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Computing the Initial Temperature of Simulated Annealing

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The classical version of simulated annealing is based on a cooling schedule. Generally, the initial temperature is set such that the acceptance ratio of bad moves is equal to a certain value 0. In this paper, we first propose a simple algorithm to compute a temperature which is compatible with a given acceptance ratio. Then, we study the properties of the acceptance probability. It is shown that this function is convex for low temperatures and concave for high temperatures. We also provide a lower bound for the number of plateaux of a simulated annealing based on a geometric cooling schedule. Finally, many numerical experiments are reported.
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Computational Optimization and Applications, 29, 369–385, 2004
c
2004 Kluwer Academic Publishers. Manufactured in The Netherlands.
Computing the Initial Temperature
of Simulated Annealing
WALID BEN-AMEUR walid.benameur@int-evry.fr
GET/INT—CNRS/SAMOVAR, Institut National des T´
el´
ecommunications, 9, rue Charles Fourier,
91011 Evry, France
Received May 6, 2003; Revised December 30, 2003
Abstract. The classical version of simulated annealing is based on a cooling schedule. Generally, the initial
temperature is set such that the acceptance ratio of bad moves is equal to a certain value χ0.Inthis paper, we first
propose a simple algorithm to compute a temperature which is compatible with a given acceptance ratio. Then,
we study the properties of the acceptance probability. It is shown that this function is convex for low temperatures
and concave for high temperatures. We also provide a lower bound for the number of plateaux of a simulated
annealing based on a geometric cooling schedule. Finally, many numerical experiments are reported.
Keywords: simulated annealing, initial temperature, acceptance ratio
Introduction
Simulated annealing is a general probabilistic local search algorithm, proposed 20 years ago
by Cerny [3] and Kirkpatrick et al. [10] to solve difficult optimization problems. Many large
instances of practical difficult problems were successfully solved by simulated annealing
(see, e.g., [2, 7–9]).
To use a simulated annealing algorithm, one has first to define a set of solutions, generally
large, representing the solutions of an optimization problem. Then a neighborhood structure
is defined. To find a good solution we move from a solution to one of its neighbors in
accordance to a probabilistic criterion. If the cost decreases then the solution is changed and
the move is accepted. Otherwise, the move is accepted only with a probability depending on
the cost increase and a control parameter called temperature. Classically, the probability to
accept bad moves, i.e. moves with increase in terms of cost, is high at the beginning to allow
the algorithm to escape from local minimum. This probability decreases in a progressive
waybyreducing the temperature. The method used to decrease the temperature is generally
called cooling schedule. The performance of the algorithm strongly depends on the choice
of the cooling schedule and the neighborhood structure.
Many theoretical papers focused on an optimal cooling schedule (see, e.g., [1, 4, 6, 12,
13]). One of the most important results may be the proof of optimality of a logarithmic
cooling schedule given in Hajek [6]. However, the number of iterations needed to guarantee
to find of a global optimum is generally very large (see, e.g., [1]). The transition probability
Pij from state ito state jis defined as the product of a generation probability Gij and an
acceptance probability Aij.
370 BEN-AMEUR
The acceptance probability considered in this paper is the one defined by Metropolis
[11]:
Aij =exp EjEi
Tif Ej>Eiand Aij =1 otherwise (1)
where Tis the current temperature and Ei(resp. Ej)isthe energy of state i(resp. j).
A state is a solution of an optimization problem and energy is the cost function that has
to be minimized. We indifferently use energy and cost to designate the same thing.
We also assume that the homogenous Markov chain representing the simulated annealing
at a given temperature Tis irreducible (i.e. all states can be reached from any other state
with a positive probability) and aperiodic (see, e.g., [1]). These conditions are generally
satisfied.
If we assume that the generation probabilities are symmetrical (Gij =Gji), the stationary
distribution is nothing other than the Boltzmann distribution: πi=exp(Ei
T)
jexp(Ej
T).
Another generation strategy that is commonly used is given by
Gij =
1
|N(i)|if jN(i)
0 else
(2)
where N(i)isthe set of neighbors of i. The stationary distribution is then given by
πi=|N(i)|exp Ei
T
j|N(j)|exp Ej
T(3)
As previously said, one of the most important properties of simulated annealing is its hill
climbing feature. This is achieved by accepting some increasing cost moves. Consequently,
the average probability of accepting these moves is very important to evaluate the ability
of simulated annealing to escape from local minimum.
This acceptation ratio strongly depends on the temperature. To allow the simulated an-
nealing to find good solutions, one has to carefully compute the initial temperature. This
parameter plays an important role in simulated annealing, but is of course only a piece of a
large puzzle. This paper will focus on this initial temperature and some other properties of
the acceptance ratio.
Many methods have been proposed in literature to compute the initial temperature T0.It
is suggested in Kirkpatrick et al. [10] to take T0=Emax where Emax is the maximal cost
difference between any two neighboring solutions.
Another scheme based on a more precise estimation of the cost distribution is proposed
with multiple variants (see, e.g., [1, 16]). It is recommended to choose T0=Kσ2
where
Kis a constant typically ranging from 5 to 10 and σ2
is the second moment of the energy
distribution when the temperature is .σis estimated using a random generation of some
solutions.
COMPUTING INITIAL TEMPERATURE OF SIMULATED ANNEALING 371
A more classical and intuitive method is described in Kirkpatrick et al. [10]. It consists
in computing a temperature such that the acceptance ratio is approximately equal to a given
value χ0. First, we choose a large initial temperature. Then, we have to perform a number of
transitions using this temperature. The ratio of accepted transitions is compared with χ0.If
it is less than χ0, then the temperature is multiplied by 2. The procedure continues until the
observed acceptance ratio exceeds χ0. Other variants are proposed to obtain an acceptance
ratio which is close to χ0.Itis, for example, possible to divide the temperature by 3 if the
acceptance ratio is much higher than χ0. Using this kind of rules, cycles are avoided and a
good estimation of the temperature can be found.
Another procedure is proposed in Johnson et al. [7, 8]. Temperature is obtained using the
formula T0=− E
ln(χ0), where Eis an estimation of the cost increase of strictly positive
transitions. This estimation is again obtained by randomly generating some transitions.
Notice that δt
ln(χ0), where δtis the cost increase induced by a transition t,isthe temperature
allowing this transition to be accepted with a probability χ0.Inother terms, T0=− E
ln(χ0)
is the average of these temperatures over a set of random transitions.
Finally, note that to accelerate the simulated annealing, a heuristic is sometimes used
to find a good initial solution. Then, simulated annealing is applied with a low initial
temperature (see, e.g., [5, 7, 15]). An algorithm is provided by Varanelli [15] to compute
an initial temperature such that the expected cost of the best solution that can be found at
this temperature is approximately equal to the cost of the solution given by the heuristic.
Anew algorithm to compute the initial temperature is given in this paper. The algorithm
is fast and accurate. It is presented in next section. The convergence is proved in Section 1.
Some other properties of the acceptance probability are presented in Section 2. Many
numerical experiments are reported and commented in Section 3. Finally, some concluding
remarks are given in Section 4.
1. An efficient algorithm to compute the temperature
The initial temperature is often chosen such that the acceptance probability is approximately
equal to a certain value, for example, 0.8 (see, e.g., [1]). Let tbe a strictly positive transition
and let maxt(resp. mint)bethe state after (resp. before) the transition. As we assumed
that the transition is strictly positive, then Emaxt>Emint.Tosimplify notation, we use δtto
designate the cost difference EmaxtEmint. Using the generation strategy (2), the acceptance
probability is given by:
χ(T)=tpositive πmint
1
|N(mint)|exp δt
T
tpositive πmint
1
|N(mint)|
.(4)
Note that πmint
1
|N(mint)|represents the probability to generate a transition twhen the
energy states are distributed in conformance with the stationary distribution (3). Moreover,
exp(δt
T)isthe probability to accept a positive transition t. Thus, χ(T)isthe conditional
expectation of the acceptance of positive transitions.
372 BEN-AMEUR
We will use an estimation ˆχ(T)ofthis acceptance probability based on a random set S
of positive transitions. ˆχ(T)isdefined as follows:
ˆχ(T)=tSπmint
1
|N(mint)|exp δt
T
tSπmint
1
|N(mint)|
=tSexp Emaxt
T
tSexp Emint
T.(5)
Now, let us assume that we are looking for a temperature T0such that χ(T0)=χ0where
χ0]0,1[ is the wanted acceptance probability. We will propose a simple iterative method
to compute such a temperature. In fact, we will consider ˆχ(T) instead of χ(T). First, we
randomly generate a set of positive transitions S. This can be done, for example, by gener-
ating some states and a neighbor for each state. The energies Emaxtand Emintcorresponding
with the states of the subset Sare stored. Then we choose a value T1for temperature. T1
can be any positive number.
T1may be far from T0.TofindT0we use the recursive formula
Tn+1=Tnln (ˆχ(Tn))
ln(χ0)1
p
.(6)
where pis a real number 1.
When ˆχ(Tn) becomes close to χ0we can stop: Tnis a good approximation of the wanted
temperature T0.
Please note that we use at each iteration the energy values previously stored. In other
words, we do not have to generate new transitions.
Before proving the convergence of our procedure, let us give a summary of the whole
process. denotes a small real number (e.g., 103).
Computing the temperature of simulated annealing
Step 1.
(a) Estimate the number of samples Sneeded to compute ˆχ(T).
(b) Generate and store Srandom positive transitions.
(c) Set T1at any strictly positive number and set n=1.
Step 2.
(a) Compute ˆχ(Tn)=tSexp(Emaxt
Tn)
tSexp(Emint
Tn).
(b) If |ˆχ(Tn)χ0|≤, return Tn.
Otherwise
Tn+1=Tn(ln( ˆχ(Tn))
ln(χ0))1
p.
n=n+1.
–gotoStep 2(a).
End.
COMPUTING INITIAL TEMPERATURE OF SIMULATED ANNEALING 373
Steps 1(a) and (b) will be discussed later.
As said before, the value of T1can be any strictly positive number. However, to slightly
accelerate the whole process, we compute T1using the formula given in introduction Johnson
[7, 8]:
T1=− tSδt
Sln(χ0).(7)
In the rest of this section, we first prove under some assumptions the convergence of
the algorithm described above. Then we give some remarks about the sampling procedure
needed by the algorithm.
1.1. Algorithm convergence
To show the convergence of the algorithm, we will prove that TT(ln( ˆχ(T))
ln(χ0))1
pis a non
decreasing function and Tˆχ(T)isastrictly increasing function. This means that T0is
a unique fixed point of function TT(ln( ˆχ(T))
ln(χ0))1
pand min(T0,Tn)Tn+1max(T0,Tn).
Notice that if TT(ln( ˆχ(T))
ln(χ0))1
pis a non decreasing function when p=1, then it will have the
same behavior for any p1. This can be seen by computing the derivative of the logarithm
of this function: 1
T+1
p
ˆχ(T)
ˆχ(T)ln( ˆχ(T)) .Ifweassume that ˆχ(T)0, then 1
T+1
p
ˆχ(T)
ˆχ(T)ln( ˆχ(T))
clearly increases when pincreases. Therefore, we will focus on p=1.
Before giving the proofs of the wanted results, we will present an hypothesis that will be
used to simplify calculation.
Hypothesis 1.1. We assume that the energy levels Emintand the cost differences δtof the
set of transitions Sare independent.
More precisely, given a temperature T,weassume that the positive transitions are gen-
erated in conformance with the equilibrium distribution. As we focus here on S,wecon-
sider the conditional distribution where the probability to generate a transition t0is given
by πmint0
1
|N(mint0)|
tSπmint
1
|N(mint)|
.Itisnatural to assume that there is no correlation between {δi,Emini}
and {δj,Emin j}where iand jare two transitions of Sobtained by independent trials in
conformance with the conditional equilibrium distribution. However, in Hypothesis 1.1
we also assume that Eminiis independent with δi. This assumption is less easy to under-
stand. In fact, it depends on the distribution which is related to temperature. Said another
way, even if it is valid for some temperatures, it will be invalid for others. Note how-
ever that we do not need this assumption to be strictly satisfied. The convergence of the
algorithm is obtained in almost all cases when p=1. Moreover, it can be ensured by
increasing the value of the parameter p. More details will be given in the end of this
subsection.
374 BEN-AMEUR
Lemma 1.2. Assuming hypothesis 1.1is valid,then we have
i,jS,i<jexpEmini+Emin j
TEminiEmin jexp δi
Texp δj
T
i,jSexp Emaxi+Emin j
Tδi=0.
Proof: Let L(resp. R)bethe numerator (resp. denominator) of the ratio given in the
lemma. We want to show that L
R=0. In fact, Lis nothing but 1
2i,jSexp(Emini+Eminj
T)
(EminiEmin j)(exp(δi
T)exp(δj
T)).
Moreover, using formulas 3 and 2, the expectation of ( EminiEmin j)(exp(δi
T)
exp(δj
T)) is given by E((EminiEmin j)(exp(δi
T)exp(δj
T)) |i,jS)=i,jS
exp(Emini
T)
kSexp(Emink
T)×exp(
Emin j
T)
kSexp(Emink
T)×((EminiEmin j)(exp(δi
T)exp(δj
T))). Note that we
used here the fact that the transitions of Sare independent. We obtain
L=1
2
kS
expEmink
T2
EEminiEmin jexp δi
T
exp δj
T
i,jS.
On the other hand,
R=
i,jS
expEmaxi+Emin j
Tδi
=
jS
expEmin j
T
iS
exp Emaxi
Tδi
=
jS
exp Emin j
T2
iS
exp Emini
T
jSexp Emin j
Texp δi
Tδi
=
jS
exp Emin j
T2
Eexp δi
Tδi
iS
Combination of the previous expressions related to Land Rleads to
L
R=1
2
EEminiEmin jexp δi
Texp δj
Ti,jS
Eexp δi
TδiiS
COMPUTING INITIAL TEMPERATURE OF SIMULATED ANNEALING 375
Now using Hypothesis 1.1, one can deduce that
EEminiEmin jexp δi
Texp δj
T
i,jS
=EEminiEmin ji,jSEexp δi
Texp δj
T
i,jS
Finally,
L
R=1
2
EEminiEmin ji,jSEexp δi
Texp δj
Ti,jS
Eexp δi
TδiiS
=0
which means that L
R=0.
Note that it is possible to build a particular small example for which both Hypothesis 1.1
and Lemma 1.2 are not valid. However, our experimental results (Section 3) show that the
algorithm works very well in practice, and the convergence is obtained in almost all cases.
More details will be given in the end of this subsection.
Proposition 1.3. Assuming Hypothesis 1.1is valid,then the derivative of ˆχ(T)is given
by:
ˆχ(T)=1
T2iSexp Emaxi
Tδi
iSexp Emini
T.
Proof: Let us calculate ˆχ(T).
ˆχ(T)
=iSEmaxiexp Emaxi
TjSexp Emin j
TiSEminiexp Emini
TjSexp Emax j
T
T2iSexp Emini
T2
=i,jexp Emaxi+Eminj
TEmaxiEmin j
T2iSexp Emini
T2
=i,jexp Emaxi+Eminj
TEminiEmin j+δi
T2iSexp Emini
T2
=i,jexp Emaxi+Eminj
TEminiEmin j+i,jexp Emaxi+Emin j
Tδi
T2iSexp Emini
T2
=i<jexp Emini+Eminj
TEminiEmin jexp δi
Texp δj
T+i,jexp Emaxi+Eminj
Tδi
T2iSexp Emini
T2
376 BEN-AMEUR
Using Lemma 1.2, the previous expression becomes:
ˆχ(T)=i,jexp Emaxi+Emin j
Tδi
T2iSexp Emini
T2
=1
T2iSexp Emaxi
Tδi
iSexp Emini
T
Please note that even if Hypothesis 1.1 is not valid, we can be satisfied with a small value
of the ratio L
Rof Lemma 1.2 to obtain a good approximate value of ˆχ(T).
Proposition 1.3 tells us that ˆχ(T)>0. To finish our proof of convergence, we have to
show that TTln(ˆχ(T))
ln(χ0)is a non decreasing function.
Proposition 1.4. Assuming Hypothesis 1.1is valid,then (Tln( ˆχ(T)))0.
Proof: Derivative of Tln( ˆχ(T)) is given by ln( ˆχ(T)) +Tˆχ(T)
ˆχ(T).
Using expression (5), one can write:
1
ˆχ(T)=iSexp Emini
T
iSexp Emaxi
T
1
ˆχ(T)=iSexp Emaxi
Texp δi
T
iSexp Emaxi
T
=
iS
exp Emaxi
T
jSexp Emax j
Texp δi
T
By concavity of logarithm, one can deduce that ln( 1
ˆχ(T))iS
exp(Emaxi
T)
jSexp(Emaxj
T)
δi
T.Said
another way, we have
ln( ˆχ(T)) <
iS
exp Emaxi
T
jSexp Emax j
Tδi
T.
On the other hand, using Proposition 1.3, we obtain:
Tˆχ(T)
ˆχ(T)=1
TiSexp Emaxi
Tδi
iSexp Emaxi
T.
Combination of the previous two results leads to (Tln( ˆχ(T)))0.
Propositions 1.3 and 1.4 clearly imply the convergence of the algorithm: (Tn)nNis
monotonous and bounded.
COMPUTING INITIAL TEMPERATURE OF SIMULATED ANNEALING 377
Note that even if the results of this subsection are based on Hypothesis 1.1, they are
useful in a general context. Let us give an insight into this point. First, to show that ˆχ(T)
is an increasing function, we only need to have the ratio of Lemma 1.2 close to 0. In other
terms, we do not really require Hypothesis 1.1 to be strictly satisfied. Second, we already
said in the beginning of this subsection that the derivative of the logarithm of the function
TT(ln( ˆχ(T))
ln(χ0))1
pincreases when pincreases. Said another way, if we get some convergence
problems when p=1 due to the inaccuracy of Hypothesis 1.1, we can sufficiently increase
pto allow TT(ln( ˆχ(T))
ln(χ0))1
pto be an increasing function. Moreover, our experimental
results (Section 3) show that in most of cases p=1issufficient. We needed to take p=2
in about 1 run per 1000 to guarantee the convergence. However, to strictly guarantee the
convergence, we can slightly modify the algorithm of Section 1. If an oscillation is detected
(i.e., (Tn+1Tn)(TnTn1)<0) then we multiply pby 2 and we continue the algorithm.
1.2. On the sampling procedure
The first steps of the algorithm (1(a) and (b)) can be called the sampling procedure.
Even if the convergence of the algorithm is shown for a set Sof random transitions
satisfying Hypothesis 1.1 (and experimentally in Section 3), the set Smust be representative
to allow the algorithm to give a temperature which is close to the wanted temperature.
Obviously, the exact temperature is given when Scontains all positive transitions. However,
it is generally not possible to consider all transitions.
We will not give a definitive description of the sampling procedure: we think that it
depends on the nature and the size of the problem that we are solving.
One can, for example, begin with a small value of S, compute the temperature, and
increase the number of transitions until the temperature becomes stable.
It is also possible to use the temperature T1of Eq. (7) to perform a first simulated annealing
plateau. All positive transitions considered during this plateau can be stored and then used
to compute a more accurate temperature using our algorithm.
Numerical experiments that will be presented in Section 3, are based, for each value
of S,onarandom generation of independent transitions. Notice that when we use the
transitions encountered during a plateau, transitions may not be independent.
2. Other properties
More properties of the acceptance probability are given in this section.
Proposition 2.1. Assuming Hypothesis 1.1is valid,then ˆχ(T)1
eT .
Proof: It was shown in Proposition 1.3 that ˆχ(T)=1
T2iSexp(Emaxi
T)δi
iSexp(Emini
T).
It implies that ˆχ(T)=1
TiSexp(Emini
T)exp(δi
T)δi
T
iSexp(Emini
T)
Moreover, the function xxexp(x)isbounded by 1/e. Using this upper bound in
the previous approximation leads to the wanted result.
378 BEN-AMEUR
An important straightforward corollary dealing with the evolution of the acceptance
probability is given below.
Corollary 2.2. Assuming Hypothesis 1.1is valid,then ˆχ(T+T)ˆχ(T)1
eln(1+T
T).
Proof: A simple integration of the inequality ˆχ(T)1
eT gives the wanted result.
Using the fact that ln(1 +x)x, one can deduce that ˆχ(T+T)ˆχ(T)1
e
T
T.
Corollary 2.2 implies that even if you divide the temperature by 2, you can not expect to
reduce the acceptance probability by more than ln(2)
e0,255.
It is also possible to use the previous corollary to have an indication about the number
of iterations of a classical simulated annealing with a geometric cooling schedule. As-
sume that the temperature is multiplied by α<1atthe end of each plateau. In most of
cases, the initial temperature is chosen such that the acceptance probability of positive
moves is equal to χ0. The stopping criterion can also be a low acceptance probability χf.
Using Corollary 2.2, one can easily show that the number of plateaux Nis higher than
e(χ0χf)
ln(1).
Proposition 2.3. Assuming Hypothesis 1.1is valid,then the number of plateaux is higher
than e(χ0χf)
ln(1).
Assume, for example, that χ0=0.9, χf=0.05 and α=0.95. The number of plateaux
is then higher than 46. If α=0.99, we need more than 230 plateaux. More precisely, if
α=1where 1, then the number of plateaux is approximately higher than e(χ0χf)
.
Note that one of the advantages of the upper bound 1
eT given in Proposition 2.1 is its
independence with energy. However, this upper bound is bad for low temperatures. In fact,
one can easily see that ˆχ(T)C
T2exp(
T) where Cis a constant depending on the
energies and the transitions and is the difference between the smallest Emaxiand the
smallest Emini. This clearly implies that ˆχ(T)isapproximately equal to 0 when Tis close
to 0.
To finish our study of the acceptance probability, let us consider the second derivative
ˆχ(T).
First, another simple lemma will be stated.
Lemma 2.4. Assuming Hypothesis 1.1is valid,then
i,jS,i<jexp Emini+Emin j
TEminiEmin jδiexp δi
Tδjexp δj
T
i,jSexp Emaxi+Emin j
Tδi2=0.
This lemma can be easily proved using the same kind of arguments as those given to
prove the validity of Lemma 1.2.
COMPUTING INITIAL TEMPERATURE OF SIMULATED ANNEALING 379
Proposition 2.5. Assuming Hypothesis 1.1is valid,then the second derivative is given
by:
ˆχ(T)=1
T4iSexp Emaxi
Tδi(δi2T)
iSexp Emini
T.
Proof: A simple derivation of T2ˆχ(T) using Proposition 1.3 gives the following:
(T2ˆχ(T))
=1
T2iSEmaxiexp Emaxi
TδijSexp Eminj
TiSEminiexp Emini
TδijSexp Emaxj
T
iSexp Emini
T2
=1
T2i,jS,i<jexp Emini+Eminj
TEminiEmin j(δiexp δi
Tδjexp δj
T)
iSexp Emini
T2
+1
T2i,jSexp Emaxi+Eminj
Tδi2
iSexp Emini
T2
Using Lemma 2.4, we obtain:
(T2ˆχ(T))=1
T2i,jSexp Emaxi+Emin j
Tδi2
iSexp Emini
T2
=1
T2iSexp Emaxi
Tδi2
iSexp Emini
T
Using again Proposition 1.3 gives the wanted result.
One can easily see that the expression given above is positive when Tis close to 0 and
negative when Tis sufficiently high.
Corollary 2.6. Assuming Hypothesis 1.1is valid,the probability to accept positive tran-
sitions is convex for low temperatures and concave for high temperatures.
Finally, we give here simple bounds for the second derivative ˆχ(T).
Corollary 2.7. Assuming Hypothesis 1.1is valid,then
1
T2(2 22) exp(22) ˆχ(T)1
T2(2 +22) exp(22).
380 BEN-AMEUR
Proof: Proposition 2.5 tells us that ˆχ(T)=1
T4iSexp(Emaxi
T)δi(δi2T)
iSexp(Emini
T). Simple calculation
leads to:
ˆχ(T)=1
T2iSexp Emini
Texp δi
Tδi
Tδi
T2
iSexp Emini
T
=1
T2iSexp Emini
Tfδi
T
iSexp Emini
T
=1
T2Efδi
T
iS
where fdenotes the function xR+x(x2) exp(x). One can easily see that the
minimum of fis obtained for x=22 and the maximum is reached for x=2+
2. Thus, for any x0wehave(222) exp(22) f(x)(2 +22)
exp(22).
Combinations of these inequalities and the expression of ˆχ(T) leads to the wanted
result.
Notice that (2 22) exp(22) ≈−0.462 and (2 +22) exp(22) 0.159.
3. Numerical experiments
To illustrate the results given in the previous sections, extensive numerical experiments are
carried out. Two kind of problems are considered: random problems and traveling salesman
problems (TSP).
3.1. Random problems
A random problem is represented by a symmetric graph G=(V,E) where Vis the set of
vertices corresponding with the solutions of the problem, and Eis the set of edges repre-
senting the neighborhood relationship. Each solution (vertex) has a random cost. Simulated
annealing is applied to find a minimum cost solution.
The graphs used to represent random problems are described using three parameters:
the number of vertices V, the graph density d=E
VV1
2
and an upper bound Ufor the
maximum degree. Two sets of problems are considered in this section where (V,d,U)=
(5 ×104,104,30) in the first case and (2 ×106,106,5) in the second one. Notice that
these problems are small and can be solved by enumeration. However, the aim of this
section is only to study the procedure proposed in this paper to compute the temperature of
a simulated annealing.
Due to space limitation, we only give a summary of the results: more details will be
provided on Kluwer’s web site.
COMPUTING INITIAL TEMPERATURE OF SIMULATED ANNEALING 381
We consider 4 different values of the number of samples S: 20, 100, 500 and 2500.
We also try 8 values of the acceptance probability χ0: 0.99, 0.9, 0.7, 0.5, 0.3, 0.1, 0.05 and
0.01. Positive transitions are randomly and independently generated. For each value of χ0
and Sthe algorithm of Section 1 is used to give a temperature. Convergence was always
obtained with p=1(Formula 6).
Simulated annealing is then applied using the given temperature, without any decrease,
to provide the experimental acceptance probability ¯χ.Wealso apply simulated annealing
using the temperature T1defined by Eq. (7) to obtain χ(T1) defined as the experimental
acceptance probability corresponding with T1. Recall that this temperature is commonly
used by simulated annealing practitioners.
All experiments are repeated 200 times (200 runs for each value of χ0and S). Results
are expressed in terms of average and standard deviation.
The ratios corresponding with Lemmas 1.2 and 2.4 are also considered here in order
to check the validity of our hypothesis. We also focus on the number of iterations of the
algorithm needed to compute the temperature. This number is null if the temperature T1
given by Eq. (7) obtained in Step 1(c) is the final result of the algorithm. The precision term
used in the algorithm is here equal to 103.
To summarize, we focus on the average and the standard deviation of the following quan-
tities: the experimental acceptance probability ¯χ, the experimental acceptance probability
obtained with T1of Eq. (7), the ratios corresponding with Lemmas 1.2 and 2.4 and the
number of iterations of the algorithm of Section 1.
First, the algorithm used to compute the temperature converges at each run. The average
and the standard deviation values corresponding with Lemmas 1.2 and 2.4 are generally
low. They are not null because Hypothesis 1.1 is not always valid. When both χ0and S
are very low, Lemma 1.2 does not seem to be satisfied. In fact, the denominator of the
fraction defined in Lemma 1.2 is close to 0 when Tis very low. Moreover, Hypothesis 1.1 is
unlikely to be satisfied when |S|is very small. Nevertheless, as previously said, convergence
is always obtained, even if Hypothesis 1.1 is not satisfied.
The number of iterations needed by the algorithm to achieve convergence is small for
high values of χ0.Infact, when χ0is high, temperature T1seems to be a good one. This is
shown by χ(T1) which is very close to χ0when χ0is high.
χ(T1) becomes far from χ0when χ0is low. We also observed that the standard deviation
of χ(T1) generally decreases when Sincreases. However, the average value of χ(T1) can
be considered as stable. This is due to the fact that T1is based on the average of the cost
variations. Although, ¯χis generally close to χ0more than χ(T1).
Moreover, our numerical experiments show that for high values of χ0,asmall value of
Scan be sufficient to obtain a temperature achieving the goal. However, if χ0is low, we
need higher values of S.
The difference between χ0and ¯χdecreases when the problem size decreases. The problem
size considered in the first case is smaller than the second problem size and the results are
slightly better: the standard deviation of ¯χis lower in the first case than in the second one.
Said another way, when the problem size islarger, we may need more transitions to compute
a temperature.
382 BEN-AMEUR
Table 1.T1=500 and p=1.
χ0.9352 0.5114 0.5007 0.5000
T500.0 48.3411 46.7738 46.6768
Table 2.T1=5 and p=1.
χ0.0461 0.2661 0.4687 0.4978 0.4999 0.5000
T5.0 22.1953 42.3929 46.3526 46.6492 46.6687
Table 3.T1=500 and p=2.
χ0.9352 0.8067 0.6818 0.5993 0.5536 0.5286 0.5152 0.5081
0.5043 0.5023 0.5012 0.5006 0.5003 0.5002 0.5000
T500.0 155.4690 86.5412 64.3252 55.2274 51.0096 48.9224 47.8533
47.2957 47.0020 46.8464 46.7639 46.72 46.6966 46.6842
Table 4.T1=5 and p=2.
χ0.0461 0.1141 0.2208 0.3291 0.4043 0.4482 0.4722 0.4852
0.4921 0.4958 0.4976 0.4988 0.4994 0.4997 0.4998 0.4999
T5.0 10.5345 18.6429 27.5214 34.8496 39.8314 42.8594 44.5904
45.5479 46.0682 46.3483 46.4984 46.5785 46.6213 46.6441 46.6562
Finally, some sequences of temperatures and acceptance ratios obtained by the algo-
rithm are given in Tables 1–4. We intend to compute a temperature corresponding with
an acceptance ratio χ0=0.5. Instead of using T1, defined in Eq. (7), we take T1=500 to
perform the experiments of Tables 1 and 3, and T1=5 for the experiments of Tables 2 and
4. The precision is here 104.Parameter pused in the recursive formula 6 is equal to 1 in
Tables 1 and 2. We take p=2inTables 3 and 4.
We can see that the convergence of the algorithm is slower for p=2 than for p=1. This
can be easily understood from formula 6. However, recall that when pincreases then the
derivative of the function TT(ln( ˆχ(T))
ln(χ0))1
pis more likely to be positive (Section 1.1). Said
another way, as we know that Hypothesis 1.1 is not always valid, it may be more advisable
to take p>1. Although we did not need to take p>1tocompute the temperature in the
case of these random problems, we will see in the next section that this may be necessary
in very few of cases.
3.2. Traveling salesman problems
The algorithm of Section 1 is applied here for the traveling salesman problem. We consider
transitions based on the very classical 2-OPT moves (see, for example [9]). Two sets of
COMPUTING INITIAL TEMPERATURE OF SIMULATED ANNEALING 383
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0,35
0,45
0,53
0,64
0,97
1,09
1,47
1,87
2,26
2,74
3,35
4,39
5,42
6,77
8,36
10,4
12,7
16,1
20,2
25,2
30,9
39,3
49
61,1
77
95,1
116
148
183
233
293
Temperature
acceptance probability
Figure 1.Evolution of χ(T) for TSP(100).
randomly generated Euclidean instances are used: 20-city and 100-city problems. Notice
that these problems can now be solved by efficient cutting plane algorithms.
Three values of Sare considered: 20, 2500 and 62500. Eight values of χ0are used:
0.99, 0.9, 0.7, 0.5, 0.3, 0.1, 0.05 and 0.01. 200 experiments are performed for each value of
χ0and S.
Comments given in the previous subsection about random problems are still valid here.
However, we noticed that S=2500 was sufficient to give a good approximation of the
temperature in the previous case, but does not seem to be sufficient here for some values
of χ0.Inother words, when the size of problems increases, we need larger size samples to
obtain a good approximation of the temperature.
The procedure used to compute the temperature does not converge in about 1 run per
1000 when p=1. If p=2, the algorithm always converges. In fact, when the algorithm
is applied, it is easy to check whether there is an oscillation in terms of temperature. In this
case, we multiply pby 2 and we continue the algorithm.
The experimental acceptance ratio is plotted as a function of temperature. The graph of
figure 1 corresponds with the 100-city problem. This ratio is, as claimed in Corollary 2.6,
convex for low temperatures and concave for high temperatures.
Finally, we studied the experimental number of plateaux of simulated annealing when a
geometric cooling schedule (α=0.95) is used. Different values of the initial acceptance ratio
χ0and the final acceptance ratio χfare considered. The number of Plateaux is compared with
the lower bound of Proposition 2.3. This lower bound seems to be good for intermediate
acceptance ratios and bad for extremal acceptance ratios (either very low or very high
ratios).
384 BEN-AMEUR
4. Conclusion
A simple algorithm is proposed to compute a temperature such that the acceptance ratio of
increasing cost moves is equal to a given value χ0.Wealso presented some properties of
the acceptance ratio.
We think that this algorithm can be used as a component of either classical or modern
simulated annealing schemes for which the cooling schedule is not necessarily monotonous.
The procedure proposed in this paper can be modified in different ways. First, the formula
linking Tnand Tn+1can be changed. Said another way, even if the algorithm is very fast, one
can find another formula allowing a faster convergence. Second, We assumed in this paper
that transitions are accepted in accordance with the Metropolis criterion. A further research
direction may consist in introducing some modifications and studying the convergence of
the algorithm when other acceptance probabilities are considered.
Finally, we considered the acceptance ratio of positive transitions. Although, we may want
to focus on the acceptance of all transitions. A similar algorithm allowing the computation
of a temperature that is compatible with a given acceptation probability of all transitions is
now under study.
Acknowledgment
Iwould like to thank an anonymous referee for his valuable comments.
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