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2    	/*                                                                           */
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4    	/*         SCIP --- Solving Constraint Integer Programs                      */
5    	/*                                                                           */
6    	/*  Copyright (c) 2002-2023 Zuse Institute Berlin (ZIB)                      */
7    	/*                                                                           */
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19   	/*                                                                           */
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22   	/*                                                                           */
23   	/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
24   	
25   	/**@file   sepa_eccuts.h
26   	 * @ingroup SEPARATORS
27   	 * @brief  edge concave cut separator
28   	 * @author Benjamin Mueller
29   	 *
30   	 * We call \f$ f \f$ an edge-concave function on a polyhedron \f$P\f$ iff it is concave in all edge directions of
31   	 * \f$P\f$. For the special case \f$ P = [\ell,u]\f$ this is equivalent to \f$f\f$ being concave componentwise.
32   	 *
33   	 * Since the convex envelope of an edge-concave function is a polytope, the value of the convex envelope for a
34   	 * \f$ x \in [\ell,u] \f$ can be obtained by solving the following LP:
35   	 *
36   	 * \f{align}{
37   	 *     \min \, & \sum_i \lambda_i f(v_i)  \\
38   	 *     s.t. \, & \sum_i \lambda_i v_i = x \\
39   	 *             & \sum_i \lambda_i = 1
40   	 * \f}
41   	 * where \f$ \{ v_i \} \f$ are the vertices of the domain \f$ [\ell,u] \f$. Let \f$ (\alpha, \alpha_0) \f$ be the dual
42   	 * solution of this LP. It can be shown that \f$ \alpha' x + \alpha_0 \f$ is a facet of the convex envelope of \f$ f \f$
43   	 * if \f$ x \f$ is in the interior of \f$ [\ell,u] \f$.
44   	 *
45   	 * We use this as follows:  We transform the problem to the unit box \f$ [0,1]^n \f$ by using a linear affine
46   	 * transformation \f$ T(x) = Ax + b \f$ and perturb \f$ T(x) \f$ if it is not an interior point.
47   	 * This has the advantage that we do not have to update the matrix of the LP for different edge-concave functions.
48   	 *
49   	 * For a given quadratic constraint \f$ g(x) := x'Qx + b'x + c \le 0 \f$ we decompose \f$ g \f$ into several
50   	 * edge-concave aggregations and a remaining part, e.g.,
51   	 *
52   	 * \f[
53   	 *             g(x) = \sum_{i = 1}^k f_i(x) + r(x)
54   	 * \f]
55   	 *
56   	 * where each \f$ f_i \f$ is edge-concave. To separate a given solution \f$ x \f$, we compute a facet of the convex
57   	 * envelope \f$ \tilde f \f$ for each edge-concave function \f$ f_i \f$ and an underestimator \f$ \tilde r \f$
58   	 * for \f$ r \f$. The resulting cut looks like:
59   	 *
60   	 * \f[
61   	 *             \tilde f_i(x) + \tilde r(x) \le 0
62   	 * \f]
63   	 *
64   	 * We solve auxiliary MIP problems to identify good edge-concave aggregations. From the literature it is known that the
65   	 * convex envelope of a bilinear edge-concave function \f$ f_i \f$ differs from McCormick iff in the graph
66   	 * representation of \f$ f_i \f$ there exist a cycle with an odd number of positive weighted edges. We look for a
67   	 * subgraph of the graph representation of the quadratic function \f$ g(x) \f$ with the previous property using a model
68   	 * based on binary flow arc variables.
69   	 *
70   	 * This separator is currently disabled by default. It requires additional
71   	 * tuning to be enabled by default. However, it may be useful to enable
72   	 * it on instances with nonconvex quadratic constraints, in particular boxQPs.
73   	 */
74   	
75   	/*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
76   	
77   	#ifndef __SCIP_SEPA_ECCUTS_H__
78   	#define __SCIP_SEPA_ECCUTS_H__
79   	
80   	
81   	#include "scip/def.h"
82   	#include "scip/type_retcode.h"
83   	#include "scip/type_scip.h"
84   	
85   	#ifdef __cplusplus
86   	extern "C" {
87   	#endif
88   	
89   	/** creates the edge-concave separator and includes it in SCIP
90   	 *
91   	 * @ingroup SeparatorIncludes
92   	 */
93   	SCIP_EXPORT
94   	SCIP_RETCODE SCIPincludeSepaEccuts(
95   	   SCIP*                 scip                /**< SCIP data structure */
96   	   );
97   	
98   	#ifdef __cplusplus
99   	}
100  	#endif
101  	
102  	#endif
103