going beyond the physical interpretationsuppose that (xis an arbitrary function to be minimizedwith taking values in some discrete or continuous set the gibbs pdf corresponding to (xis defined as gibbs pdf (xet ft (xzt xp provided that the normalization constant zt : exp(- ( )/ is finite note that this is simply the boltzmann pdf with the boltzmann constant removed as the pdf becomes more and more peaked around the set of global minimizers of the idea of simulated annealing is to create sequence of points that are approximately distributed according to pdfs ft ( )ft ( )where is sequence of "temperaturesthat decreases (is "cooled"to -known as the annealing schedule if each xt were sampled exactly from ftt then xt would converge to global minimum of (xas howeverin practice sampling is approximate and convergence to global minimum is not assured generic simulated annealing algorithm is as follows annealing schedule algorithm simulated annealing inputannealing schedule ,function initial value outputapproximations to the global minimizer xand minimum value ( set and while not stopping do approximately simulate xt from ftt ( - + return xt (xt popular annealing schedule is geometric coolingwhere - for given initial temperature and cooling factor ( appropriate values for and are problem-dependent and this has traditionally required tuning on the part of the user possible stopping criterion is to stop after fixed number of iterationsor when the temperature is "small enoughapproximate sampling from gibbs distribution is most often carried out via markov chain monte carlo for each iteration tthe markov chain should theoretically run for large number of steps to accurately sample from the gibbs pdf ftt howeverin practiceone often only runs single step of the markov chainbefore updating the temperatureas in algorithm below to sample from gibbs distribution ft this algorithm uses random walk metropolishastings sampler from ( )the acceptance probability of proposal is thus et (yt ( ( )- ( ) min (xymin et (xhenceif (ys ( )then the proposal is aways accepted otherwisethe proposal is accepted with probability exp( ( (ys ( ))geometric cooling cooling factor