1 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */ 2 /* */ 3 /* This file is part of the program and library */ 4 /* SCIP --- Solving Constraint Integer Programs */ 5 /* */ 6 /* Copyright (C) 2002-2021 Konrad-Zuse-Zentrum */ 7 /* fuer Informationstechnik Berlin */ 8 /* */ 9 /* SCIP is distributed under the terms of the ZIB Academic License. */ 10 /* */ 11 /* You should have received a copy of the ZIB Academic License */ 12 /* along with SCIP; see the file COPYING. If not visit scipopt.org. */ 13 /* */ 14 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */ 15 16 /**@file benderscut_int.h 17 * @ingroup BENDERSCUTS 18 * @brief Generates a Laporte and Louveaux Benders' decomposition integer cut 19 * @author Stephen J. Maher 20 * 21 * The classical Benders' decomposition algorithm is only applicable to problems with continuous second stage variables. 22 * Laporte and Louveaux (1993) developed a method for generating cuts when Benders' decomposition is applied to problem 23 * with discrete second stage variables. However, these cuts are only applicable when the master problem is a pure 24 * binary problem. 25 * 26 * The integer optimality cuts are a point-wise underestimator of the optimal subproblem objective function value. 27 * Similar to benderscuts_opt.c, an auxiliary variable, \f$\varphi\f$. is required in the master problem as a lower 28 * bound on the optimal objective function value for the Benders' decomposition subproblem. 29 * 30 * Consider the Benders' decomposition subproblem that takes the master problem solution \f$\bar{x}\f$ as input: 31 * \f[ 32 * z(\bar{x}) = \min\{d^{T}y : Ty \geq h - H\bar{x}, y \mbox{ integer}\} 33 * \f] 34 * If the subproblem is feasible, and \f$z(\bar{x}) > \varphi\f$ (indicating that the current underestimators are not 35 * optimal) then the Benders' decomposition integer optimality cut can be generated from the optimal solution of the 36 * subproblem. Let \f$S_{r}\f$ be the set of indicies for master problem variables that are 1 in \f$\bar{x}\f$ and 37 * \f$L\f$ a known lowerbound on the subproblem objective function value. 38 * 39 * The resulting cut is: 40 * \f[ 41 * \varphi \geq (z(\bar{x}) - L)(\sum_{i \in S_{r}}(x_{i} - 1) + \sum_{i \notin S_{r}}x_{i} + 1) 42 * \f] 43 * 44 * Laporte, G. & Louveaux, F. V. The integer L-shaped method for stochastic integer programs with complete recourse 45 * Operations Research Letters, 1993, 13, 133-142 46 */ 47 48 /*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/ 49 50 #ifndef __SCIP_BENDERSCUT_INT_H__ 51 #define __SCIP_BENDERSCUT_INT_H__ 52 53 54 #include "scip/def.h" 55 #include "scip/type_benders.h" 56 #include "scip/type_retcode.h" 57 #include "scip/type_scip.h" 58 59 #ifdef __cplusplus 60 extern "C" { 61 #endif 62 63 /** creates the integer optimality cut for Benders' decomposition cut and includes it in SCIP 64 * 65 * @ingroup BenderscutIncludes 66 */ 67 SCIP_EXPORT 68 SCIP_RETCODE SCIPincludeBenderscutInt( 69 SCIP* scip, /**< SCIP data structure */ 70 SCIP_BENDERS* benders /**< Benders' decomposition */ 71 ); 72 73 #ifdef __cplusplus 74 } 75 #endif 76 77 #endif 78