Simulated Annealing methods have been applied successfully in the description of a variety of global extremization problems. GSA method have attracted significant attention due to their suitability for large scale optimization problems, especially for those in which a desired global minimum is hidden among many local minima. The basic aspect of the Simulated Annealing method is that it is analogous to thermodynamics, especially concerning the way that liquids freeze and crystalline, or that metals cool and anneal.
The GSA methods is based on the correlation between the minimization of a cost function (conformational energy) and the geometries randomly obtained through a slow cooling. In this technique, an artificial temperature is introduced and gradually cooled in complete analogy with the well known annealing technique, frequently used in metallurgy when a molten metal reaches its crystalline state (global minimum of the thermodynamics energy). In our case the temperature is intended as an external noise.
The artificial temperature (or set of temperatures) acts as an extremely convenient stochastic source for eventual detrapping from local minima. Near the end of the process the system hopefully is within the attractive basin of the global minimum. The challenge is to cool the temperature as fast as possible and still have the guarantee that no irreversible trapping at any local minimum has occurred. More precisely, we search for the quickest annealing (approaching a quenching) which maintains the probability of finishing within the global minimum equal to one.
The procedure to search the minima (global and local) or to map the
energy hyper surface consist in comparing the conformational energy for
two consecutive random geometries 
and 
obtained from the GSA routine.
is a N-dimensional vector that contains all atomic coordinates (N) to be
optimized. The geometries, for two consecutive step, are related by
where 
is a random perturbation on the atomic position.
To generate the random vector
the present GSA routine use a procedure like usual Simulated Annealing,
but differently from this one, we have calculated the 
using a numerical integration of the visiting distribution probability
. Here 
is a random vector obtained from an equi-probability distribution and 
is the inverse of the integral of
.
In summary, the whole algorithm for mapping the global minimum of the energy is:
(i) Fix the parameters (q 
; q 
) (we recall that (q
; q
) = (1; 1) and (1; 2) respectively correspond to the Boltzmann and Cauchy
machines). Start, at t = 1, with an arbitrary Z matrix and a high
enough value for T(1) (visiting temperature) and cool as follows:
where t is the discrete time corresponding to computer iteration,
and q
(q
) is the acceptance index (visiting index).
(ii) Then randomly generate
from
as given by the visiting distribution probability 
as follows:
where 
is the gamma function. This procedure assures that the system can both
escape from any local minimum and can explore the entire energy hipersurface.
This change is due to the isotropy of the space used.
(iii) Then calculate the conformational energy E(
) by using the THOR program:
with 
(t) = 
(t).
Or the algorithm:
if E(
) 
(
), replace
by
;
if E(
) >E(
), run a random number r 
[0,1]:
if r 
(acceptance probability) retain
; otherwise, replace
by
.
(iv) Calculate the new temperature
(t) using (14) and go back to (ii) until the
E(
) is reached within the desired precision.
