1    	/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
2    	/*                                                                           */
3    	/*                  This file is part of the program and library             */
4    	/*         SCIP --- Solving Constraint Integer Programs                      */
5    	/*                                                                           */
6    	/*  Copyright (c) 2002-2023 Zuse Institute Berlin (ZIB)                      */
7    	/*                                                                           */
8    	/*  Licensed under the Apache License, Version 2.0 (the "License");          */
9    	/*  you may not use this file except in compliance with the License.         */
10   	/*  You may obtain a copy of the License at                                  */
11   	/*                                                                           */
12   	/*      http://www.apache.org/licenses/LICENSE-2.0                           */
13   	/*                                                                           */
14   	/*  Unless required by applicable law or agreed to in writing, software      */
15   	/*  distributed under the License is distributed on an "AS IS" BASIS,        */
16   	/*  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. */
17   	/*  See the License for the specific language governing permissions and      */
18   	/*  limitations under the License.                                           */
19   	/*                                                                           */
20   	/*  You should have received a copy of the Apache-2.0 license                */
21   	/*  along with SCIP; see the file LICENSE. If not visit scipopt.org.         */
22   	/*                                                                           */
23   	/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
24   	
25   	/**@file   sepa_minor.h
26   	 * @ingroup SEPARATORS
27   	 * @brief  principal minor separator
28   	 * @author Benjamin Mueller
29   	 *
30   	 * This separator detects all principal minors of the matrix \f$ xx' \f$ for which all auxiliary variables \f$ X \f$
31   	 * exist, i.e., two indices \f$ i \neq j \f$ such that \f$ X_{ii} \f$, \f$ X_{jj} \f$, and \f$ X_{ij} \f$ exist. Because
32   	 * \f$ X - xx' \f$ is required to be positive semi-definite, it follows that the matrix
33   	 *
34   	 * \f[
35   	 *    A(x,X) = \begin{bmatrix} 1 & x_i & x_j \\ x_i & X_{ii} & X_{ij} \\ x_j & X_{ij} & X_{jj} \end{bmatrix}
36   	 * \f]
37   	 *
38   	 * is also required to be positive semi-definite. Let \f$ v \f$ be a negative eigenvector for \f$ A(x^*,X^*) \f$ in a
39   	 * point \f$ (x^*,X^*) \f$, which implies that \f$ v' A(x^*,X^*) v < 0 \f$. To cut off \f$ (x^*,X^*) \f$, the separator
40   	 * computes the globally valid linear inequality \f$ v' A(x,X) v \ge 0 \f$.
41   	 *
42   	 *
43   	 * To identify which entries of the matrix X exist, we (the separator) iterate over the available nonlinear constraints.
44   	 * For each constraint, we explore its expression and collect all nodes (expressions) of the form
45   	 * - \f$x^2\f$
46   	 * - \f$y \cdot z\f$
47   	 *
48   	 * Then, we go through the found bilinear terms \f$(yz)\f$ and if the corresponding \f$y^2\f$ and \f$z^2\f$ exist, then we have found
49   	 * a minor.
50   	 *
51   	 * For circle packing instances, the minor cuts are not really helpful (see [Packing circles in a square: a theoretical
52   	 * comparison of various convexification techniques](http://www.optimization-online.org/DB_HTML/2017/03/5911.html)).
53   	 * Furthermore, the performance was negatively affected, thus circle packing constraint are identified and ignored in
54   	 * the above algorithm. This behavior is controlled with the parameter "separating/minor/ignorepackingconss".
55   	 */
56   	
57   	/*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
58   	
59   	#ifndef __SCIP_SEPA_MINOR_H__
60   	#define __SCIP_SEPA_MINOR_H__
61   	
62   	
63   	#include "scip/scip.h"
64   	
65   	#ifdef __cplusplus
66   	extern "C" {
67   	#endif
68   	
69   	/** creates the minor separator and includes it in SCIP
70   	 *
71   	 * @ingroup SeparatorIncludes
72   	 */
73   	SCIP_EXPORT
74   	SCIP_RETCODE SCIPincludeSepaMinor(
75   	   SCIP*                 scip                /**< SCIP data structure */
76   	   );
77   	
78   	/**@addtogroup SEPARATORS
79   	 *
80   	 * @{
81   	 */
82   	
83   	/** @} */
84   	
85   	#ifdef __cplusplus
86   	}
87   	#endif
88   	
89   	#endif
90