easyGParetoptim {GPareto}

EGO algorithm for multiobjective optimization

Description

User-friendly wrapper of the function GParetoptim. Generates initial DOEs and kriging models (objects of class km), and executes nsteps iterations of multiobjective EGO methods.

Usage

easyGParetoptim(
  fn,
  ...,
  cheapfn = NULL,
  budget,
  lower,
  upper,
  par = NULL,
  value = NULL,
  noise.var = NULL,
  control = list(method = "SMS", trace = 1, inneroptim = "pso", maxit = 100, seed = 42),
  ncores = 1
)

Arguments

fn

the multi-objective function to be minimized (vectorial output), found by a call to match.fun, see details,

...

additional parameters to be given to the objective fn.

cheapfn

optional additional fast-to-evaluate objective function (handled next with class fastfun), which does not need a kriging model, handled by a call to match.fun,

budget

total number of calls to the objective function,

lower

vector of lower bounds for the variables to be optimized over,

upper

vector of upper bounds for the variables to be optimized over,

par

initial design of experiments. If not provided, par is taken as a maximin LHD with budget/3 points,

value

initial set of objective observations fn(par). Computed if not provided. Not that value may NOT contain any cheapfn value,

noise.var

optional noise variance, for noisy objectives fn. If not NULL, either a scalar (constant noise, identical for all objectives), a vector (constant noise, different for each objective) or a function (type closure) with vectorial output (variable noise, different for each objective). Alternatively, set noise.var="given_by_fn", see details.

control

an optional list of control parameters. See "Details",

ncores

number of CPU available (> 1 makes mean parallel TRUE). Only used with discrete optimization for now.

Details

Does not require specific knowledge on kriging models (objects of class km).

The problem considered is of the form: min f(x) = f_1(x), ..., f_p(x). The control argument is a list that can supply any of the following optional components:

• method: choice of multiobjective improvement function: "SMS", "EHI", "EMI" or "SUR" (see crit_SMS, crit_EHI, crit_EMI, crit_SUR),

• trace: if positive, tracing information on the progress of the optimization is produced (1 (default) for general progress, >1 for more details, e.g., warnings from genoud),

• inneroptim: choice of the inner optimization algorithm: "genoud", "pso" or "random" (see genoud and psoptim),

• maxit: maximum number of iterations of the inner loop,

• seed: to fix the random variable generator,

• refPoint: reference point for hypervolume computations (for "SMS" and "EHI" methods),

• extendper: if no reference point refPoint is provided, for each objective it is fixed to the maximum over the Pareto front plus extendper times the range. Default value to 0.2, corresponding to 1.1 for a scaled objective with a Pareto front in [0,1]^n.obj.

If noise.var="given_by_fn", fn must return a list of two vectors, the first being the objective functions and the second the corresponding noise variances. See examples in GParetoptim.

For additional details or other possible arguments, see GParetoptim.

Display of results and various post-processings are available with plotGPareto.

Value

A list with components:

• par: all the non-dominated points found,

• value: the matrix of objective values at the points given in par,

• history: a list containing all the points visited by the algorithm (X) and their corresponding objectives (y),

• model: a list of objects of class km, corresponding to the last kriging models fitted.

Note that in the case of noisy problems, value and history$y.denoised are denoised values. The original observations are available in the slot history$y.

Author(s)

Victor Picheny (INRA, Castanet-Tolosan, France)

Mickael Binois (Mines Saint-Etienne/Renault, France)

References

M. T. Emmerich, A. H. Deutz, J. W. Klinkenberg (2011), Hypervolume-based expected improvement: Monotonicity properties and exact computation, Evolutionary Computation (CEC), 2147-2154.

V. Picheny (2015), Multiobjective optimization using Gaussian process emulators via stepwise uncertainty reduction, Statistics and Computing, 25(6), 1265-1280.

T. Wagner, M. Emmerich, A. Deutz, W. Ponweiser (2010), On expected-improvement criteria for model-based multi-objective optimization. Parallel Problem Solving from Nature, 718-727, Springer, Berlin.

J. D. Svenson (2011), Computer Experiments: Multiobjective Optimization and Sensitivity Analysis, Ohio State university, PhD thesis.

M. Binois, V. Picheny (2019), GPareto: An R Package for Gaussian-Process-Based Multi-Objective Optimization and Analysis, Journal of Statistical Software, 89(8), 1-30, doi:10.18637/jss.v089.i08.

Examples

#---------------------------------------------------------------------------
# 2D objective function, 4 cases
#---------------------------------------------------------------------------
## Not run: 
set.seed(25468)
n_var <- 2 
fname <- ZDT3
lower <- rep(0, n_var)
upper <- rep(1, n_var)

#---------------------------------------------------------------------------
# 1- Expected Hypervolume Improvement optimization, using pso
#---------------------------------------------------------------------------
res <- easyGParetoptim(fn=fname, lower=lower, upper=upper, budget=15, 
                   control=list(method="EHI", inneroptim="pso", maxit=20))
par(mfrow=c(1,2))
plotGPareto(res)
title("Pareto Front")
plot(res$history$X, main="Pareto set", col = "red", pch = 20)
points(res$par, col="blue", pch = 17)

#---------------------------------------------------------------------------
# 2- SMS Improvement optimization using random search, with initial DOE given
#---------------------------------------------------------------------------
library(DiceDesign)
design.init   <- maximinESE_LHS(lhsDesign(10, n_var, seed = 42)$design)$design
response.init <- t(apply(design.init, 1, fname))
res <- easyGParetoptim(fn=fname, par=design.init, value=response.init, lower=lower, upper=upper, 
                       budget=15, control=list(method="SMS", inneroptim="random", maxit=100))
par(mfrow=c(1,2))
plotGPareto(res)
title("Pareto Front")
plot(res$history$X, main="Pareto set", col = "red", pch = 20)
points(res$par, col="blue", pch = 17)

#---------------------------------------------------------------------------
# 3- Stepwise Uncertainty Reduction optimization, with one fast objective function
#---------------------------------------------------------------------------
fname <- camelback
cheapfn <- function(x) {
if (is.null(dim(x))) return(-sum(x))
else return(-rowSums(x))
}
res <- easyGParetoptim(fn=fname, cheapfn=cheapfn, lower=lower, upper=upper, budget=15, 
                   control=list(method="SUR", inneroptim="pso", maxit=20))
par(mfrow=c(1,2))
plotGPareto(res)
title("Pareto Front")
plot(res$history$X, main="Pareto set", col = "red", pch = 20)
points(res$par, col="blue", pch = 17)

#---------------------------------------------------------------------------
# 4- Expected Hypervolume Improvement optimization, using pso, noisy fn
#---------------------------------------------------------------------------
noise.var <- c(0.1, 0.2)
funnoise <- function(x) {ZDT3(x) + sqrt(noise.var)*rnorm(n=2)}
res <- easyGParetoptim(fn=funnoise, lower=lower, upper=upper, budget=30, noise.var=noise.var,
                       control=list(method="EHI", inneroptim="pso", maxit=20))
par(mfrow=c(1,2))
plotGPareto(res)
title("Pareto Front")
plot(res$history$X, main="Pareto set", col = "red", pch = 20)
points(res$par, col="blue", pch = 17)

#---------------------------------------------------------------------------
# 5- Stepwise Uncertainty Reduction optimization, functional noise
#---------------------------------------------------------------------------
funnoise <- function(x) {ZDT3(x) + sqrt(abs(0.1*x))*rnorm(n=2)}
noise.var <- function(x) return(abs(0.1*x))

res <- easyGParetoptim(fn=funnoise, lower=lower, upper=upper, budget=30, noise.var=noise.var,
                     control=list(method="SUR", inneroptim="pso", maxit=20))
par(mfrow=c(1,2))
plotGPareto(res)
title("Pareto Front")
plot(res$history$X, main="Pareto set", col = "red", pch = 20)
points(res$par, col="blue", pch = 17)

## End(Not run)