CGAL 4.14 - Linear and Quadratic Programming Solver
QP_solver/infeasibility_certificate.cpp
// example: extracting and verifying a proof of infeasibility from the solution
#include <cassert>
#include <CGAL/QP_models.h>
#include <CGAL/QP_functions.h>
// choose exact integral type
#ifdef CGAL_USE_GMP
#include <CGAL/Gmpz.h>
typedef CGAL::Gmpz ET;
#else
#include <CGAL/MP_Float.h>
typedef CGAL::MP_Float ET;
#endif
// program and solution types
<int**, // for A
int*, // for b
int*> // for c
Program;
// we demonstrate Farkas Lemma: either the system
// A x <= b
// x >= 0
// has a solution, or there exists y such that
// y >= 0
// y^TA >= 0
// y^Tb < 0
// In the following instance, the first system has no solution,
// since adding up the two inequalities gives x_2 <= -1:
// x_1 - 2x_2 <= 1
// -x_1 + 3x_2 <= -2
// x_1, x_2 >= 0
int main() {
int Ax1[] = { 1, -1}; // column for x1
int Ax2[] = {-2, 3}; // column for x2
int* A[] = {Ax1, Ax2}; // A comes columnwise
int b[] = {1, -2}; // right-hand side
r[] = {CGAL::SMALLER, CGAL::SMALLER}; // constraints are "<="
int c[] = {0, 0}; // zero objective function
// now construct the linear program; the first two parameters are
// the number of variables and the number of constraints (rows of A)
Program lp (2, 2, A, b, r, c);
// solve the program, using ET as the exact type
// get certificate for infeasibility
assert (s.is_infeasible());
Solution::Infeasibility_certificate_iterator y =
s.infeasibility_certificate_begin();
// check y >= 0
assert (ET(y[0]) >= 0);
assert (ET(y[1]) >= 0);
// check y^T A >= 0
assert (ET(y[0]) * A[0][0] + ET(y[1]) * A[0][1] >= 0);
assert (ET(y[0]) * A[1][0] + ET(y[1]) * A[1][1] >= 0);
// check y^T b < 0
assert (ET(y[0]) * b[0] + ET(y[1]) * b[1] < 0);
return 0;
}