\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - Linear and Quadratic Programming Solver

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QP_solver/unboundedness_certificate.cpp
// example: extracting and verifying a proof of unboundedness from the solution
#include <cassert>
#include <CGAL/basic.h>
#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
typedef CGAL::Nonnegative_linear_program_from_iterators
<int**, // for A
int*, // for b
CGAL::Comparison_result*, // for r
int*> // for c
Program;
typedef CGAL::Quadratic_program_solution<ET> Solution;
// we demonstrate the unboundedness certificate: either the feasible
// linear program
// min c^T x
// A x <= b
// x >= 0
// is bounded, or there exists w such that
// w >= 0
// A w <= 0
// c^t w < 0,
// in which case the objective function becomes arbitrarily small
// along the ray {x* + tw | t >= 0}, x* any feasible solution.
//
// In the following instance, the linear program is unbounded, since
// the ray {(t,t)| t>= 1} is feasible, and the objective function becomes
// arbitrarily small on it:
// min -x_1 - 2x_2
// x_1 - 2x_2 <= -1
// -x_1 + x_2 <= 2
// x_1, x_2 >= 0
int main() {
int Ax1[] = { 1, -1}; // column for x1
int Ax2[] = {-2, 1}; // column for x2
int* A[] = {Ax1, Ax2}; // A comes columnwise
int b[] = {-1, 2}; // right-hand side
CGAL::Comparison_result
r[] = {CGAL::SMALLER, CGAL::SMALLER}; // constraints are "<="
int c[] = {-1, -2}; // 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
Solution s = CGAL::solve_nonnegative_linear_program(lp, ET());
// get certificate for unboundedness
assert (s.is_unbounded());
Solution::Unboundedness_certificate_iterator w =
s.unboundedness_certificate_begin();
// check w >= 0
assert (ET(w[0]) >= 0);
assert (ET(w[1]) >= 0);
// check A w <= 0
assert (A[0][0] * ET(w[0]) + A[1][0] * ET(w[1]) <= 0);
assert (A[0][1] * ET(w[0]) + A[1][1] * ET(w[1]) <= 0);
// check c^T w < 0
assert (c[0] * ET(w[0]) + c[1] * ET(w[1]) < 0);
return 0;
}