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% STK_HALFPINTL computes an intersection of lower half-planes
%
% CALL: [A, B, Z] = stk_halfpintl (A, B)
%
% computes the intersection of the lower half-planes defined by the vector
% of slopes A and the vector of intercepts B. The output vectors A and B
% contain the slopes and intercept of the lines that actually contribute to
% the boundary of the intersection, sorted in such a way that the k^th
% element corresponds to the k^th piece of the piecewise affine boundary.
% The output Z contains the intersection points (shorter by one element).
%
% ALGORITHM
%
% The algorithm implemented in this function is described in [1, 2].
%
% REFERENCE
%
% [1] P. I. Frazier, W. B. Powell, and S. Dayanik. The Knowledge-Gradient
% Policy for Correlated Normal Beliefs. INFORMS Journal on Computing
% 21(4):599-613, 2009.
%
% [2] W. Scott, P. I. Frazier and W. B. Powell. The correlated knowledge
% gradient for simulation optimization of continuous parameters using
% Gaussian process regression. SIAM J. Optim, 21(3):996-1026, 2011.
% Copyright Notice
%
% Copyright (C) 2017 CentraleSupelec
%
% Author: Julien Bect <julien.bect@centralesupelec.fr>
% Copying Permission Statement
%
% 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 [a, b, z] = stk_halfpintl (a, b)
% a: slopes
% b: intercepts
m = length (a);
assert (isequal (size (a), [m 1]));
assert (isequal (size (b), [m 1]));
if m == 1
z = [];
return
end
% 1) Sort by decreasing slopes (and increasing intercept in case of equality)
tmp = [a b];
tmp = sortrows (tmp, [-1 2]);
a_in = tmp(:, 1);
b_in = tmp(:, 2);
% 2) Prepare output lists
a_out = nan (m, 1); % *at most* m lines in the output list
b_out = nan (m, 1);
z_out = nan (m - 1, 1); % the maximal number of intersections in m - 1, then
a_out(1) = a_in(1);
b_out(1) = b_in(1);
k_in = 2; % index of the next input element to be analyzed
k_out = 1; % index of the last element stored in the output list
% 3) Process input list
while k_in <= m
if a_in(k_in) == a_out(k_out) % equality of slopes
k_in = k_in + 1;
else % inequality: a_in(k_in) < a_out(k_out)
% Compute intersection
z = (b_in(k_in) - b_out(k_out)) / (a_out(k_out) - a_in(k_in));
if (k_out == 1) || (z > z_out(k_out-1))
% Insert the new element at the end of the output list
z_out(k_out) = z;
k_out = k_out + 1;
a_out(k_out) = a_in(k_in);
b_out(k_out) = b_in(k_in);
k_in = k_in + 1;
else
% Remove the last element of the output list
k_out = k_out - 1;
end
end % if
end % while
a = a_out(1:k_out);
b = b_out(1:k_out);
z = z_out(1:(k_out - 1));
end % function
%!test % case #1
%! a = 1;
%! b = 1;
%! [a_out, b_out, z_out] = stk_halfpintl (a, b);
%! assert (a_out == 1)
%! assert (b_out == 1)
%! assert (isempty (z_out))
%!test % case #2: two lines, slopes not equal, already sorted
%! a = [1; -1];
%! b = [0; 2];
%! [a_out, b_out, z_out] = stk_halfpintl (a, b);
%! assert (isequal (a_out, [1; -1]))
%! assert (isequal (b_out, [0; 2]))
%! assert (z_out == 1)
%!test % case #3: same as #2, but not sorted
%! a = [-1; 1];
%! b = [ 2; 0];
%! [a_out, b_out, z_out] = stk_halfpintl (a, b);
%! assert (isequal (a_out, [1; -1]))
%! assert (isequal (b_out, [0; 2]))
%! assert (z_out == 1)
%!test % case #4: two lines, equal slopes, already sorted
%! a = [0; 0];
%! b = [1; 2];
%! [a_out, b_out, z_out] = stk_halfpintl (a, b);
%! assert (a_out == 0)
%! assert (b_out == 1)
%! assert (isempty (z_out))
%!test % case #5: same as #4, but not sorted
%! a = [0; 0];
%! b = [2; 1];
%! [a_out, b_out, z_out] = stk_halfpintl (a, b);
%! assert (a_out == 0)
%! assert (b_out == 1)
%! assert (isempty (z_out))
%!test % case #6: add a dominated line to #2 (the result does not change)
%! a = [1; -1; 0];
%! b = [0; 2; 1];
%! [a_out, b_out, z_out] = stk_halfpintl (a, b);
%! assert (isequal (a_out, [1; -1]))
%! assert (isequal (b_out, [0; 2]))
%! assert (z_out == 1)
%!test % case #7: permutation of #6
%! a = [1; 0; -1];
%! b = [0; 1; 2];
%! [a_out, b_out, z_out] = stk_halfpintl (a, b);
%! assert (isequal (a_out, [1; -1]))
%! assert (isequal (b_out, [0; 2]))
%! assert (z_out == 1)
%!test % case #8: another permutation of #6
%! a = [0; 1; -1];
%! b = [1; 0; 2];
%! [a_out, b_out, z_out] = stk_halfpintl (a, b);
%! assert (isequal (a_out, [1; -1]))
%! assert (isequal (b_out, [0; 2]))
%! assert (z_out == 1)
%!test % case #9: same as #8, with some duplicated lines added
%! a = [0; 1; 0; -1; 0; -1; 1];
%! b = [1; 0; 1; 2; 1; 2; 0];
%! [a_out, b_out, z_out] = stk_halfpintl (a, b);
%! assert (isequal (a_out, [1; -1]))
%! assert (isequal (b_out, [0; 2]))
%! assert (z_out == 1)
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