CPoptim {CPoptim}

CPoptim

Description

Minimise a given objective function in a bounded search space.

Usage

CPoptim(rFunction,lower,upper,maxFE,sampleSize)

Arguments

rFunction

the function to minimise

lower

a vector providing the lower bounds of the search space

upper

a vector providing the upper bounds of the search space

maxFE

number of function evaluations to compute (default 5000*dim)

sampleSize

sample size per partition (default 1000)

Details

Convex Partition is a black-box optimisation algorithm for single objective functions with real parameters. The basic principle is to progressively estimate and exploit a regression tree similar to CART (Classification and Regression Tree) of the objective function. This model is computed by recursively partitioning the function domain into subsets and estimating local statistics for each one. The subsets most likely to contain the optimum are selected by the Bayesian inference method proposed in de Paz et al. (2024).

The optimisation strategy consists of iterating two phases: 1) Selecting the most promising subset to contain extreme values according to the current model, and 2) Updating the model by sampling over the most promising subsets.

Value

CPoptim returns two lists: sample and subsets. If the function call is not proper, CPoptim returns NULL.

sample contains three matrices that summarise information about each evaluated point. The i-th row of these matrices contains:

sample$x

the i-th point that was evaluated

sample$y

the function evaluation for the i-th point

sample$subsetID

the subset-id to which the i-th point belongs

subsets summarises information about each defined subset. The i-th row of each matrix contains

subsets$lower

the lower bounds of the i-th subset

subsets$upper

the upper bounds of the i-th subset

subsets$mean

the mean value of the objective fun. in the i-th subset

subsets$std

the standard deviation of the objective fun. in the i-th subset

subsets$aPriori

the relative (length, area, volume, etc.) of the i-th subset

subsets$aPosteriori

the posteriori probability that the i-th subset contains the optimum

Author(s)

The design is inspired by the algorithm proposed in de Paz et al. (2024). However, instead of the original regression tree based on simplexes, this implementation is based on hyper-rectangular subsets (a model similar to the continuous Classification and Regression Trees) Loh (2011).

References

de Paz, Erick G.G., et al. (2024). A Regression Tree as Acquisition Function for Low-dimensional Optimisation. Pattern Recognition. MCPR 2024. Lecture Notes in Computer Science, vol 14755. Springer, Cham. doi: 10.1007/978-3-031-62836-8_3

Loh, Wei-Yin (2011). Classification and regression trees. WIREs Data Mining Knowl Discov 2011 1 14-23. John Wiley & Sons, Inc. doi: 10.1002/widm.8

Examples

## An illustrative function
sphere<-function(X) sum(X^2)^.5
bounds<-rep(5,10)

## Calling the CPoptim function
obj<-CPoptim(sphere, -bounds, +bounds)

## Ploting the convergence curve
plot(obj$sample$y,t='l')

## Selecting the best X evaluated
optimum.x<-obj$sample$x[which.min(obj$sample$y),]