Given how the problem has been presented, its resolution by SA makes that mathe- matically speaking, at each temperature level, it can be modeled as a homogeneous Markov chain. Indeed the algorithm can be characterized by a transition matrix TT representing all transition probabilities for all the possible solutions at temperature T . The whole algorithm is described by an inhomogeneous Markov process that consists of the sequence of all the particular chains at each temperature level. For example, if it is assumed that there is a finite number N of possible solutions, then the transition matrix at temperature T is given by:
TT = 1 −N1 P k6=1p1,k 1 Np1,2 · · · 1 Np1,N 1 Np2,1 1 − 1 N P k6=2p2,k · · · 1 Np2,N .. . ... . .. ... 1 NpN,1 1 NpN,2 · · · 1 − 1 N P k6=1pN,k (4.19)
Being pi,j the transition probability from solution Si to solution Sj defined by the Metropolis criterion in Equation (4.18). The terms have been conveniently normalized so that each row of the matrix sums 1. This yields the probability of not changing a certain solution Si, which is given by 1 − N1 Pk6=ipi,k. It is clear that TT(i, j) = 0 if Sj6∈ N (Si), where N (Si) is the subset of new solutions that can be generated from Si, i.e. its neighbor solutions. The transition matrix not only allows determining the probability of reaching each solution for one step of the simulation, but also across n steps through the matrix TnT.
Thanks to this mapping, it has been demonstrated in [AK89] that the Markov chain associated with SA converges at T to a stationary probability distribution defined by the probability vector πT:
lim n→∞T n T(i, j) = πT(j) ⇒ lim n→∞T n T = πT (4.20)
Thus, for any fixed temperature T > 0, if SA is executed for “a long time”, the distribution of the associated Markov chain will be very close to πT. Once this equilibrium is reached, the temperature can be decreased again to further improve the solution. In particular, two general modes of convergence exist towards the global minimum or minima:
1. In the first mode, convergence is guaranteed if T is decreased until zero but executing each Markov chain for an infinite time, so that the stationary dis- tribution is reached at every level.
2. In the second mode, convergence is achieved if temperature is reduced so that it approaches to zero in a logarithmic manner, i.e. if temperature at level n fulfills Tn≥ T0/ ln(1 + n), where T0 is the initial temperature level and n is a
counter of the already executed levels.
If one of these update rules is hold, then πT converges to the optimal vector π?T which represents a uniform probability distribution among all the global minima. Although both modes of convergence require infinite computing time, many works have used fast cooling algorithms and demonstrate that SA is a highly efficient metaheuristic for a wide variety of combinatorial problems (see Section2.2.2). The slower the temperature is decreased, the higher the probability of reaching the global optimum.
Given this, for a slow enough cooling strategy, if stationarity is achieved at level n, in the next one (n + 1) the number of required iterations will be lower because both Markov chains are going to be very similar and stationarity appears sooner. Hence, and outlining the importance of having a long enough iteration at T0 the authors in [AK89] propose as the most appropriate cooling update the following:
Tn+1=
Tn 1 +Tnln(1+δ)
3σn
(4.21)
Other options have been used in the literature being the geometric cooling the most typical one because of its easy implementation: Tn+1 = βTn, β < 1. In the present work, however, the temperature has been updated as proposed in Equation (4.21) because it is mathematically demonstrated that it preserves the convergence theory as much as possible.
The aggressiveness in the reduction of Tncan be controlled with δ so that simu- lation time can be adjusted to the available one. On the other hand, σn represents the standard deviation of the cost evolution with temperature Tnand its importance is described in next paragraphs.
The update should be executed whenever a certain stationarity is detected for a given level. However, monitoring this condition demands a very high computational cost and more practical empirical rules have been derived. For example, for the previous cooling algorithm, it is found in [AK89] that a good stationarity condition is executing the Metropolis criterion as many times as the dimension of N (set of neighbor solutions) for a given solution. In the case of study, this value can be estimated by the product of NBS and the number of possible CPICH powers: 1 + (PT X,CP ICH,max− PT X,CP ICH,min) /∆PT X,CP ICH. The resultant value is in fact an upper bound since several combinations are not valid solutions because they do not respect constraints. This rule-of-thumb works properly and an exact election of this value is not so critical because the temperature reduction is inversely proportional to σn, thus if the update is done too soon, σn will be high and the decrease will be lower and viceversa. That means the cooling schema would react to some extent if the equilibrium condition had not been very accurately chosen. This is graphically shown by Figure 4.6 which represents Equation (4.21) for different values of σn.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 1 2 T n+ 1 Tn s=1 s=0.5 s=0.25 s=0.1 σ = 1 σ = 0.5 σ = 0.25 σ = 0.1
Figure 4.6: Graphical representation of the cooling schema for different values of σn.
temperature, the higher the reduction as well. When the algorithm is supposed to be near the optimum, the temperature decrease is slower. In fact, an infinite time is required to reach T = 0 and therefore a convergence criteria different from that must be defined. In this case, it is taken into account the ratio of accepted solutions by the Metropolis criterion at a certain temperature level. The algorithm is considered to have finished when this ratio is less than 1%. At this point cost variations are very low and no more significant gains are obtained.