# 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 /* */
7 /* fuer Informationstechnik Berlin */
8 /* */
10 /* */
12 /* along with SCIP; see the file COPYING. If not visit scip.zib.de. */
13 /* */
14 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
15
16 /**@file sepa_minor.h
17  * @ingroup SEPARATORS
18  * @brief principal minor separator
19  * @author Benjamin Mueller
20  *
21  * This separator detects all principal minors of the matrix \f$xx' \f$ for which all auxiliary variables \f$X \f$
22  * 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
23  * \f$X - xx' \f$ is required to be positive semi-definite, it follows that the matrix
24  *
25  * \f[
26  * A(x,X) = \begin{bmatrix} 1 & x_i & x_j \\ x_i & X_{ii} & X_{ij} \\ x_j & X_{ij} & X_{jj} \end{bmatrix}
27  * \f]
28  *
29  * is also required to be positive semi-definite. Let \f$v \f$ be a negative eigenvector for \f$A(x^*,X^*) \f$ in a
30  * 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
31  * computes the globally valid linear inequality \f$v' A(x,X) v \ge 0 \f$.
32  *
33  *
34  * To identify which entries of the matrix X exist, we (the separator) iterate over the available nonlinear constraints.
35  * For each constraint, we explore its expression and collect all nodes (expressions) of the form
36  * - \f$x^2\f$
37  * - \f$y \cdot z\f$
38  *
39  * 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
40  * a minor.
41  *
42  * For circle packing instances, the minor cuts are not really helpful (see [Packing circles in a square: a theoretical
43  * comparison of various convexification techniques](http://www.optimization-online.org/DB_HTML/2017/03/5911.html)).
44  * Furthermore, the performance was negatively affected, thus circle packing constraint are identified and ignored in
45  * the above algorithm. This behavior is controlled with the parameter "separating/minor/ignorepackingconss".
46  */
47
48 /*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
49
50 #ifndef __SCIP_SEPA_MINOR_H__
51 #define __SCIP_SEPA_MINOR_H__
52
53
54 #include "scip/scip.h"
55
56 #ifdef __cplusplus
57 extern "C" {
58 #endif
59
60 /** creates the minor separator and includes it in SCIP
61  *
62  * @ingroup SeparatorIncludes
63  */
64 SCIP_EXPORT
66  SCIP* scip /**< SCIP data structure */
67  );
68
70  *
71  * @{
72  */
73
74 /** @} */
75
76 #ifdef __cplusplus
77 }
78 #endif
79
80 #endif
enum SCIP_Retcode SCIP_RETCODE
Definition: type_retcode.h:54
SCIP_RETCODE SCIPincludeSepaMinor(SCIP *scip)
Definition: sepa_minor.c:871
SCIP callable library.