mirror of https://github.com/CGAL/cgal
77 lines
2.5 KiB
C++
77 lines
2.5 KiB
C++
// example: extracting and verifying a proof of optimality 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 optimality certificate: if the
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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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// has an optimal solution x*, then there exists y (in fact an optimal
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// solution of the dual linear program) such that
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// y >= 0
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// y^T A >= -c^T,
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// y^T b = -c^T x*
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//
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// In the following instance, the linear program has an optimal solution,
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// since it is feasible (take (1,0)) and bounded, due to the first
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// constraint:
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// min x_1 + 2x_2
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// x_1 + 3x_2 <= 1
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// -x_1 + x_2 <= -1
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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[] = { 3, 1}; // column for x2
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int* A[] = {Ax1, Ax2}; // A comes columnwise
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int b[] = { 1, -1}; // 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 optimality
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assert (s.is_optimal());
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Solution::Variable_value_iterator x =
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s.variable_values_begin();
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Solution::Optimality_certificate_iterator y =
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s.optimality_certificate_begin();
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// check y >= 0
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assert (y[0] >= 0);
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assert (y[1] >= 0);
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// check y^T A >= -c^T
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assert (y[0] * A[0][0] + y[1] * A[0][1] >= -c[0]);
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assert (y[0] * A[1][0] + y[1] * A[1][1] >= -c[1]);
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// check y^T b = c^T x*
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assert (y[0] * b[0] + y[1] * b[1] == -c[0] * x[0] - c[1] * x[1]);
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return 0;
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}
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