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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
81   	SCIP_RETCODE SCIPincludeBenderscutFeas(
82   	   SCIP*                 scip,               /**< SCIP data structure */
83   	   SCIP_BENDERS*         benders             /**< Benders' decomposition */
84   	   );
85   	
86   	#ifdef __cplusplus
87   	}
88   	#endif
89   	
90   	#endif
91