Scippy

SCIP

Solving Constraint Integer Programs

benderscut_int.h
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4 /* SCIP --- Solving Constraint Integer Programs */
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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
78  SCIP* scip, /**< SCIP data structure */
79  SCIP_BENDERS* benders /**< Benders' decomposition */
80  );
81 
82 #ifdef __cplusplus
83 }
84 #endif
85 
86 #endif
SCIP_RETCODE SCIPincludeBenderscutInt(SCIP *scip, SCIP_BENDERS *benders)
enum SCIP_Retcode SCIP_RETCODE
Definition: type_retcode.h:63
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