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/* Copyright (c) 1997-2024
Ewgenij Gawrilow, Michael Joswig, and the polymake team
Technische Universität Berlin, Germany
https://polymake.org
This program 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 2, or (at your option) any
later version: http://www.gnu.org/licenses/gpl.txt.
This program 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.
--------------------------------------------------------------------------------
*/
#pragma once
#include "polymake/client.h"
#include "polymake/linalg.h"
#include "polymake/Matrix.h"
#include "polymake/polytope/solve_LP.h"
namespace polymake { namespace polytope {
// find a point in the true interior of the polyhedron spanned by the facets f
// works with inequalities too
template <typename MatrixType, typename Coord>
Vector<Coord> inner_point_from_facets(const GenericMatrix<MatrixType, Coord>& F)
{
/* The LP should maximize eps in F * x >= eps, 0 <= eps <= 1
The LP looks like this:
|0 -1 | |
|. . | | | z | |0|
|. . | F | |eps| |.|
|0 -1 | | * |x_0| >= |.|
|--------------| |...| |.|
|0 1 | 0 . . 0| |x_n| |0|
|1 -1 | 0 . . 0|
The solver sets z=0 or z=1 always
second-to-last row guarantees eps >= 0
last row guarantees z-eps>=0
we catch the eps=0 case below.
*/
Int c = F.cols();
Int r = F.rows();
const Matrix<Coord> ineqs = (zero_vector<Coord>(r) | (-ones_vector<Coord>(r) | F))
/ unit_vector<Coord>(c+2,1) // eps >= 0
/ (unit_vector<Coord>(c+2,0) - unit_vector<Coord>(c+2,1)); // eps <= 1
const auto S = solve_LP(ineqs, unit_vector<Coord>(c+2, 1), true); // maximize eps
if (S.status != LP_status::valid || S.solution[1] == 0)
throw std::runtime_error("Polyhedron has empty interior.");
Vector<Coord> ip = S.solution.slice(sequence(2, F.cols()));
if (!is_zero(ip[0])) ip.dehomogenize();
return ip;
}
} }
// Local Variables:
// mode:C++
// c-basic-offset:3
// indent-tabs-mode:nil
// End:
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