Optimize (minimize) a noisy black-box function (i.e., a function which
may not be differentiable, and may not execute deterministically).
A b* tgp model is used as a surrogate for
adaptive sampling via improvement (and other) statistics. Note that
this function is intended as a skeleton to be tailored as required
for a particular application
optim.step.tgp(f, rect, model = btgp, prev = NULL, X = NULL,
Z = NULL, NN = 20 * length(rect), improv = c(1, 5), cands = c("lhs", "tdopt"),
method = c("L-BFGS-B", "Nelder-Mead", "BFGS", "CG", "SANN", "optimize"), ...)
optim.ptgpf(start, rect, tgp.obj,
method=c("L-BFGS-B", "Nelder-Mead", "BFGS", "CG", "SANN", "optimize"))A function to be optimized, having only one free argument
matrix indicating the domain of the argument of
f over which an optimal should be searched; must have
ncol(rect) = 2 and nrow agreeing with the argument
of f indicating the dimension of the data. For 1-d data,
a vector of length 2 is allowed
The b* regression model used as a surrogate
for optimization; see btgp, and others,
for more detail
The output from a previous call to optim.step.tgp;
this should be a list with entries as described the “Value”
section below
data.frame, matrix, or vector of current
inputs X, to be augmented
Vector of current output responses Z of length
equal to the leading dimension (rows) of X, i.e.,
length(Z) == nrow(X), to be augmented
Number of candidate locations (XX) at which to
sample from the improvement statistic
Indicates the improv argument provided
to a b* model function for sampling from the
improvement statistic at the NN candidate locations
(XX); see btgp, and others, for more detail
The type of candidates (XX)
at which samples from the improvement statistics are gathered.
The default setting of "lhs" is recommended. However,
a sequential treed D-optimal design can be used with "tdopt"
for a more global exploration; see tgp.design for
more details
Further arguments to the b* model
function
A starting value for optimization of the MAP predictive
(kriging) surface of a "tgp"-class object. A good starting
value is the X or XX location found to be a minimum
in the mean predictive surface contained in "tgp"-class
object
The list return has the following components.
A matrix with nrow(rect) columns whose
rows contain recommendations for input locations to add into
the design
A one-row data.frame indicating the
the X-location and objective value of the current best guess
of the solution to the (kriging) surrogate optimization along with the
maximum values of the improvement statistic
the "tgp"-class object output from the
model function
optim.step.tgp executes one step in a search for the global
optimum (minimum) of a noisy function (Z~f(X)) in a bounded
rectangle (rect). The procedure essentially fits a tgp
model and samples from
the posterior distribution of improvement
statistics at NN+1 candidates locations.
NN of the candidates come from
cands, i.e., "lhs" or "tdopt", plus one which
is the location of the minima found in a previous run via
prev by using optim (with a particular
method or optimize instead) on the MAP
model predictive surface using the "tgp"-class object
contained therein.
The improv[2] with the the highest expected improvement are
recommended for adding into the design on output.
optim.ptgpf is the subroutine used by
optim.step.tgp to find optimize on the MAP (surrogate)
predictive surface for the "tgp"-class object contained in
prev.
Please see vignette("tgp2") for a detailed illustration
Gramacy, R. B. (2020) Surrogates: Gaussian Process Modeling, Design and Optimization for the Applied Sciences. Boca Raton, Florida: Chapman Hall/CRC. (See Chapter 7.) https://bobby.gramacy.com/surrogates/
Matthew Taddy, Herbert K.H. Lee, Genetha A. Gray, and Joshua D. Griffin. (2009) Bayesian guided pattern search for robust local optimization. Technometrics, 51(4), pp. 389-401
btgp, etc., optim,
optimize, tgp.design,
predict.tgp, dopt.gp
# NOT RUN {
## optimize the simple exponential function
f <- function(x) { exp2d.Z(x)$Z }
## create the initial design with D-optimal candidates
rect <- rbind(c(-2,6), c(-2,6))
Xcand <- lhs(500, rect)
X <- dopt.gp(50, X=NULL, Xcand)$XX
Z <- f(X)
## do 10 rounds of adaptive sampling
out <- progress <- NULL
for(i in 1:10) {
## get recommendations for the next point to sample
out <- optim.step.tgp(f, X=X, Z=Z, rect=rect, prev=out)
## add in the inputs, and newly sampled outputs
X <- rbind(X, out$X)
Z <- c(Z, f(out$X))
## keep track of progress and best optimum
progress <- rbind(progress, out$progress)
print(progress[i,])
}
## plot the progress so far
par(mfrow=c(2,2))
plot(out$obj, layout="surf")
plot(out$obj, layout="as", as="improv")
matplot(progress[,1:nrow(rect)], main="optim results",
xlab="rounds", ylab="x[,1:2]", type="l", lwd=2)
plot(log(progress$improv), type="l", main="max log improv",
xlab="rounds", ylab="max log(improv)")
# }