3.2.3.2. The simulated annealing method

The Simulated Annealing (SA) algorithm is a probabilistic metaheuristic which can solve a global optimisation problem. It is here applied to the construction of maximin Latin Hypercube Designs (maximin LHS). The SA algorithm consists in exploring the space of LHS through elementary random perturbations of both rows and columns in order to converge to maximin ones. We have implemented in Uranie the algorithm of Morris and Mitchell [DCI13, MM95], which is driven by the following parameters

• \(T_0\) is the initial temperature

• the decreasing of the temperature is controlled by \(c\)

• the number of iterations in the outer loop \(I\)

• the number of iterations in the inner loop \(I_{inner}\)

It is important to keep in mind that the performances of the simulated annealing method can strongly depend on \(c\) and thus changing parametrisation can lead to disappointing results. Below, in Table 3.1, we provide some parametrisation examples working well with respect both to the number of the input variables \(d\) and the size of the requested design \(N\).

Table 3.1 Proposed list of parameters value for simulated annealing algorithm, depending on the number of points requested (\(N\)) and the number of inputs under consideration (\(d\))

\(d\)	\(N = 10\times d\)		\(N = 20\times d\)		\(N = 30\times d\)
2,3,4	\(c\)	0.99	\(c\)	0.99	\(c\)	0.99
	\(T_{0}\)	0.1	\(T_{0}\)	0.1	\(T_{0}\)	0.1
	\(I\)	300	\(I\)	300	\(I\)	300
	\(I_{inner}\)	300	\(I_{inner}\)	300	\(I_{inner}\)	300
5,6,7	\(c\)	0.99	\(c\)	0.99	\(c\)	0.99
	\(T_{0}\)	0.001	\(T_{0}\)	0.001	\(T_{0}\)	0.001
	\(I\)	300	\(I\)	300	\(I\)	300
	\(I_{inner}\)	300	\(I_{inner}\)	300	\(I_{inner}\)	300
8,9,10	\(c\)	0.99	\(c\)	0.99	\(c\)	0.99
	\(T_{0}\)	0.0001	\(T_{0}\)	0.0001	\(T_{0}\)	0.0001
	\(I\)	300	\(I\)	300	\(I\)	300
	\(I_{inner}\)	300-1000	\(I_{inner}\)	300-1000	\(I_{inner}\)	300-1000