# SCIP

Solving Constraint Integer Programs

benderscut_int.h
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1 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
2 /* */
3 /* This file is part of the program and library */
4 /* SCIP --- Solving Constraint Integer Programs */
5 /* */
7 /* fuer Informationstechnik Berlin */
8 /* */
10 /* */
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
69  SCIP* scip, /**< SCIP data structure */
70  SCIP_BENDERS* benders /**< Benders' decomposition */
71  );
72
73 #ifdef __cplusplus
74 }
75 #endif
76
77 #endif
SCIP_RETCODE SCIPincludeBenderscutInt(SCIP *scip, SCIP_BENDERS *benders)
enum SCIP_Retcode SCIP_RETCODE
Definition: type_retcode.h:54
type definitions for return codes for SCIP methods
type definitions for SCIP&#39;s main datastructure
type definitions for Benders&#39; decomposition methods
common defines and data types used in all packages of SCIP