Annealing schedule

In order to facilitate control of the specified temperatures, the annealing schedule can be adjusted so that lower cooling rates are applied at the beginning h = 2/3 hy, rate hy in the second stage and h = 2/iy in the last. The overall residence time in the annealing range remains unchanged. 

These considerations may be applied to both temporary and permanent stresses when assuming that permanent stress has the same value (with an opposite sign) as that which has been relieved by viscous flow at the onset of cooling. This principle serves as a basis for calculating safe annealing schedules. 

The annealing schedule is represented graphically by a curve which shows the respective lehr temperatures and the calculated cooling rates (cf. Fig. 112). The dashed line shows the theoretically calculated schedule while the full lines represent the cooling employed in practice with respect to the possible temperature control in the lehr. 

FIG. 112. Annealing schedule for glass tubes 100 mm in diameter with 10 mm wall thickness (Schill et al., 1968). 

With sufficiently large NumTemp and Maxit and sufficiently small (1-x), P, and , the annealing schedule ensures slow, steady convergence to a near-global optimum clustering. For the runs reported in this paper, the settings were NumTemp=200, Maxlt=4ra, (1- )=0.25, 8=0.125, and =0.00000000001. 

An important difference between the annealing schedules for global and local searches, in addition to those outlined in the previous section, is that during global searches the temperature is held constant once the ratio of acceptances to total number of trials falls below a threshold of 30%. This ensures that the search is able to escape local minima and allows continued sampling over a greater portion of the search space. This contrasts with local searches where the temperature continues to decrease until convergence is detected. 

The success and efficiency of simulated annealing depends on the choice of the annealing schedule [58], that is, the sequence of numerical values Ti > T2 > > T for the temperature. Note that multiplication of the temperature T by a factor s is formally equivalent to scaling the target E by 1/s. This applies to both the Monte Carlo as well as the molecular dynamics implementation of simulated annealing. This is immediately obvious upon inspection of the Metropolis Monte Carlo acceptance criterion (Eq. 10). For molecular dynamics this can be seen as follows. Let E be scaled by a factor 1/s while maintaining a constant temperature during the simulation. 

The advantage of the procedure as just described is that it allows one to experiment with different annealing schedules and perhaps develop some physical intuition for the process. On the other hand, it is possible to use a standard method of applying a simulated annealing schedule to an optimization problem, as discussed by O Mard [5]. 

Several variants of the basic SA algorithm have been developed, which differ in the choice of the CF, in the design of the annealing schedule or in the procedure for the generation of the trial configurations. For example, Andreev et al. reduce T at a preset rate and, for each T, perform several moves which increase as the acceptance ratio decreases David et introduced the following relevant novelties, included in the computer program DASH ... 

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