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// Copyright (c) 1997-2007 ETH Zurich (Switzerland). // All rights reserved. // // This file is part of CGAL (www.cgal.org). // // $URL: https://github.com/CGAL/cgal/blob/v6.1/QP_solver/include/CGAL/QP_solver/QP_solution_impl.h $ // $Id: include/CGAL/QP_solver/QP_solution_impl.h b26b07a1242 $ // SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial // // // Author(s) : Bernd Gaertner #ifndef CGAL_QP_SOLUTION_IMPL_H #define CGAL_QP_SOLUTION_IMPL_H #include #include namespace CGAL { // checks whether the solution actually solves program p // this performs exactly the checks described in the doc // of the class Quadratic_program_solution // ----------------------------------------------------- template template bool Quadratic_program_solution::solves_program (const Program& p, Is_linear is_linear, Is_nonnegative is_nonnegative) { CGAL_qpe_assertion_msg(!is_void(), "Solution not initialized"); // first check whether the dimensions agree int n = static_cast(variable_numerators_end() - variable_numerators_begin()); if (n != p.get_n()) return error ("wrong number of variables"); int m = static_cast(solver()->lambda_end() - solver()->lambda_begin()); if (m != p.get_m()) return error("wrong number of constraints"); // now distinguish between the three possible cases switch (status()) { case QP_OPTIMAL: { std::vector ax_minus_b (m, et0); // d(Ax-b) std::vector two_Dx (n, et0); // d(2Dx) return // the following fills ax_minus_b... is_feasible (p, ax_minus_b, is_nonnegative) && // feasible? // the following fills two_Dx... is_value_correct (p, two_Dx, is_linear) && // obj. value? is_optimal_1 (p) && // condition 1? // ...and the following uses ax_minus_b is_optimal_2 (p, ax_minus_b) && // condition 2? // ...and the following uses two_Dx is_optimal_3 (p, two_Dx, is_linear, is_nonnegative); // condition 3? } case QP_INFEASIBLE: { std::vector lambda_a (n, et0); // lambda^TA return is_infeasible_1 (p) && // condition 1? // the following fills lambda_a... is_infeasible_2 (p, lambda_a, is_nonnegative) && // condition 2? // ...and the following uses lambda_a is_infeasible_3 (p, lambda_a, is_nonnegative); // condition 3? } case QP_UNBOUNDED: { std::vector ax_minus_b (m, et0); return is_feasible (p, ax_minus_b, is_nonnegative) && // feasible? is_unbounded_1 (p) && // condition 1? is_unbounded_2 (p, is_nonnegative) && // condition 2? is_unbounded_3 (p, is_linear); // condition 3? } default: return error ("solution in undefined state"); } } // tests whether Ax ~ b is satisfied and computes d(Ax-b); // precondition: ax_minus_b has length m and is zero // ------------------------------------------------------ template template bool Quadratic_program_solution::are_constraints_feasible (const Program& p, typename std::vector& ax_minus_b) { typedef typename Program::B_iterator B_column_iterator; typedef typename Program::R_iterator R_column_iterator; // fill ax_minus_b (length m) int m = p.get_m(); B_column_iterator b = p.get_b(); add_Az (p, variable_numerators_begin(), ax_minus_b); // d(Ax) ET d = variables_common_denominator(); for (int i=0; i et0) return error("inequality (<=) violated"); break; case EQUAL: // == if (ax_minus_b[i] != et0) return error("inequality (==) violated"); break; case LARGER: // >= if (ax_minus_b[i] < et0) return error("inequality (>=) violated"); break; } return true; } // tests whether Ax ~ b, x >= 0 is satisfied and computes d(Ax-b) // precondition: ax_minus_b has length m and is zero // -------------------------------------------------------------- template template bool Quadratic_program_solution::is_feasible (const Program& p, typename std::vector& ax_minus_b, Tag_true /*is_nonnegative*/) { // test Ax ~ b if (!are_constraints_feasible (p, ax_minus_b)) return false; // test x >= 0 if (variables_common_denominator() <= et0) return error ("common variable denominator is negative"); for (Variable_numerator_iterator v = variable_numerators_begin(); v < variable_numerators_end(); ++v) if (*v < et0) return error("bound (>= 0) violated"); return true; } // tests whether Ax ~ b, l <= x <= u is satisfied and computes d(Ax-b) // precondition: ax_minus_b has length m and is zero // ------------------------------------------------------------------- template template bool Quadratic_program_solution::is_feasible (const Program& p, typename std::vector& ax_minus_b, Tag_false /*is_nonnegative*/) { // test Ax ~ b if (!are_constraints_feasible (p, ax_minus_b)) return false; // (*) // test l <= x <= u ET d = variables_common_denominator(); if (d <= et0) return error ("common variable denominator is negative"); typedef typename Program::FL_iterator FL_column_iterator; typedef typename Program::L_iterator L_column_iterator; typedef typename Program::FU_iterator FU_column_iterator; typedef typename Program::U_iterator U_column_iterator; FL_column_iterator fl = p.get_fl(); FU_column_iterator fu = p.get_fu(); L_column_iterator l = p.get_l(); U_column_iterator u = p.get_u(); for (Variable_numerator_iterator v = variable_numerators_begin(); v < variable_numerators_end(); ++v, ++fl, ++l, ++fu, ++u) { if (*fl && (*v < ET(*l) * d)) return error("bound (>=l) violated"); if (*fu && (*v > ET(*u) * d)) return error("bound (<=u) violated"); } return true; } // checks whether constraint type <= (>=) implies lambda >= (<=) 0 // --------------------------------------------------------------- template template bool Quadratic_program_solution::is_optimal_1 (const Program& p) { if (variables_common_denominator() <= et0) return error ("wrong sign of optimality certificate"); typedef typename Program::R_iterator R_column_iterator; int m = p.get_m(); R_column_iterator r = p.get_r(); Optimality_certificate_numerator_iterator opt = optimality_certificate_numerators_begin(); for (int i=0; i et0) return error("constraint (>=) with lambda > 0"); } return true; } // checks whether constraint type <= (>=) implies lambda >= (<=) 0 // --------------------------------------------------------------- template template bool Quadratic_program_solution::is_infeasible_1 (const Program& p) { int m = p.get_m(); typedef typename Program::R_iterator R_column_iterator; R_column_iterator r = p.get_r(); Infeasibility_certificate_iterator inf = infeasibility_certificate_begin(); for (int i=0; i et0) return error("constraint (>=) with lambda > 0"); } return true; } // checks whether lambda and Ax-b are complementary // ------------------------------------------------ template template bool Quadratic_program_solution::is_optimal_2 (const Program& p, const typename std::vector& ax_minus_b) { int m = p.get_m(); Optimality_certificate_numerator_iterator opt = optimality_certificate_numerators_begin(); for (int i=0; i= (==) 0 if x_j = (>) 0 // ----------------------------------------------------------- template template bool Quadratic_program_solution::is_optimal_3 (const Program& p, std::vector& q, Tag_true /*is_linear*/, Tag_true /*is_nonnegative*/) { int n = p.get_n(); // q = 0 add_c (p, q); // d c^T add_zA (p, optimality_certificate_numerators_begin(), q); // d lambda^T A Variable_numerator_iterator v = variable_numerators_begin(); for (int j=0; j et0 && q[j] > et0) return error("(c^T + lambda^TA) and x are not complementary"); } return true; } // checks whether (c^T + lambda^TA + 2x^TD)_j >= (==) 0 if x_j = (>) 0 // ------------------------------------------------------------------- template template bool Quadratic_program_solution::is_optimal_3 (const Program& p, std::vector& q, Tag_false /*is_linear*/, Tag_true /*is_nonnegative*/) { int n = p.get_n(); // q = 2Dx add_c (p, q); // d * c^T add_zA (p, optimality_certificate_numerators_begin(), q); // d * lambda^T A Variable_numerator_iterator v = variable_numerators_begin(); for (int j=0; j et0 && q[j] > et0) return error("(c^T + lambda^TA + 2Dx) and x are not complementary"); } return true; } // checks whether (c^T + lambda^TA )_j >= (==, <=) 0 if // x_j = l_j < u_j (l_j < x_j < u_j, l_j < u_j = x_j) // --------------------------------------------------- template template bool Quadratic_program_solution::is_optimal_3 (const Program& p, std::vector& q, Tag_true /*is_linear*/, Tag_false /*is_nonnegative*/) { int n = p.get_n(); // q = 0 add_c (p, q); // d * c^T add_zA (p, optimality_certificate_numerators_begin(), q); // d * lambda^T A typedef typename Program::FL_iterator FL_column_iterator; typedef typename Program::L_iterator L_column_iterator; typedef typename Program::FU_iterator FU_column_iterator; typedef typename Program::U_iterator U_column_iterator; FL_column_iterator fl = p.get_fl(); FU_column_iterator fu = p.get_fu(); L_column_iterator l = p.get_l(); U_column_iterator u = p.get_u(); Variable_numerator_iterator v = variable_numerators_begin(); ET d = variables_common_denominator(); if (d <= et0) return error ("common variable denominator is negative"); for (int j=0; j ET(*l) * d) && (!*fu || *v < ET(*u) * d) && q[j] != et0) return error("l_j < x_j < u_j but (c^T + lambda^TA )_j != 0"); if (*fu && *v == ET(*u) * d && (!*fl || *l < *u) && q[j] > et0) return error("x_j = u_j > l_j but (c^T + lambda^TA )_j > 0"); } return true; } // checks whether (c^T + lambda^TA +2x^*D)_j >= (==, <=) 0 if // x_j = l_j < u_j (l_j < x_j < u_j, l_j < u_j = x_j) // ---------------------------------------------------------- template template bool Quadratic_program_solution::is_optimal_3 (const Program& p, std::vector& q, Tag_false /*is_linear*/, Tag_false /*is_nonnegative*/) { int n = p.get_n(); // q = d * 2Dx add_c (p, q); // d * c^T add_zA (p, optimality_certificate_numerators_begin(), q); // d * lambda^T A typedef typename Program::FL_iterator FL_column_iterator; typedef typename Program::L_iterator L_column_iterator; typedef typename Program::FU_iterator FU_column_iterator; typedef typename Program::U_iterator U_column_iterator; FL_column_iterator fl = p.get_fl(); FU_column_iterator fu = p.get_fu(); L_column_iterator l = p.get_l(); U_column_iterator u = p.get_u(); Variable_numerator_iterator v = variable_numerators_begin(); ET d = variables_common_denominator(); if (d <= et0) return error ("common variable denominator is negative"); for (int j=0; j ET(*l) * d) && (!*fu || *v < ET(*u) * d) && q[j] != et0) return error("l_j < x_j < u_j but (c^T + lambda^TA + 2Dx)_j != 0"); if (*fu && *v == ET(*u) * d && (!*fl || *l < *u) && q[j] > et0) return error("x_j = u_j > l_j but (c^T + lambda^TA + 2Dx)_j > 0"); } return true; } // checks whether objective function value is c^T x + c_0, but // leaves the vector q alone // -------------------------------------------------------------- template template bool Quadratic_program_solution::is_value_correct (const Program& p, std::vector& /*q*/, Tag_true /*is_linear*/) { // check objective value c^T x + c_0 ET d = variables_common_denominator(); if (d <= et0) return error ("common variable denominator is negative"); Variable_numerator_iterator v = variable_numerators_begin(); ET obj = d * ET(p.get_c0()); // d * c_0 typedef typename Program::C_iterator C_column_iterator; C_column_iterator c = p.get_c(); int n = p.get_n(); for (int j=0; j(obj, d) != objective_value()) return error ("optimal objective value c^T x + c_0 incorrect"); return true; } // checks whether objective function value is x^TDx + c^T x + c_0 // and fills the vector q with 2Dx // -------------------------------------------------------------- template template bool Quadratic_program_solution::is_value_correct (const Program& p, std::vector& q, Tag_false /*is_linear*/) { ET d = variables_common_denominator(); if (d <= et0) return error ("common variable denominator is negative"); Variable_numerator_iterator v = variable_numerators_begin(); int n = p.get_n(); add_two_Dz (p, v, q); // 2d * Dx ET et2(2); ET obj = et2 * d * d * ET(p.get_c0()); // 2d^2 * c_0 typedef typename Program::C_iterator C_column_iterator; C_column_iterator c = p.get_c(); for (int j=0; j(obj, et2*d*d) != objective_value()) return error ("optimal objective value x^TDx + c^T x + c_0 incorrect"); return true; } // checks whether lambda^TA >= 0 // ----------------------------- template template bool Quadratic_program_solution::is_infeasible_2 (const Program& p, typename std::vector& lambda_a, Tag_true /*is_nonnegative*/) { // fill lambda_a add_zA (p, infeasibility_certificate_begin(), lambda_a); int n = p.get_n(); for (int j=0; j= 0 (<= 0) if u_j = infty (l_j = -infty) // ------------------------------------------------------------------ template template bool Quadratic_program_solution::is_infeasible_2 (const Program& p, typename std::vector& lambda_a, Tag_false /*is_nonnegative*/) { // fill lambda_a add_zA (p, infeasibility_certificate_begin(), lambda_a); int n = p.get_n(); typedef typename Program::FL_iterator FL_column_iterator; typedef typename Program::FU_iterator FU_column_iterator; FL_column_iterator fl = p.get_fl(); FU_column_iterator fu = p.get_fu(); for (int j=0; j et0) return error ("l_j = -infty but (lambda^TA)_j > 0"); } return true; } // checks whether lambda^T b < 0 // ----------------------------- template template bool Quadratic_program_solution::is_infeasible_3 (const Program& p, const typename std::vector& /*lambda_a*/, Tag_true /*is_nonnegative*/) { ET lambda_b(0); int m = p.get_m(); typedef typename Program::B_iterator B_column_iterator; B_column_iterator b = p.get_b(); Infeasibility_certificate_iterator inf = infeasibility_certificate_begin(); for (int i=0; i= et0) return error ("lambda_b >= 0"); return true; } // checks whether lambda^T b < \sum_{j: lambda^TA_j < 0}lambda^TA_j u_j + // \sum_{j: lambda^TA_j > 0}lambda^TA_j l_j // ---------------------------------------------------------------------- template template bool Quadratic_program_solution::is_infeasible_3 (const Program& p, const typename std::vector& lambda_a, Tag_false /*is_nonnegative*/) { ET lambda_b(0); int m = p.get_m(); typedef typename Program::B_iterator B_column_iterator; B_column_iterator b = p.get_b(); Infeasibility_certificate_iterator inf = infeasibility_certificate_begin(); for (int i=0; i et0) sum += lambda_a[j] * ET(*l); } // now compare if (lambda_b >= sum) return error("lambda^T b >= sum"); return true; } // checks whether (Aw)_i <= (>=, ==) 0 if i-th constraint is <= (>=, ==) // --------------------------------------------------------------------- template template bool Quadratic_program_solution::is_unbounded_1 (const Program& p) { int m = p.get_m(); std::vector aw (m, et0); add_Az (p, unboundedness_certificate_begin(), aw); typedef typename Program::R_iterator R_column_iterator; R_column_iterator r = p.get_r(); for (int i=0; i et0) return error ("i-th constraint <= but (Aw)_i > 0"); break; case EQUAL: if (aw[i] != et0) return error ("i-th constraint == but (Aw)_i != 0"); break; case LARGER: if (aw[i] < et0) return error ("i-th constraint >= but (Aw)_i < 0"); break; } return true; } // checks whether w >= 0 // --------------------- template template bool Quadratic_program_solution::is_unbounded_2 (const Program& p, Tag_true /*is_nonnegative*/) { int n = p.get_n(); Unboundedness_certificate_iterator unb = unboundedness_certificate_begin(); for (int j=0; j= (<=) 0 if l_j (u_j) is finite // --------------------------------------------------- template template bool Quadratic_program_solution::is_unbounded_2 (const Program& p, Tag_false /*is_nonnegative*/) { int n = p.get_n(); typedef typename Program::FL_iterator FL_column_iterator; typedef typename Program::FU_iterator FU_column_iterator; FL_column_iterator fl = p.get_fl(); FU_column_iterator fu = p.get_fu(); Unboundedness_certificate_iterator unb = unboundedness_certificate_begin(); for (int j=0; j et0) return error ("some u_j is finite but w_j > 0"); } return true; } // checks whether c^Tw < 0 // ----------------------- template template bool Quadratic_program_solution::is_unbounded_3 (const Program& p, Tag_true /*is_linear*/) { ET cw(0); int n = p.get_n(); typedef typename Program::C_iterator C_column_iterator; C_column_iterator c = p.get_c(); Unboundedness_certificate_iterator unb = unboundedness_certificate_begin(); for (int j=0; j= et0) return error("c^Tw >= 0"); return true; } // checks whether w^TDw = 0 and (c^T + 2x^TD)w < 0 // ----------------------------------------------- template template bool Quadratic_program_solution::is_unbounded_3 (const Program& p, Tag_false /*is_linear*/) { int n = p.get_n(); std::vector two_dw (n, et0); add_two_Dz(p, unboundedness_certificate_begin(), two_dw); // 2Dw // check w^TDw = 0 Unboundedness_certificate_iterator unb = unboundedness_certificate_begin(); ET wdw(0); for (int j=0; j= 0) return error("c^T + 2x^TD)w >= 0"); return true; } // computes the product of A with the n-vector given by z // and adds the result to v; precondition: v has size m // ------------------------------------------------------ template template void Quadratic_program_solution::add_Az (const Program& p, Z_iterator z, typename std::vector& v) { // iterator types typedef typename Program::A_iterator A_matrix_iterator; typedef typename std::iterator_traits::value_type A_column_iterator; // now compute the product A_matrix_iterator a = p.get_a(); int n = p.get_n(); for (int j=0; j template void Quadratic_program_solution::add_two_Dz (const Program& p, Z_iterator z, typename std::vector& v) { // we compute the contribution from the enries of D on or below // the diagonal rowwise, and the remaining entries columnwise typedef typename Program::D_iterator D_matrix_iterator; typedef typename std::iterator_traits::value_type D_row_iterator; int n = p.get_n(); // make sure that we only handle the nonzero terms of z std::vector z_indices; Z_iterator l = z; for (int i=0; i::const_iterator k = z_indices.begin(); k < z_indices.end(); ++k) { int j = *k; if (j <= i) v[i] += *(z+j) * ET (*(d_i+j)); } } // the columnwise contribution: above the diagonal d = p.get_d(); for (std::vector::const_iterator k = z_indices.begin(); k < z_indices.end(); ++k) { int j = *k; // handle column j D_row_iterator d_j = *(d+j); // column j above diagonal for (int i=0; i template void Quadratic_program_solution::add_zA (const Program& p, Z_iterator z, typename std::vector& v) { typedef typename Program::A_iterator A_matrix_iterator; typedef typename std::iterator_traits::value_type A_column_iterator; A_matrix_iterator a = p.get_a(); int n = p.get_n(); int m = p.get_m(); // make sure that we only handle the nonzero terms of z std::vector z_indices; // indices of nonzero entries Z_iterator l = z; for (int i=0; i::const_iterator k = z_indices.begin(); k < z_indices.end(); ++k) v[j] += *(z+*k) * ET(*(a_j+*k)); } } // adds d c^T to v; precondition: v has length n // --------------------------------------------- template template void Quadratic_program_solution::add_c (const Program& p, typename std::vector& v) { int n = p.get_n(); typedef typename Program::C_iterator C_column_iterator; C_column_iterator c = p.get_c(); ET d = variables_common_denominator(); for (int j=0; j