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4    	/*         SCIP --- Solving Constraint Integer Programs                      */
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24   	
25   	/**@file   benderscut_feasalt.h
26   	 * @ingroup BENDERSCUTS
27   	 * @brief  Alternative feasibility cuts for Benders' decomposition
28   	 * @author Stephen J. Maher
29   	 *
30   	 * The alternative feasibility cut for Benders' decomposition uses the optimality cut generation code to generate a cut
31   	 * that minimises the violation of the constraints.
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 some of the constraints are violated.
37   	 * In this case, we define an alternative/auxiliary subproblem to find a solution that minimises the constraint
38   	 * violations. Such a problem is given by
39   	 * \f[
40   	 * \min\{\mathbb{1}{T}v : Ty + v \geq h - H\bar{x}, y \geq 0, v \geq 0\}
41   	 * \f]
42   	 *
43   	 * This auxiliary problem is guaranteed to always be feasible. Given a solution to this problem, it is possible to
44   	 * generate a classical Benders' optimality cut. For such a cut, the reader is referred to \ref benderscut_opt.h.
45   	 *
46   	 * If the Benders' decomposition subproblem contains non-linear constraints, an equivalent auxiliary subproblem can be
47   	 * formed to generate an alternative feasibility cut.
48   	 */
49   	
50   	/*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
51   	
52   	#ifndef __SCIP_BENDERSCUT_FEASALT_H__
53   	#define __SCIP_BENDERSCUT_FEASALT_H__
54   	
55   	
56   	#include "scip/def.h"
57   	#include "scip/type_benders.h"
58   	#include "scip/type_retcode.h"
59   	#include "scip/type_scip.h"
60   	
61   	#ifdef __cplusplus
62   	extern "C" {
63   	#endif
64   	
65   	/** creates the Alternative Feasibility Benders' decomposition cuts and includes it in SCIP
66   	 *
67   	 *  @ingroup BenderscutIncludes
68   	 */
69   	SCIP_EXPORT
70   	SCIP_RETCODE SCIPincludeBenderscutFeasalt(
71   	   SCIP*                 scip,               /**< SCIP data structure */
72   	   SCIP_BENDERS*         benders             /**< Benders' decomposition */
73   	   );
74   	
75   	#ifdef __cplusplus
76   	}
77   	#endif
78   	
79   	#endif
80