# SCIP

Solving Constraint Integer Programs

benderscut_feas.h
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1 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
2 /* */
3 /* This file is part of the program and library */
4 /* SCIP --- Solving Constraint Integer Programs */
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24
25 /**@file benderscut_feas.h
26  * @ingroup BENDERSCUTS
27  * @brief Standard feasibility cuts for Benders' decomposition
28  * @author Stephen J. Maher
29  *
30  * The classical Benders' decomposition feasibility cuts arise from an infeasible instance of the Benders' decomposition
31  * subproblem.
32  * Consider the linear Benders' decomposition subproblem that takes the master problem solution \f$\bar{x}\f$ as input:
33  * \f[
34  * z(\bar{x}) = \min\{d^{T}y : Ty \geq h - H\bar{x}, y \geq 0\}
35  * \f]
36  * If the subproblem is infeasible as a result of the solution \f$\bar{x}\f$, then the Benders' decomposition
37  * feasibility cut can be generated from the dual ray. Let \f$w\f$ be the vector corresponding to the dual ray of the
38  * Benders' decomposition subproblem. The resulting cut is:
39  * \f[
40  * 0 \geq w^{T}(h - Hx)
41  * \f]
42  *
43  * Next, consider the nonlinear Benders' decomposition subproblem that takes the master problem solution \f$\bar{x}\f$ as input:
44  * \f[
45  * z(\bar{x}) = \min\{d^{T}y : g(\bar{x}, y) \leq 0, y \geq 0\}
46  * \f]
47  * If the subproblem is infeasible as a result of the solution \f$\bar{x}\f$, then the Benders' decomposition
48  * feasibility cut can be generated from a minimal infeasible solution, i.e., a solution of the NLP
49  * \f[
50  * \min\left\{\sum_i u_i : g(\bar{x}, y) \leq u, y \geq 0, u \geq 0\right\}
51  * \f]
52  * Let \f$\bar{y}\f$, \f$w\f$ be the vectors corresponding to the primal and dual solution of this auxiliary NLP.
53  * The resulting cut is:
54  * \f[
55  * 0 \geq w^{T}\left(g(\bar{x},\bar{y}) + \nabla_x g(\bar{x},\bar{y}) (x - \bar{x})\right)
56  * \f]
57  * Note, that usually NLP solvers already provide a minimal infeasible solution when declaring the Benders'
58  * decomposition subproblem as infeasible.
59  */
60
61 /*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
62
63 #ifndef __SCIP_BENDERSCUT_FEAS_H__
64 #define __SCIP_BENDERSCUT_FEAS_H__
65
66
67 #include "scip/def.h"
68 #include "scip/type_benders.h"
69 #include "scip/type_retcode.h"
70 #include "scip/type_scip.h"
71
72 #ifdef __cplusplus
73 extern "C" {
74 #endif
75
76 /** creates the Standard Feasibility Benders' decomposition cuts and includes it in SCIP
77  *
78  * @ingroup BenderscutIncludes
79  */
80 SCIP_EXPORT
82  SCIP* scip, /**< SCIP data structure */
83  SCIP_BENDERS* benders /**< Benders' decomposition */
84  );
85
86 #ifdef __cplusplus
87 }
88 #endif
89
90 #endif
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
SCIP_RETCODE SCIPincludeBenderscutFeas(SCIP *scip, SCIP_BENDERS *benders)
type definitions for Benders&#39; decomposition methods
common defines and data types used in all packages of SCIP