# Sequential multipoint Expected improvement (qEI) maximizations and model re-estimation Executes `nsteps` iterations of the multipoint EGO method to an object of class `km`. At each step, a kriging model (including covariance parameters) is re-estimated based on the initial design points plus the points visited during all previous iterations; then a new batch of points is obtained by maximizing the multipoint Expected Improvement criterion (`qEI`). ```r qEGO.nsteps( fun, model, npoints, nsteps, lower = rep(0, model@d), upper = rep(1, model@d), crit = "exact", minimization = TRUE, optimcontrol = NULL, cov.reestim = TRUE, ... ) ``` ## Arguments - `fun`: the objective function to be optimized, - `model`: an object of class `km` , - `npoints`: an integer repesenting the desired batchsize, - `nsteps`: an integer representing the desired number of iterations, - `lower`: vector of lower bounds for the variables to be optimized over, - `upper`: vector of upper bounds for the variables to be optimized over, - `crit`: "exact", "CL" : a string specifying the criterion used. "exact" triggers the maximization of the multipoint expected improvement at each iteration (see `max_qEI`), "CL" applies the Constant Liar heuristic, - `minimization`: logical specifying if we want to minimize or maximize `fun`, - `optimcontrol`: an optional list of control parameters for the qEI optimization (see details or `max_qEI`), - `cov.reestim`: optional boolean specifying if the kriging hyperparameters should be re-estimated at each iteration, - `...`: optional arguments for `fun`. ## Returns A list with components: - **par**: a data frame representing the additional points visited during the algorithm, - **value**: a data frame representing the response values at the points given in `par`, - **npoints**: an integer representing the number of parallel computations, - **nsteps**: an integer representing the desired number of iterations (given in argument), - **lastmodel**: an object of class `km` corresponding to the last kriging model fitted, - **history**: a vector of size `nsteps` representing the current known optimum at each step. ## Details The parameters of list `optimcontrol` are : - `optimcontrol$method` : "BFGS" (default), "genoud" ; a string specifying the method used to maximize the criterion (irrelevant when `crit` is "CL" because this method always uses genoud), - when `crit="CL"` : + `optimcontrol$parinit` : optional matrix of initial values (must have model@d columns, the number of rows is not constrained), + `optimcontrol$L` : "max", "min", "mean" or a scalar value specifying the liar ; "min" takes `model@min`, "max" takes `model@max`, "mean" takes the prediction of the model ; When L is `NULL`, "min" is taken if `minimization==TRUE`, else it is "max". + The parameters of function `genoud`. Main parameters are : `"pop.size"` (default : [N=3*2^model@d for dim<6 and N=32*model@d otherwise]), `"max.generations"` (default : 12), `"wait.generations"` (default : 2) and `"BFGSburnin"` (default : 2). - when `optimcontrol$method = "BFGS"` : + `optimcontrol$nStarts` (default : 4), + `optimcontrol$fastCompute` : if TRUE (default), a fast approximation method based on a semi-analytic formula is used, see [Marmin 2014] for details, + `optimcontrol$samplingFun` : a function which sample a batch of starting point (default : `sampleFromEI`), + `optimcontrol$parinit` : optional 3d-array of initial (or candidate) batches (for all `k`, parinit[,,k] is a matrix of size `npoints*model@d` representing one batch). The number of initial batches (length(parinit[1,1,])) is not contrained and does not have to be equal to `nStarts`. If there is too few initial batches for `nStarts`, missing batches are drawn with `samplingFun` (default : `NULL`), - when `optimcontrol$method = "genoud"` : + `optimcontrol$fastCompute` : if TRUE (default), a fast approximation method based on a semi-analytic formula is used, see [Marmin 2014] for details, + `optimcontrol$parinit` : optional matrix of candidate starting points (one row corresponds to one point), + The parameters of the `genoud` function. Main parameters are `"pop.size"` (default : `[50*(model@d)*(npoints)]`), `"max.generations"` (default : 5), `"wait.generations"` (default : 2), `"BFGSburnin"` (default : 2). ## Examples ```r set.seed(123) ##################################################### ### 2 ITERATIONS OF EGO ON THE BRANIN FUNCTION, ### ### STARTING FROM A 9-POINTS FACTORIAL DESIGN ### ##################################################### # a 9-points factorial design, and the corresponding response d <- 2 n <- 9 design.fact <- expand.grid(seq(0,1,length=3), seq(0,1,length=3)) names(design.fact)<-c("x1", "x2") design.fact <- data.frame(design.fact) names(design.fact)<-c("x1", "x2") response.branin <- apply(design.fact, 1, branin) response.branin <- data.frame(response.branin) names(response.branin) <- "y" # model identification fitted.model1 <- km(~1, design=design.fact, response=response.branin, covtype="gauss", control=list(pop.size=50,trace=FALSE), parinit=c(0.5, 0.5)) # EGO n steps library(rgenoud) nsteps <- 2 # increase to 10 for a more meaningful example lower <- rep(0,d) upper <- rep(1,d) npoints <- 3 # The batchsize oEGO <- qEGO.nsteps(model = fitted.model1, branin, npoints = npoints, nsteps = nsteps, crit="exact", lower, upper, optimcontrol = NULL) print(oEGO$par) print(oEGO$value) plot(c(1:nsteps),oEGO$history,xlab='step',ylab='Current known minimum') ## Not run: # graphics n.grid <- 15 # increase to 21 for better picture x.grid <- y.grid <- seq(0,1,length=n.grid) design.grid <- expand.grid(x.grid, y.grid) response.grid <- apply(design.grid, 1, branin) z.grid <- matrix(response.grid, n.grid, n.grid) contour(x.grid, y.grid, z.grid, 40) title("Branin function") points(design.fact[,1], design.fact[,2], pch=17, col="blue") points(oEGO$par, pch=19, col="red") text(oEGO$par[,1], oEGO$par[,2], labels=c(tcrossprod(rep(1,npoints),1:nsteps)), pos=3) ## End(Not run) ``` ## References C. Chevalier and D. Ginsbourger (2014) Learning and Intelligent Optimization - 7th International Conference, Lion 7, Catania, Italy, January 7-11, 2013, Revised Selected Papers, chapter Fast computation of the multipoint Expected Improvement with applications in batch selection, pages 59-69, Springer. D. Ginsbourger, R. Le Riche, L. Carraro (2007), A Multipoint Criterion for Deterministic Parallel Global Optimization based on Kriging. The International Conference on Non Convex Programming, 2007. D. Ginsbourger, R. Le Riche, and L. Carraro. Kriging is well-suited to parallelize optimization (2010), In Lim Meng Hiot, Yew Soon Ong, Yoel Tenne, and Chi-Keong Goh, editors, **Computational Intelligence in Expensive Optimization Problems**, Adaptation Learning and Optimization, pages 131-162. Springer Berlin Heidelberg. S. Marmin. Developpements pour l'evaluation et la maximisation du critere d'amelioration esperee multipoint en optimisation globale (2014). Master's thesis, Mines Saint-Etienne (France) and University of Bern (Switzerland). J. Mockus (1988), **Bayesian Approach to Global Optimization**. Kluwer academic publishers. M. Schonlau (1997), **Computer experiments and global optimization**, Ph.D. thesis, University of Waterloo. ## See Also `qEI`, `max_qEI`, `qEI.grad` ## Author(s) Sebastien Marmin Clement Chevalier David Ginsbourger

qEGO.nsteps function