# discrepSA_LHS: Simulated annealing (SA) routine for Latin Hypercube Sample... In DiceDesign: Designs of Computer Experiments

## Description

The objective is to produce low-discrepancy LHS. SA is an efficient algorithm to produce space-filling designs. It has been adapted here to main discrepancy criteria.

## Usage

 `1` ```discrepSA_LHS(design, T0=10, c=0.95, it=2000, criterion="C2", profile="GEOM", Imax=100) ```

## Arguments

 `design` a matrix (or a data.frame) corresponding to the design of experiments `T0` The initial temperature `c` A constant parameter regulating how the temperature goes down `it` The number of iterations `criterion` The criterion to be optimized. One can choose three different L2-discrepancies: the C2 (centered) discrepancy ("C2"), the L2-star discrepancy ("L2star") and the W2 (wrap-around) discrepancy ("W2") `profile` The temperature down-profile, purely geometric called "GEOM", geometrical according to the Morris algorithm called "GEOM_MORRIS" or purely linear called "LINEAR" `Imax` A parameter given only if you choose the Morris down-profile. It adjusts the number of iterations without improvement before a new elementary perturbation

## Details

This function implements a classical routine to produce optimized LHS. It is based on the work of Morris and Mitchell (1995). They have proposed a SA version for LHS optimization according to mindist criterion. Here, it has been adapted to some discrepancy criteria taking in account new ideas about the reevaluations of a discrepancy value after a LHS elementary perturbation (in order to avoid computing all terms in the discrepancy formulas).

## Value

A list containing:

 `InitialDesign` the starting design `T0` the initial temperature of the SA algorithm `c` the constant parameter regulating how the temperature goes down `it` the number of iterations `criterion` the criterion to be optimized `profile` the temperature down-profile `Imax` The parameter given in the Morris down-profile `design` the matrix of the final design (low-discrepancy LHS) `critValues` vector of criterion values along the iterations `tempValues` vector of temperature values along the iterations `probaValues` vector of acceptation probability values along the iterations

## Author(s)

G. Damblin & B. Iooss

## References

Damblin G., Couplet M., and Iooss B. (2013). Numerical studies of space filling designs: optimization of Latin Hypercube Samples and subprojection properties, Journal of Simulation, 7:276-289, 2013.

M. Morris and J. Mitchell (1995) Exploratory designs for computationnal experiments. Journal of Statistical Planning and Inference, 43:381-402.

R. Jin, W. Chen and A. Sudjianto (2005) An efficient algorithm for constructing optimal design of computer experiments. Journal of Statistical Planning and Inference, 134:268-287.

Latin Hypercube Sample(`lhsDesign`),discrepancy criteria(`discrepancyCriteria`), geometric criterion (`mindist``phiP`), optimization (`maximinSA_LHS`,`maximinESE_LHS` ,`discrepESE_LHS`)
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```dimension <- 2 n <- 10 X <- lhsDesign(n, dimension)\$design ## Optimize the LHS with C2 criterion Xopt <- discrepSA_LHS(X, T0=10, c=0.99, it=2000, criterion="C2") plot(Xopt\$design) plot(Xopt\$critValues, type="l") ## Optimize the LHS with C2 criterion and GEOM_MORRIS profile ## Not run: Xopt2 <- discrepSA_LHS(X, T0=10, c=0.99, it=1000, criterion="C2", profile="GEOM_MORRIS") plot(Xopt2\$design) ## End(Not run) ```