presol_qpkktref.h

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/**@file   presol_qpkktref.h
 * @ingroup PRESOLVERS
 * @brief  qpkktref presolver
 * @author Tobias Fischer
 *
 * This presolver tries to add the KKT conditions as additional (redundant) constraints to the (mixed-binary) quadratic
 * program
 *  \f[
 *  \begin{array}{ll}
 *  \min         & x^T Q x + c^T x + d \\
 *               & A x \leq b, \\
 *               & x \in \{0, 1\}^{p} \times R^{n-p}.
 * \end{array}
 * \f]
 *
 * We first check if the structure of the program is like (QP), see the documentation of the function
 * checkConsQuadraticProblem().
 *
 * If the problem is known to be bounded (all variables have finite lower and upper bounds), then we add the KKT
 * conditions. For a continuous QPs the KKT conditions have the form
 * \f[
 *  \begin{array}{ll}
 *   Q x + c + A^T \mu = 0,\\
 *   Ax \leq b,\\
 *   \mu_i \cdot (Ax - b)_i = 0,    & i \in \{1, \dots, m\},\\
 *   \mu \geq 0.
 * \end{array}
 * \f]
 * where \f$\mu\f$ are the Lagrangian variables. Each of the complementarity constraints \f$\mu_i \cdot (Ax - b)_i = 0\f$
 * is enforced via an SOS1 constraint for \f$\mu_i\f$ and an additional slack variable \f$s_i = (Ax - b)_i\f$.
 *
 * For mixed-binary QPs, the KKT-like conditions are
 * \f[
 *  \begin{array}{ll}
 *   Q x + c + A^T \mu + I_J \lambda = 0,\\
 *   Ax \leq b,\\
 *   x_j \in \{0,1\}                    & j \in J,\\
 *   (1 - x_j) \cdot z_j = 0            & j \in J,\\
 *   x_j \cdot (z_j - \lambda_j) = 0    & j \in J,\\
 *   \mu_i \cdot (Ax - b)_i = 0         & i \in \{1, \dots, m\},\\
 *   \mu \geq 0,
 * \end{array}
 * \f]
 * where \f$J = \{1,\dots, p\}\f$, \f$\mu\f$ and \f$\lambda\f$ are the Lagrangian variables, and \f$I_J\f$ is the
 * submatrix of the \f$n\times n\f$ identity matrix with columns indexed by \f$J\f$. For the derivation of the KKT-like
 * conditions, see
 *
 *  Branch-And-Cut for Complementarity and Cardinality Constrained Linear Programs,@n
 *  Tobias Fischer, PhD Thesis (2016)
 *
 * Algorithmically:
 *
 * - we handle the quadratic term variables of the quadratic constraint like in the method
 *   presolveAddKKTQuadQuadraticTerms()
 * - we handle the bilinear term variables of the quadratic constraint like in the method presolveAddKKTQuadBilinearTerms()
 * - we handle the linear term variables of the quadratic constraint like in the method presolveAddKKTQuadLinearTerms()
 * - we handle linear constraints in the method presolveAddKKTLinearConss()
 * - we handle aggregated variables in the method presolveAddKKTAggregatedVars()
 *
 * we have a hashmap from each variable to the index of the dual constraint in the KKT conditions.
 */
 
/*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
 
#ifndef __SCIP_PRESOL_QPKKTREF_H__
#define __SCIP_PRESOL_QPKKTREF_H__
 
#include "scip/def.h"
#include "scip/type_retcode.h"
#include "scip/type_scip.h"
 
#ifdef __cplusplus
extern "C" {
#endif
 
/** creates the QP KKT reformulation presolver and includes it in SCIP
 *
 * @ingroup PresolverIncludes
 */
SCIP_EXPORT
SCIP_RETCODE SCIPincludePresolQPKKTref(
   SCIP*                 scip                /**< SCIP data structure */
   );
 
#ifdef __cplusplus
}
#endif
 
#endif