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4    	/*         SCIP --- Solving Constraint Integer Programs                      */
5    	/*                                                                           */
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24   	
25   	/**@file   sepa_convexproj.h
26   	 * @ingroup SEPARATORS
27   	 * @brief  convexproj separator
28   	 * @author Felipe Serrano
29   	 *
30   	 * This separator receives a point \f$ x_0 \f$ to separate, projects it onto a convex relaxation
31   	 * of the current problem and then generates gradient cuts at the projection.
32   	 *
33   	 * In more detail, the separator builds and stores a convex relaxation of the problem
34   	 * \f[
35   	 *      C = \{ x \colon g_j(x) \le 0 \, \forall j=1,\ldots,m \}
36   	 * \f]
37   	 * where each \f$ g_j \f$ is a convex function and computes the projection by solving
38   	 * \f{align}{
39   	 *      \min \; & || x - x_0 ||^2 \\
40   	 *      s.t. \; & g_j(x) \le 0 & \forall j=1,\ldots,m.
41   	 * \f}
42   	 *
43   	 * By default, if enabled, the separator runs only if the convex relaxation has at least one nonlinear convex function.
44   	 *
45   	 * The separator generates cuts for constraints which were violated by the solution we want to separate and active
46   	 * at the projection. If the projection problem is not solved to optimality, it still tries to add a cut at the
47   	 * best solution found. In case that the projection problem is solved to optimality, it is guaranteed that a cut
48   	 * separates the point. To see this, remember that \f$ z \f$ is the projection if and only if
49   	 * \f[
50   	 *      \langle x - z, z - x_0 \rangle \ge 0 \, \forall x \in C \\
51   	 * \f]
52   	 * This inequality is violated for \f$ x = x_0 \f$. On the other hand, one of the optimality conditions of the
53   	 * projection problem at the optimum looks like
54   	 * \f[
55   	 *      2 (z - x_0) + \sum_j \lambda_j \nabla g_j(z) = 0.
56   	 * \f]
57   	 * Now suppose that the no gradient cut at \f$ z \f$ separates \f$ x_0 \f$, i.e.,
58   	 * \f[
59   	 *      g_j(z) + \langle \nabla g_j(z), x_0 - z \rangle \le 0.
60   	 * \f]
61   	 * Multiplying each inequality with \f$ \lambda_j \ge 0 \f$ and summing up, we get the following contradiction:
62   	 * \f[
63   	 *      \langle -2(z - x_0), x_0 - z \rangle \le 0.
64   	 * \f]
65   	 *
66   	 * This separator is currently disabled by default. It requires additional
67   	 * tuning to be enabled by default. However, it may be useful to enable
68   	 * it on instances with convex nonlinear constraints if SCIP spends
69   	 * many iterations in the separation loop without doing sufficient progress.
70   	 */
71   	
72   	/*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
73   	
74   	#ifndef __SCIP_SEPA_CONVEXPROJ_H__
75   	#define __SCIP_SEPA_CONVEXPROJ_H__
76   	
77   	
78   	#include "scip/def.h"
79   	#include "scip/type_retcode.h"
80   	#include "scip/type_scip.h"
81   	
82   	#ifdef __cplusplus
83   	extern "C" {
84   	#endif
85   	
86   	/** creates the convexproj separator and includes it in SCIP
87   	 *
88   	 * @ingroup SeparatorIncludes
89   	 */
90   	SCIP_EXPORT
91   	SCIP_RETCODE SCIPincludeSepaConvexproj(
92   	   SCIP*                 scip                /**< SCIP data structure */
93   	   );
94   	
95   	#ifdef __cplusplus
96   	}
97   	#endif
98   	
99   	#endif
100