mirror of https://github.com/CGAL/cgal
76 lines
2.5 KiB
C++
76 lines
2.5 KiB
C++
// example: extracting and verifying a proof of unboundedness from the solution
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#include <cassert>
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#include <CGAL/QP_models.h>
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#include <CGAL/QP_functions.h>
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// choose exact integral type
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#ifdef CGAL_USE_GMP
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#include <CGAL/Gmpz.h>
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typedef CGAL::Gmpz ET;
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#else
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#include <CGAL/MP_Float.h>
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typedef CGAL::MP_Float ET;
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#endif
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// program and solution types
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typedef CGAL::Nonnegative_linear_program_from_iterators
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<int**, // for A
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int*, // for b
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CGAL::Comparison_result*, // for r
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int*> // for c
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Program;
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typedef CGAL::Quadratic_program_solution<ET> Solution;
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// we demonstrate the unboundedness certificate: either the feasible
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// linear program
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// min c^T x
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// A x <= b
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// x >= 0
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// is bounded, or there exists w such that
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// w >= 0
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// A w <= 0
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// c^t w < 0,
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// in which case the objective function becomes arbitrarily small
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// along the ray {x* + tw | t >= 0}, x* any feasible solution.
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//
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// In the following instance, the linear program is unbounded, since
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// the ray {(t,t)| t>= 1} is feasible, and the objective function becomes
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// arbitrarily small on it:
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// min -x_1 - 2x_2
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// x_1 - 2x_2 <= -1
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// -x_1 + x_2 <= 2
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// x_1, x_2 >= 0
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int main() {
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int Ax1[] = { 1, -1}; // column for x1
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int Ax2[] = {-2, 1}; // column for x2
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int* A[] = {Ax1, Ax2}; // A comes columnwise
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int b[] = {-1, 2}; // right-hand side
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CGAL::Comparison_result
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r[] = {CGAL::SMALLER, CGAL::SMALLER}; // constraints are "<="
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int c[] = {-1, -2}; // objective function
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// now construct the linear program; the first two parameters are
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// the number of variables and the number of constraints (rows of A)
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Program lp (2, 2, A, b, r, c);
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// solve the program, using ET as the exact type
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Solution s = CGAL::solve_nonnegative_linear_program(lp, ET());
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// get certificate for unboundedness
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assert (s.is_unbounded());
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Solution::Unboundedness_certificate_iterator w =
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s.unboundedness_certificate_begin();
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// check w >= 0
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assert (ET(w[0]) >= 0);
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assert (ET(w[1]) >= 0);
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// check A w <= 0
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assert (A[0][0] * ET(w[0]) + A[1][0] * ET(w[1]) <= 0);
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assert (A[0][1] * ET(w[0]) + A[1][1] * ET(w[1]) <= 0);
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// check c^T w < 0
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assert (c[0] * ET(w[0]) + c[1] * ET(w[1]) < 0);
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return 0;
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}
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