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/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
/* */
/* This file is part of the program and library */
/* SCIP --- Solving Constraint Integer Programs */
/* */
/* Copyright 2002-2022 Zuse Institute Berlin */
/* */
/* Licensed under the Apache License, Version 2.0 (the "License"); */
/* you may not use this file except in compliance with the License. */
/* You may obtain a copy of the License at */
/* */
/* http://www.apache.org/licenses/LICENSE-2.0 */
/* */
/* Unless required by applicable law or agreed to in writing, software */
/* distributed under the License is distributed on an "AS IS" BASIS, */
/* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. */
/* See the License for the specific language governing permissions and */
/* limitations under the License. */
/* */
/* You should have received a copy of the Apache-2.0 license */
/* along with SCIP; see the file LICENSE. If not visit scipopt.org. */
/* */
/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
/**@file sepa_minor.h
* @ingroup SEPARATORS
* @brief principal minor separator
* @author Benjamin Mueller
*
* This separator detects all principal minors of the matrix \f$ xx' \f$ for which all auxiliary variables \f$ X \f$
* 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
* \f$ X - xx' \f$ is required to be positive semi-definite, it follows that the matrix
*
* \f[
* A(x,X) = \begin{bmatrix} 1 & x_i & x_j \\ x_i & X_{ii} & X_{ij} \\ x_j & X_{ij} & X_{jj} \end{bmatrix}
* \f]
*
* is also required to be positive semi-definite. Let \f$ v \f$ be a negative eigenvector for \f$ A(x^*,X^*) \f$ in a
* point \f$ (x^*,X^*) \f$, which implies that \f$ v' A(x^*,X^*) v < 0 \f$. To cut off \f$ (x^*,X^*) \f$, the separator
* computes the globally valid linear inequality \f$ v' A(x,X) v \ge 0 \f$.
*
*
* To identify which entries of the matrix X exist, we (the separator) iterate over the available nonlinear constraints.
* For each constraint, we explore its expression and collect all nodes (expressions) of the form
* - \f$x^2\f$
* - \f$y \cdot z\f$
*
* 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
* a minor.
*
* For circle packing instances, the minor cuts are not really helpful (see [Packing circles in a square: a theoretical
* comparison of various convexification techniques](http://www.optimization-online.org/DB_HTML/2017/03/5911.html)).
* Furthermore, the performance was negatively affected, thus circle packing constraint are identified and ignored in
* the above algorithm. This behavior is controlled with the parameter "separating/minor/ignorepackingconss".
*/
/*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
#ifndef __SCIP_SEPA_MINOR_H__
#define __SCIP_SEPA_MINOR_H__
#include "scip/scip.h"
#ifdef __cplusplus
extern "C" {
#endif
/** creates the minor separator and includes it in SCIP
*
* @ingroup SeparatorIncludes
*/
SCIP_EXPORT
SCIP_RETCODE SCIPincludeSepaMinor(
SCIP* scip /**< SCIP data structure */
);
/**@addtogroup SEPARATORS
*
* @{
*/
/** @} */
#ifdef __cplusplus
}
#endif
#endif