# Maximization of multipoint expected improvement criterion (qEI) Maximization of the `qEI` criterion. Two options are available : Constant Liar (CL), and brute force qEI maximization with Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, or GENetic Optimization Using Derivative (genoud) algorithm. ```r max_qEI( model, npoints, lower, upper, crit = "exact", minimization = TRUE, optimcontrol = NULL ) ``` ## Arguments - `model`: an object of class `km` , - `npoints`: an integer representing the desired number of iterations, - `lower`: vector of lower bounds, - `upper`: vector of upper bounds, - `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 the qEI to be maximized is used in minimiziation or in maximization, - `optimcontrol`: an optional list of control parameters for optimization. See details. ## Returns A list with components: - **par**: A matrix containing the `npoints` input vectors found. - **value**: A value giving the qEI computed in `par`. ## Details - CL is a heuristic method. First, the regular Expected Improvement EI is maximized (`max_EI`). Then, for the next points, the Expected Improvement is maximized again, but with an artificially updated Kriging model. Since the response values corresponding to the last best point obtained are not available, the idea of CL is to replace them by an arbitrary constant value L (a "lie") set by the user (default is the minimum of all currently available observations). - The BFGS algorithm is implemented in the standard function `optim`. Analytical formulae of `qEI` and its gradient `qEI.grad` are used. The `nStarts` starting points are by default sampled with respect to the regular EI (`sampleFromEI`) criterion. - The "genoud" method calls the function `genoud` using analytical formulae of `qEI` and its gradient `qEI.grad`. 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$parinit` : optional matrix of candidate starting points (one row corresponds to one point), + The parameters of the `genoud` function. Main parameters are `"pop.size"` (default : `[50*(model@d)*(npoints)]`), `"max.generations"` (default : 5), `"wait.generations"` (default : 2), `"BFGSburnin"` (default : 2). ## Examples ```r set.seed(000) # 3-points EI maximization. # 9-points factorial design, and the corresponding response d <- 2 n <- 9 design.fact <- expand.grid(seq(0,1,length=3), seq(0,1,length=3)) names(design.fact)<-c("x1", "x2") design.fact <- data.frame(design.fact) names(design.fact)<-c("x1", "x2") response.branin <- apply(design.fact, 1, branin) response.branin <- data.frame(response.branin) names(response.branin) <- "y" lower <- c(0,0) upper <- c(1,1) # number of point in the bacth batchSize <- 3 # model identification fitted.model <- km(~1, design=design.fact, response=response.branin, covtype="gauss", control=list(pop.size=50,trace=FALSE), parinit=c(0.5, 0.5)) # maximization of qEI # With a multistarted BFGS algorithm maxBFGS <- max_qEI(model = fitted.model, npoints = batchSize, lower = lower, upper = upper, crit = "exact",optimcontrol=list(nStarts=3,method = "BFGS")) # comparison print(maxBFGS$value) ## Not run: # With a genetic algorithme using derivatives maxGen <- max_qEI(model = fitted.model, npoints = batchSize, lower = lower, upper = upper, crit = "exact", optimcontrol=list(nStarts=3,method = "genoud",pop.size=100,max.generations = 15)) # With the constant liar heuristic maxCL <- max_qEI(model = fitted.model, npoints = batchSize, lower = lower, upper = upper, crit = "CL",optimcontrol=list(pop.size=20)) print(maxGen$value) print(maxCL$value) ## 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. 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`, `qEI.grad` ## Author(s) Sebastien Marmin Clement Chevalier David Ginsbourger

max_qEI function