MSSearch: Local search algorithm for high quality design generation

Description

The MSSearch function can be used to obtain an optimal multi-stratum experimental design considering one or more optimality criteria, up to a maximum of six criteria simultaneously.
This function implements the procedure MS-Opt proposed by Sambo, Borrotti, Mylona e Gilmour (2016) as an extension of the Coordinate-Exchange (CE) algorithm for constructing approximately optimal designs. This innovative procedure is able to handle all possible multi-stratum experimental structures and, instead of minimizing a single objective function as in the original CE algorithm, it seeks to minimize the following scalarization of the objective functions for all considered criteria: $$f_W = \sum_{c \in C}{\alpha_c f_c(d; \eta)=\overline{\alpha} \cdot \overline{f}},$$ with $$\sum_{c \in C} \alpha_c = 1,$$ where $C$ is the set of criteria to be minimized, $f_c$ is the objective function for the $c$ criterion and $\overline{\alpha}$ is the vector that controls the relative weights of the objective functions.

Usage

MSSearch(msopt, alpha, ...)

Value

MSSearch returns a list, whose elements are:

optsol: A design matrix. The best solution found.
optscore: A vector containing the criteria scores for optsol.
feval: An integer representing the number of score function evaluations (number of $f_W$ evaluations over all iterations).
trend: A vector of length r. The $i$-th element is the value that minimizes $f_W$ for the $i$-th iteration.

Arguments

msopt: A list as returned by the MSOpt function.
alpha: A vector of weights, whose elements must sum to one. length(alpha) must be equal to the number of criteria considered, that is it must be equal to the length of the criteria element of msopt.
...: optional arguments (see Details).

Details

MSSearch by default does not apply any normalization to the individual objective functions $f_c$ before the calculation of f_w is performed. However, it is possible to subject the vector of objective functions $\overline{f}$ to the following transformation: $$\overline{f}_{norm} = \frac{\overline{f} - CritTR}{CritSC},$$ by specifying $CritTR$ and $CritSC$ vectors as additional parameters, as described below.

Additional arguments can be specified as follows:

'Start', sol: A string and a matrix, used in pair. They provide a starting solution (or an initial design) to the algorithm. By default the initial solution is randomly generated following the SampleDesign() procedure described in Sambo, Borrotti, Mylona and Gilmour (2016).
'Restarts', r : A string and an integer, used in pair. When r=1, the default value, the procedure implemented in MSSearch results in a local search algorithm that optimizes the objective function $f_W$ starting from one initial design in the design space. These parameters allows to restart the algorithm r times. If no initial design is passed a different starting solution is generated for each iteration, letting the probability to find a global minimum be higher. Mssearch returns the solution that minimizes $f_W$ across all the r iterations.
'Normalize', c(CritTR, CritSC): A string and a vector, used in pair. By specifying the CritTR and CritSC vectors, the user can establish the normalization factors to be applied to each objective function before evaluating $f_W$. CritTR and CritSC are vectors of length equal to the number of criteria, whose default elements are 0 and 1 respectively.

References

M. Borrotti and F. Sambo and K. Mylona and S. Gilmour. A multi-objective coordinate-exchange two-phase local search algorithm for multi-stratum experiments. Statistics & Computing, 2016.

Examples

Run this code

library(multiDoE)

## To check the number of digits to be printed.
backup_options <- options()
options(digits = 10)

## Definition of parameters for experimental setup
facts <- list(1, 2:5)
units <- list(21, 2)
level <- 3
etas <- list(1)
model2 <- "quadratic"

## Single-objective optimization
criteria_S <- c('I')
msopt_S <- MSOpt(facts, units, level, etas, criteria_S, model2)
# \donttest{
mssearch_S <- MSSearch(msopt_S, alpha = 1, "Restarts", 100)
# }

## Multi-objective optimization
criteria_M <- c('Id', 'Ds', 'As')
msopt_M <- MSOpt(facts, units, level, etas, criteria_M, model2)
# \donttest{
mssearch_M <- MSSearch(msopt_M, alpha = c(1/2, 1/4, 1/4), "Restarts", 100)
# }

options(backup_options)

## To reduce the computational cost of MSSearch function, you may reduce the number of restarts.

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