q - Rosario Toscano - Free

the current temperature; otherwise it is rejected i.e. q remains unchanged. Equivalently,
the new point qnew is accepted if it satisfies J ( qnew)
≤ J ( q)
−T
log( r)
.
4. Repeat steps 2 and 3 until the sequence of accepted points have reached a state of
equilibrium.
5. The temperature T is lowered to a new temperature Tnew in accordance with the annealing
schedule, set T = Tnew
and return to step 2. This process is continued until some stopping
rule is satisfied.
There are many way in which this algorithm can be implemented. In what follows, we give
some practical rules widely used for an efficient implementation of the simulated annealing
algorithm.
Choice of the initial temperature. The initial temperature must be chosen sufficiently large so
that any point of the search domain D has a reasonable chance of being visited. However, if Tinit
is too large then a too long time is spent in a state of “high energy” (i.e. high values of the cost
function). Many methods have been proposed in the literature to determine the initial temperature
(see for instance Ben-Ameur, 2004). A well accepted approach consist in computing an initial
temperature such that the acceptance ratio is approximately equal to a given value τ 0 . This can be
done as follows. Generate at random η samples uniformly distributed in D : q ∈D
, i = 1, K,
η ,
and choose a rate of acceptance τ 0 , then evaluate the initial temperature using:
( ) / log( 0)
τ
Tinit = − ΔJ
max , where (Δ J ) max is defined as: ( Δ J ) max = max J ( q ) − min J ( q ) . By the
1≤i≤η
1≤i≤η
way, we can use the η samples to select an initial decision vector q as follows: q = min J ( q ) .
i
i
i
arg
1≤i≤η
Generation of a new candidate solution (step 2 of the SA algorithm). Generally, a new
candidate solution qnew is generated by adding a random perturbation to the current solution q.
There are many way to do that, a common rule for continuous optimization problem is to add a
nq-dimensional Gaussian random variable to the current value q (see spall): q + (Σ)
,
q new
= g
where g is a zero-mean Gaussian random vector with covariance matrix Σ, which must be fixed
by the user. Another approach consists in changing only one component of q at a time (Brooks &
Morgan, 1995). This is done by first selecting one of the components of q at random, and then
randomly selecting a new value for that variable within its bounds. In Bohachevsky et al 1986, a
spherical uniform perturbation is adopted. More precisely, the new candidate point qnew is
obtained by first generating a random direction vector θ, with θ 1 , then multiplying it by a
fixed step size β, and finally summing the resulting vector to q, i.e. = q + βθ . The value of
the step size β must be set by the user. In a similar way one can also adopt the following rule:
qnew = q + CU , where U is a vector of uniform random number in the range (− 1,
1)
and C is a
constant diagonal matrix whose elements define the maximum change allowed in each
component of q. The matrix C is also user defined. The methods presented above are not
limitative and some other approaches have been proposed in the literature see for instance
Vanderbilt & Louie (1984), Parks (1990) to cite only a few.
2 =
q new
i