Simulated Annealing

Difficulty in Searching Global Optima

startingpoint

descenddirection

local minima

global minima

barrier to local search

Simulated Annealing

Origin:

The annealing process of heated solids.

Intuition:

By allowing occasional ascent in the search process, we might be able to escape the trap of local minima.

Consequences of the Occasional Ascents

Help escaping the local optima.

desired effect

Might pass global optima after reaching it

adverse effect

Simulated Annealing Inspired from ancient process of forging iron Annealing refers to the fast heating of a metal and then

cooling it slowly The method was first proposed by Metropolis (1953) Kirkpatrick et al. (1982) later improved the SA method

and applied it to optimization problems

Heat the solid state metal to a high temperature.

Cool it down very slowly according to a specific schedule.

If the heating temperature is sufficiently high to ensure random state and the cooling process is slow enough to ensure thermal equilibrium, then the atoms will place themselves in a pattern that corresponds to the global energy minimum of a perfect crystal.

Simulated Annealing

Simulated Annealing SA differs from hill climbing in such a way

that a move is selected at random and then it is decided whether to accept it.

In SA, better moves are always accepted. Worse moves are accepted probabilistically.

Simulated Annealing The probability of accepting a worse state

is a function of both the temperature of the system and the change in the cost function

As the temperature decreases, the probability of accepting worse moves decreases

If T=0, no worse moves are accepted (i.e. hill climbing)

Control of Annealing Process

Acceptance of a search step (Metropolis Criterion):

Assume the performance change in the search direction is .

Accept a ascending step only if it pass a random test,

Always accept a descending step, i.e. 0

1,0exp randomT

At each temperature, search is allowed to proceed for a certain number of steps, L(k).

Control of Annealing Process

Cooling Schedule:

T, the annealing temperature, is the parameter that control the frequency of acceptance of ascending steps.

We gradually reduce temperature T(k).

The choice of parametersis called the cooling schedule.

kLkT ,

SA Glossary Solution: An answer to a problem without

respect to its apparent “value” Neighbor: A solution which is “next” to a given

solution in the solution space (or neighborhood) Fitness: The “value” of a solution Temperature: The current average rate at

which less fit solutions are accepted Annealing Schedule: The function which

lowers (or raises) the temperature the algorithm uses during its search

SA (Pseudo Code)

Algorithm SIMULATED-ANNEALINGBegin

temp = INIT-TEMP;place = INIT-PLACEMENT;while (temp > FINAL-TEMP) do

while (inner_loop_criterion = FALSE) donew_place = PERTURB(place);ΔC = COST(new_place) - COST(place);if (ΔC < 0) then

place = new_place;else if (RANDOM(0,1) < e-(ΔC/temp)) then

place = new_place;temp = SCHEDULE(temp);

End.

What Annealing Schedule? This is a complicated question!

There are linear schedules, geometric schedules, oscillating schedules, schedules based on the rate of improvement

Some SA algorithms use a GA to design their annealing schedule!

A very basic, yet effective, annealing schedule is one that is geometric, but stepped T = T * 0.9996 However, this is only done every M iterations, which is

either determined by the rate of improvement or by a hard figure (say every 1000 iterations)

Quantitative analysis is the only way to determine the most effective schedule

Cooling Schedule

Speed at which temperature is reduced has an effect

Too fast and the optima are not found

Too slow and time is wasted

1) A beginning temperature.2) An ending temperature.3) An assumption of the number of iterations per temperature.4) A cooling rate.

SA Parameters

Starting Temperature Must be hot enough to allow moves to

almost every neighborhood state Must not be so hot that we conduct a

random search for a long period of time Problem is finding a suitable starting

temperature

Implementation of Simulated Annealing

Select a local search scheme

Determine the cooling schedule

For example:

• Set L = n, the number of variables in the problem.

• Set T(0) such that exp(-/T(0)) 1.• Set T(k+1) = T(k), where is a

constant smaller but close to 1.