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_int.h 26 * @ingroup BENDERSCUTS 27 * @brief Generates a Laporte and Louveaux Benders' decomposition integer cut 28 * @author Stephen J. Maher 29 * 30 * The classical Benders' decomposition algorithm is only applicable to problems with continuous second stage variables. 31 * Laporte and Louveaux (1993) developed a method for generating cuts when Benders' decomposition is applied to problem 32 * with discrete second stage variables. However, these cuts are only applicable when the master problem is a pure 33 * binary problem. 34 * 35 * The integer optimality cuts are a point-wise underestimator of the optimal subproblem objective function value. 36 * Similar to benderscuts_opt.c, an auxiliary variable, \f$\varphi\f$. is required in the master problem as a lower 37 * bound on the optimal objective function value for the Benders' decomposition subproblem. 38 * 39 * Consider the Benders' decomposition subproblem that takes the master problem solution \f$\bar{x}\f$ as input: 40 * \f[ 41 * z(\bar{x}) = \min\{d^{T}y : Ty \geq h - H\bar{x}, y \mbox{ integer}\} 42 * \f] 43 * If the subproblem is feasible, and \f$z(\bar{x}) > \varphi\f$ (indicating that the current underestimators are not 44 * optimal) then the Benders' decomposition integer optimality cut can be generated from the optimal solution of the 45 * subproblem. Let \f$S_{r}\f$ be the set of indicies for master problem variables that are 1 in \f$\bar{x}\f$ and 46 * \f$L\f$ a known lowerbound on the subproblem objective function value. 47 * 48 * The resulting cut is: 49 * \f[ 50 * \varphi \geq (z(\bar{x}) - L)(\sum_{i \in S_{r}}(x_{i} - 1) + \sum_{i \notin S_{r}}x_{i} + 1) 51 * \f] 52 * 53 * Laporte, G. & Louveaux, F. V. The integer L-shaped method for stochastic integer programs with complete recourse 54 * Operations Research Letters, 1993, 13, 133-142 55 */ 56 57 /*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/ 58 59 #ifndef __SCIP_BENDERSCUT_INT_H__ 60 #define __SCIP_BENDERSCUT_INT_H__ 61 62 63 #include "scip/def.h" 64 #include "scip/type_benders.h" 65 #include "scip/type_retcode.h" 66 #include "scip/type_scip.h" 67 68 #ifdef __cplusplus 69 extern "C" { 70 #endif 71 72 /** creates the integer optimality cut for Benders' decomposition cut and includes it in SCIP 73 * 74 * @ingroup BenderscutIncludes 75 */ 76 SCIP_EXPORT 77 SCIP_RETCODE SCIPincludeBenderscutInt( 78 SCIP* scip, /**< SCIP data structure */ 79 SCIP_BENDERS* benders /**< Benders' decomposition */ 80 ); 81 82 #ifdef __cplusplus 83 } 84 #endif 85 86 #endif 87