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% STK_TESTFUN_HARTMAN4 computes the "Hartman4" function
%
% CALL: Y = stk_testfun_hartman4 (X)
%
% computes the value Y of the Hartman4 function at X.
%
% The Hartman4 function is a test function in dimension 4,
% which is usually minimized over [0, 1]^4.
%
% HISTORICAL REMARKS
%
% This function belongs to a general class of test functions introduced
% by Hartman [1]. The particular set of coefficients used in the
% Hartman4 function seems to have been introduced by [2].
%
% Note that the test function used in [2] is a scaled version of the
% one implemented here, which can be recovered as follows:
%
% y = (1.1 + stk_testfun_hartman4 (x)) / 0.839;
%
% Picheny & co-authors [2] refer to Dixon & Szego [3] for this test
% function, but it turns out that [3] only contains two sorts of
% "Hartman functions", in dimensions three and six.
%
% In fact, this function appears to have been obtained by truncating
% the sum at the fourth coordinate in the six-dimensional Hartman
% function of [3].
%
% GLOBAL MINIMUM
%
% According to [4], the function has one global minimum at
%
% x = [0.1873 0.1906 0.5566 0.2647].
%
% The corresponding function value, with our definition of the test
% function, is:
%
% f(x) = -3.729722308557300.
%
% Slightly better function values can be found in the neighborhood of
% this point. For instance, with
%
% x = [0.18744768 0.19414868 0.558005333 0.26476409]
%
% we get
%
% f(x) = -3.729840440436292.
%
% REFERENCES
%
% [1] J. K. Hartman (1973). Some experiments in global optimization.
% Naval Research Logistics Quarterly, 20(3):569-576.
%
% [2] V. Picheny, T. Wagner & D. Ginsbourger (2013). A benchmark
% of kriging-based infill criteria for noisy optimization.
% Structural and Multidisciplinary Optimization, 48:607-626.
%
% [3] L. C. W. Dixon & G. P. Szego (1978). Towards Global
% Optimization 2, North-Holland, Amsterdam, The Netherlands
%
% [4] V. Picheny, D. Ginsbourger & O. Roustant (2021). DiceOptim:
% Kriging-Based Optimization for Computer Experiments. R package
% version 2.1.1. URL: https://CRAN.R-project.org/package=DiceOptim.
% Author
%
% Julien Bect <julien.bect@centralesupelec.fr>
% Copying Permission Statement (this file)
%
% To the extent possible under law, CentraleSupelec has waived all
% copyright and related or neighboring rights to
% stk_testfun_hartman4.m. This work is published from France.
%
% License: CC0 <http://creativecommons.org/publicdomain/zero/1.0/>
% Copying Permission Statement (STK toolbox as a whole)
%
% This file is part of
%
% STK: a Small (Matlab/Octave) Toolbox for Kriging
% (https://github.com/stk-kriging/stk/)
%
% STK is free software: you can redistribute it and/or modify it under
% the terms of the GNU General Public License as published by the Free
% Software Foundation, either version 3 of the License, or (at your
% option) any later version.
%
% STK is distributed in the hope that it will be useful, but WITHOUT
% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
% or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
% License for more details.
%
% You should have received a copy of the GNU General Public License
% along with STK. If not, see <http://www.gnu.org/licenses/>.
function y = stk_testfun_hartman4 (x)
a = [ ...
[ 10.00 0.05 3.00 17.00 ]; ...
[ 3.00 10.00 3.50 8.00 ]; ...
[ 17.00 17.00 1.70 0.05 ]; ...
[ 3.50 0.10 10.00 10.00 ]];
p = [ ...
[ 0.1312 0.2329 0.2348 0.4047]; ...
[ 0.1696 0.4135 0.1451 0.8828]; ...
[ 0.5569 0.8307 0.3522 0.8732]; ...
[ 0.0124 0.3736 0.2883 0.5743]];
c = [1.0 1.2 3.0 3.2];
y = stk_testfun_hartman_generic (x, a, p, c);
end % function
%!test
%! x = [0.1873 0.1906 0.5566 0.2647 ;
%! 0.18744768 0.19414868 0.558005333 0.26476409];
%! y = stk_testfun_hartman4 (x);
%! assert (stk_isequal_tolabs (y, ...
%! [-3.729722308557300; -3.729840440436292], 1e-15));
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