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4    	/*         SCIP --- Solving Constraint Integer Programs                      */
5    	/*                                                                           */
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22   	/*                                                                           */
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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