorithm converges to the set of globally optimal solutions.
But this result does not tell us anything about the number of iterations required to achieve convergence.

14. Algorithm 2. Select an initial temperature T>0;
1.      Select an objective function E(x);
2.      Select an initial temperature T>0;
3.      Set temperature change counter t=0;
4.      Repeat
4.1  Set repetition counter n=0;
4.2  Repeat
4.2.1   Generate state xi+1, a neighbor of xi;
4.2.2   Calculate E=E(xi+1)E(xi);
4.2.3   If E<0 then xi=xi+1;
4.2.4   else if random(0,1)<exp(E/T) then xi=xi+1;
4.2.5   n=n+1;
4.3  Until n=r(t);
4.4  t=t+1;
4.5  T=T(t);
Until stopping criterion true. 

15. Kirkpatrick et al. (1983) Generic Choices
Set initial value T to be large enough
T(t+1)= T(t),  :0.8~0.99
N(t): a sufficient number of transitions
corresponding to thermal equilibrium
constant
or proportional to the size of the neighborhood
Stopping criterion
when the solution obtained at each temperature change is unaltered for a number of consecutive temperature changes. (T…T/10)

16. SA Cooling Schedule
Starting Temperature
Final Temperature
Temperature Decrement
Iterations at each temperature

17. SA Cooling Schedule - Starting Temperature
Must be hot enough to allow moves to almost neighbourhood state (else we are in danger of implementing hill climbing)
Must not be so hot that we conduct a random search for a period of time
Problem is finding a suitable starting temperature

18. SA Cooling Schedule - Final Temperature
Final Temperature - Choosing
It is usual to let the temperature decrease until it reaches zero However, this can make the algorithm run for a lot longer.
Therefore, the stopping criteria can either be a suitably low temperature or when the system is “frozen” at the current temperature (i.e. no better or worse moves are being accepted)

19. SA Cooling Schedule - Temperature Decrement
Theory states that we should allow enough iterations at each temperature so that the system stabilises at that temperature
Unfortunately, theory also states that the number of iterations at each temperature to achieve this might be exponential to the problem size
Doing a large number of iterations at a few temperatures, a small number of iterations at many temperatures or a balance between the two

20. SA Cooling Schedule - Temperature Decrement
Linear
temp = temp - x
Geometric
temp = temp * x
Experience has shown that α should be between 0.8 and 0.99, with better results being found in the higher end of the range. Of course, the higher the value of α, the longer it will take to decrement the temperature to the stopping criterion

21. SA Cooling Schedule - Iterations
Iterations at each temperature
A constant number of iterations at each temperature
The formula used by Lundy is
t = t/(1 + βt)
where β is a suitably small value

22. Conclusions It is very easy to implement.
It can be generally applied to a wide range of problems.
SA can provide high quality solutions to many problems.
Care is needed to devise an appropriate neighborhood structure and cooling scheduler to obtain an efficient algorithm.