R/hzhf_doc.R

In danielhorn/hierarchTestFuns: Test functions for numerical optimization with hierarchical parameters

#' @title Generators for HZHF test function
#'
#' @description
#' Hierarchical test functions by Horn and Zaefferer
#'
#' @param id [\code{integer(1)}] \cr
#'   ID of hzhf function to generate, element of {1, 2, ..., 15}.
#' @param in.dim [\code{integer(1)}] \cr
#'   Size of parameter space.
#' @param k [\code{integer(1)}] \cr
#'   Number of non-hierarchical parameters. Must be smaller than in.dim - 1,
#'   must be divisible by 2 for HFHF6, HZHF7, HZHF8.
#' @param c [\code{double(1)}] \cr
#'   Metaparameter of all test functions, size of hierarchical area, see details. Default is 0.5.
#' @param s [\code{double(1)}] \cr
#'   Metaparameter of all test functions, shift of hierarchical area, see details.. Default is 0.
#' @param check [\code{logical(1)}] \cr
#'   Shall parameter checks be performed befor each function evaluation? Default is TRUE
#' @return A \code{hzhf_function}, subclass of \code{smoof_function}.
#'
#' @details
#'
#' Hierarchical test functions have several parameters that only influence the
#' outcome if a certain condition is fullfilled. The HZHF test function family
#' consists of 15 hierarchical test functions with numeric parameters
#' \eqn{z[1], ... z[in.dim]}. The first \eqn{k} of these parameters always affect
#' the function value while the last \eqn{in.dim - k} ones only do if \eqn{z[1]}
#' fullfills a certain condition.
#'
#' All HZHF functions are to be minimized and follow the scheme
#'
#' \eqn{f(z) = a(z[1], ..., z[k]) +} if (\eqn{z[1] < c}) \eqn{b(z[1], ..., z[in.dim])} else s
#'
#' Here, \eqn{c} controls the size of the active region of the hierarchical
#' parameters. \eqn{s} is a shift added to the function if the hierarchical
#' parameters are inactive and controls a) whether an imputation approach
#' is promising and b) the jump height at the discontinuity.
#'
#' The function \eqn{a} and \eqn{b} are defined using different transformation
#' of the WFG toolkit (A toolkit for multi-objective test functions). The 15
#' HZHF function differ in their properties, as for example modality and
#' seperability.
#'
#' For \eqn{s = 0} and \eqn{c < 0.7} the global optimum is allways at \eqn{z[i] = 0.7 * i, i = 1, ..., in.dim}
#' (the hierarchical parameters are active).
#' If \eqn{s = 0} and \eqn{c > 0.7}  holds, the optimum is at \eqn{z[i] = 0.7 * i, i = 1, ..., k},
#' \eqn{z[i] = NA, i = k + 1, ..., in.dim}, i.e. the hierarchical parameters are inactive.
#' The best possible function value is 0 in both cases.
#'
#' If \eqn{s < 0} and the optimum is in in the inactive region, its function value
#' is shifted to \eqn{s}, but its location remains unchanged. If \eqn{s < 0} and
#' the optimum is in the active region, the former global optimum is only
#' guarenteed to be a local optimum with function value \eqn{0}. Their may be
#' other local optima in the inactive regions with function values \eqn{< 0},
#' especially at the discontinuity \eqn{z[1] = c}. Unfortunately, we can not
#' control those values.
#'
#' If \eqn{s > 0} and the optimum is in the active region, it is guarenteed
#' to be the global optimum with function value 0. If, however, \eqn{s > 0}
#' and the optimum is in the inactive region, it is only guarenteed to be
#' a local optimum with function value \eqn{s}. There may be other local optima
#' in the active regions with function values between \eqn{0} and \eqn{b},
#' especially at the discontinuity \eqn{z[1] = c}. Unfortunately, we can not
#' control those values.
#'
#' @references
#' Huband, Simon ; Hingston, Phil ; Barone, Luigi ; While, Lyndon:
#' A Review of Multiobjective Test Problems and a Scalable Test Problem
#' Toolkit. In: IEEE Trans. on Evolutionary Computation 10 (2006),
#' No. 5, pp. 477-506
#'
#' @seealso \code{\link{plot.hzhf_function}}
#'
#' @aliases hzhf HZHF
#'
#' @name hzhf

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