sepa_gauge.h

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/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
/*                                                                           */
/*                  This file is part of the program and library             */
/*         SCIP --- Solving Constraint Integer Programs                      */
/*                                                                           */
/*    Copyright (C) 2002-2020 Konrad-Zuse-Zentrum                            */
/*                            fuer Informationstechnik Berlin                */
/*                                                                           */
/*  SCIP is distributed under the terms of the ZIB Academic License.         */
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/*                                                                           */
/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */

/**@file   sepa_gauge.h
 * @ingroup SEPARATORS
 * @brief  gauge separator
 * @author Felipe Serrano
 *
 * This separator receives a point \f$ x_0 \f$ to separate and, given an interior point \f$ \bar x \f$, finds the
 * intersection between the boundary of a convex relaxation of the current problem and the segment joining \f$ x_0 \f$
 * and \f$ \bar x \f$. Then it generates gradient cuts at the intersection.
 *
 * The interior point \f$ \bar x \f$ is computed only once, by solving
 * \f[
 *      \min t \\
 * \f]
 * \f[
 *      s.t. \; g_j(x) \le t \, \forall j=1,\ldots,m
 * \f]
 * \f[
 *      l_k(x) \le 0 \, \forall k=1,\ldots,p
 * \f]
 * where each \f$ g_j \f$ is a convex function and \f$ l_k \f$ is a linear function and
 * \f[
 *      C = \{ x \colon g_j(x) \le 0 \, \forall j=1,\ldots,m, l_k(x) \le 0 \, \forall k=1,\ldots,p \}
 * \f]
 * is a convex relaxation of the current problem.
 * If we can not find an interior solution, the separator will not be executed again.
 *
 * Note that we do not try to push the linear constraints into the interior, i.e. we use \f$ l_k(x) \le 0 \f$ instead
 * of \f$ l_k(x) \le t \f$, since some of the inequalities might actually be equalities, forcing \f$ t \f$ to zero.
 * We also use an arbitrary lower bound on \f$ t \f$ to handle the case when \f$ C \f$ is unbounded.
 *
 * By default, the separator runs only if the convex relaxation has at least two nonlinear convex constraints.
 *
 * In order to compute the boundary point, we consider only nonlinear convex constraints that are violated by the point
 * we want to separate. These constraints define a convex region for which \f$ \bar x \f$ is an interior point. Then,
 * a binary search is perform on the segment \f$[\bar x, x_0]\f$ in order to find the boundary point. Gradient cuts are
 * computed for each of these nonlinear convex constraints which are active at the boundary point.
 *
 * Technical details:
 *    - We consider a constraint for the binary search, only when its violation is larger than \f$ 10^{-4} \f$, see
 *    MIN_VIOLATION in sepa_gauge.c. The reason is that if the violation is too small, chances are that the point in the
 *    boundary is in the interior for this constraint and we wouldn't generate a cut for it anyway. On the other hand,
 *    even if we generate a cut for this constraint, it is likely that the boundary point is very close to the point to
 *    separate. Hence the cut generate would be very similar to the gradient cut at the point to separate.
 *    - Before separating, if a slight perturbation of the interior point in the direction of the point to separate is
 *    gives a point outside the region, we do not separate. The reason is that the interior point we computed could be
 *    almost at the boundary and the segment \f$[\bar x, x_0]\f$ could be tangent to the region. In that case, the cuts
 *    we generate will not separate \f$ x_0 \f$ from the feasible region.
 */

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#ifndef __SCIP_SEPA_GAUGE_H__
#define __SCIP_SEPA_GAUGE_H__


#include "scip/def.h"
#include "scip/type_retcode.h"
#include "scip/type_scip.h"

#ifdef __cplusplus
extern "C" {
#endif

/** creates the gauge separator and includes it in SCIP
 *
 * @ingroup SeparatorIncludes
 */
SCIP_EXPORT
SCIP_RETCODE SCIPincludeSepaGauge(
   SCIP*                 scip                /**< SCIP data structure */
   );

#ifdef __cplusplus
}
#endif

#endif