General simulated annealing algorithm

Minimizes a function with the method of simulated annealing

anneal Minimizes a function with the method of simulated annealing (Kirkpatrick et al., 1983)

ANNEAL takes three input parameters, in this order:

LOSS is a function handle (anonymous function or inline) with a loss function, which may be of any type, and needn't be continuous. It does, however, need to return a single value.

PARENT is a vector with initial guess parameters. You must input an initial guess.

OPTIONS is a structure with settings for the simulated annealing. If no OPTIONS structure is provided, anneal uses a default structure. OPTIONS can contain any or all of the following fields (missing fields are filled with default values):

Verbosity: Controls output to the screen.
0 suppresses all output
1 gives final report only [default]
2 gives temperature changes and final report
Generator: Generates a new solution from an old one. Any function handle that takes a solution as input and gives a valid solution (i.e. some point in the solution space) as output. The default function generates a row vector which slightly differs from the input vector in one element: @(x) (x+(randperm(length(x))==length(x))*randn/100). Other examples of possible solution generators: @(x) (rand(3,1)): Picks a random point in the unit cube. @(x) (ceil([9 5].*rand(2,1))): Picks a point in a 9-by-5 discrete grid.
Note that if you use the default generator, ANNEAL only works on row vectors. For loss functions that operate on column vectors, use this generator instead of the default: @(x) (x(:)'+(randperm(length(x))==length(x))*randn/100)'
InitTemp: The initial temperature, can be any positive number. Default is 1.
StopTemp: Temperature at which to stop, can be any positive number smaller than InitTemp. Default is 1e-8.
StopVal: Value at which to stop immediately, can be any output of LOSS that is sufficiently low for you. Default is -Inf.
CoolSched: Generates a new temperature from the previous one. Any function handle that takes a scalar as input and returns a smaller but positive scalar as output. Default is @(T) (.8*T).
MaxConsRej: Maximum number of consecutive rejections, can be any positive number. Default is 1000.
MaxTries: Maximum number of tries within one temperature, can be any positive number. Default is 300.
MaxSuccess: Maximum number of successes within one temperature, can be any positive number. Default is 20.


Usage:
[MINIMUM,FVAL] = ANNEAL(LOSS,NEWSOL,[OPTIONS]);
MINIMUM is the solution which generated the smallest encountered value when input into LOSS.
FVAL is the value of the LOSS function evaluated at MINIMUM.
OPTIONS = ANNEAL();
OPTIONS is the default options structure.


Example:
The so-called six-hump camelback function has several local minima in the range -3<=x<=3 and -2<=y<=2. It has two global minima, namely f(-0.0898,0.7126) = f(0.0898,-0.7126) = -1.0316. We can define and minimise it as follows:
camel = @(x,y)(4-2.1*x.^2+x.^4/3).*x.^2+x.*y+4*(y.^2-1).*y.^2;
loss = @(p)camel(p(1),p(2));
[x f] = anneal(loss,[0 0])
We get output:
Initial temperature: 1
Final temperature: 3.21388e-007
Consecutive rejections: 1027
Number of function calls: 6220
Total final loss: -1.03163
x =
-0.0899 0.7127
f =
-1.0316
Which reasonably approximates the analytical global minimum (note that due to randomness, your results will likely not be exactly the same).