Scippy

SCIP

Solving Constraint Integer Programs

sepa_minor.h
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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 /* */
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13 /* */
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19 /* */
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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
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
enum SCIP_Retcode SCIP_RETCODE
Definition: type_retcode.h:63
SCIP_RETCODE SCIPincludeSepaMinor(SCIP *scip)
Definition: sepa_minor.c:880
SCIP callable library.