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