Next: Gravity Modeling and Inversion Up: Optimization of Non-Linear Gravity Previous: Introduction
Next: Gravity
Modeling and Inversion Up: Optimization
of Non-Linear Gravity Previous: Introduction
The application of an iterative improvement algorithm presupposes the definition of configurations, a cost function and a generation mechanism, i.e. a simple prescription to generate a transition from a configuration to another one.
The GSA method leads to the minimization of a cost function 
through a stochastic cooling. The procedure consists in comparing the cost
function for two random vectors 
(or configuration) obtained using the generalized visiting distribution.
In this sense, the hypersurface defined by
is randomly visited and the algorithm terminates when a configuration is
obtained whose cost is no worse than any of its neighbors.
Differently of the gradient methods, solutions obtained by the SA, do not depend on the initial configuration and have a cost usually close to the minimum one. The simulated annealing is a technique that has attracted significant attention in a diversity of problems, especially those where a desired global extremum is hidden among many local extrema. The GSA algorithm can now be viewed as a sequence of generalized Metropolis algorithms evaluated at a sequence of decreasing values of the control parameter.
In summary, we describe the principal steps to built the numerical procedure
[5, 8]
in order to map the global minima of a function
depend on of a multidimensional continuous parameter 
(i) Fix the acceptance and visiting indexes 
and 
(we recall that 
and (1,2) respectively correspond to the Boltzmann and Cauchy machines).
Start, at t=1, with an arbitrary value 
and a high enough value 
(visiting temperature) and cool as follows:
where t is the discrete time corresponding to the computer iteration.
(ii) Then randomly generate each component 
( 
) of 
using the visiting distribution probability 
given by
where 
is the gamma function; D is the number of components of 
. This procedure assures that the system can both escape from any local
minimum and explore the entire cost function.
(iii) Then calculate the cost function 
with the following conditions:
If
, replace 
by 
If 
, accept 
with the acceptance probability 
defined by
with 
[5].
(iv) Calculate the new temperature 
using Eq.(1) and go back to (ii) until the minimum of 
is reached within the desired precision.
