1 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */ 2 /* */ 3 /* This file is part of the program and library */ 4 /* SCIP --- Solving Constraint Integer Programs */ 5 /* */ 6 /* Copyright (c) 2002-2023 Zuse Institute Berlin (ZIB) */ 7 /* */ 8 /* Licensed under the Apache License, Version 2.0 (the "License"); */ 9 /* you may not use this file except in compliance with the License. */ 10 /* You may obtain a copy of the License at */ 11 /* */ 12 /* http://www.apache.org/licenses/LICENSE-2.0 */ 13 /* */ 14 /* Unless required by applicable law or agreed to in writing, software */ 15 /* distributed under the License is distributed on an "AS IS" BASIS, */ 16 /* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. */ 17 /* See the License for the specific language governing permissions and */ 18 /* limitations under the License. */ 19 /* */ 20 /* You should have received a copy of the Apache-2.0 license */ 21 /* along with SCIP; see the file LICENSE. If not visit scipopt.org. */ 22 /* */ 23 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */ 24 25 /**@file benderscut_feas.h 26 * @ingroup BENDERSCUTS 27 * @brief Standard feasibility cuts for Benders' decomposition 28 * @author Stephen J. Maher 29 * 30 * The classical Benders' decomposition feasibility cuts arise from an infeasible instance of the Benders' decomposition 31 * subproblem. 32 * Consider the linear Benders' decomposition subproblem that takes the master problem solution \f$\bar{x}\f$ as input: 33 * \f[ 34 * z(\bar{x}) = \min\{d^{T}y : Ty \geq h - H\bar{x}, y \geq 0\} 35 * \f] 36 * If the subproblem is infeasible as a result of the solution \f$\bar{x}\f$, then the Benders' decomposition 37 * feasibility cut can be generated from the dual ray. Let \f$w\f$ be the vector corresponding to the dual ray of the 38 * Benders' decomposition subproblem. The resulting cut is: 39 * \f[ 40 * 0 \geq w^{T}(h - Hx) 41 * \f] 42 * 43 * Next, consider the nonlinear Benders' decomposition subproblem that takes the master problem solution \f$\bar{x}\f$ as input: 44 * \f[ 45 * z(\bar{x}) = \min\{d^{T}y : g(\bar{x}, y) \leq 0, y \geq 0\} 46 * \f] 47 * If the subproblem is infeasible as a result of the solution \f$\bar{x}\f$, then the Benders' decomposition 48 * feasibility cut can be generated from a minimal infeasible solution, i.e., a solution of the NLP 49 * \f[ 50 * \min\left\{\sum_i u_i : g(\bar{x}, y) \leq u, y \geq 0, u \geq 0\right\} 51 * \f] 52 * Let \f$\bar{y}\f$, \f$w\f$ be the vectors corresponding to the primal and dual solution of this auxiliary NLP. 53 * The resulting cut is: 54 * \f[ 55 * 0 \geq w^{T}\left(g(\bar{x},\bar{y}) + \nabla_x g(\bar{x},\bar{y}) (x - \bar{x})\right) 56 * \f] 57 * Note, that usually NLP solvers already provide a minimal infeasible solution when declaring the Benders' 58 * decomposition subproblem as infeasible. 59 */ 60 61 /*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/ 62 63 #ifndef __SCIP_BENDERSCUT_FEAS_H__ 64 #define __SCIP_BENDERSCUT_FEAS_H__ 65 66 67 #include "scip/def.h" 68 #include "scip/type_benders.h" 69 #include "scip/type_retcode.h" 70 #include "scip/type_scip.h" 71 72 #ifdef __cplusplus 73 extern "C" { 74 #endif 75 76 /** creates the Standard Feasibility Benders' decomposition cuts and includes it in SCIP 77 * 78 * @ingroup BenderscutIncludes 79 */ 80 SCIP_EXPORT 81 SCIP_RETCODE SCIPincludeBenderscutFeas( 82 SCIP* scip, /**< SCIP data structure */ 83 SCIP_BENDERS* benders /**< Benders' decomposition */ 84 ); 85 86 #ifdef __cplusplus 87 } 88 #endif 89 90 #endif 91