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.         */
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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,        */
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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_gauge.h
26   	 * @ingroup SEPARATORS
27   	 * @brief  gauge separator
28   	 * @author Felipe Serrano
29   	 *
30   	 * This separator receives a point \f$ x_0 \f$ to separate and, given an interior point \f$ \bar x \f$, finds the
31   	 * intersection between the boundary of a convex relaxation of the current problem and the segment joining \f$ x_0 \f$
32   	 * and \f$ \bar x \f$. Then it generates gradient cuts at the intersection.
33   	 *
34   	 * The interior point \f$ \bar x \f$ is computed only once, by solving
35   	 * \f{align}{
36   	 *      \min \; & t \\
37   	 *      s.t. \; & g_j(x) \le t & \forall j=1,\ldots,m \\
38   	 *      & l_k(x) \le 0 & \forall k=1,\ldots,p
39   	 * \f}
40   	 * where each \f$ g_j \f$ is a convex function and \f$ l_k \f$ is a linear function and
41   	 * \f[
42   	 *      C = \{ x \colon g_j(x) \le 0 \, \forall j=1,\ldots,m, l_k(x) \le 0 \, \forall k=1,\ldots,p \}
43   	 * \f]
44   	 * is a convex relaxation of the current problem.
45   	 * If we can not find an interior solution, the separator will not be executed again.
46   	 *
47   	 * 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
48   	 * of \f$ l_k(x) \le t \f$, since some of the inequalities might actually be equalities, forcing \f$ t \f$ to zero.
49   	 * We also use an arbitrary lower bound on \f$ t \f$ to handle the case when \f$ C \f$ is unbounded.
50   	 *
51   	 * By default, the separator, if enabled, runs only if the convex relaxation has at least two nonlinear convex constraints.
52   	 *
53   	 * In order to compute the boundary point, we consider only nonlinear convex constraints that are violated by the point
54   	 * we want to separate. These constraints define a convex region for which \f$ \bar x \f$ is an interior point. Then,
55   	 * a binary search is perform on the segment \f$[\bar x, x_0]\f$ in order to find the boundary point. Gradient cuts are
56   	 * computed for each of these nonlinear convex constraints which are active at the boundary point.
57   	 *
58   	 * Technical details:
59   	 *    - We consider a constraint for the binary search only when its violation is larger than \f$ 10^{-4} \f$, see
60   	 *    MIN_VIOLATION in sepa_gauge.c. The reason is that if the violation is too small, chances are that the point in the
61   	 *    boundary is in the interior for this constraint and we wouldn't generate a cut for it anyway. On the other hand,
62   	 *    even if we generate a cut for this constraint, it is likely that the boundary point is very close to the point to
63   	 *    separate. Hence the cut generated would be very similar to the gradient cut at the point to separate.
64   	 *    - Before separating, if a slight perturbation of the interior point in the direction of the point to separate
65   	 *    gives a point outside the region, we do not separate. The reason is that the interior point we computed could be
66   	 *    almost at the boundary and the segment \f$[\bar x, x_0]\f$ could be tangent to the region. In that case, the cuts
67   	 *    we generate will not separate \f$ x_0 \f$ from the feasible region.
68   	 *
69   	 * This separator is currently disabled by default. It requires additional
70   	 * tuning to be enabled by default. However, it may be useful to enable
71   	 * it on instances with convex nonlinear constraints if SCIP spends
72   	 * many iterations in the separation loop without doing sufficient progress.
73   	 */
74   	
75   	/*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
76   	
77   	#ifndef __SCIP_SEPA_GAUGE_H__
78   	#define __SCIP_SEPA_GAUGE_H__
79   	
80   	
81   	#include "scip/def.h"
82   	#include "scip/type_retcode.h"
83   	#include "scip/type_scip.h"
84   	
85   	#ifdef __cplusplus
86   	extern "C" {
87   	#endif
88   	
89   	/** creates the gauge separator and includes it in SCIP
90   	 *
91   	 * @ingroup SeparatorIncludes
92   	 */
93   	SCIP_EXPORT
94   	SCIP_RETCODE SCIPincludeSepaGauge(
95   	   SCIP*                 scip                /**< SCIP data structure */
96   	   );
97   	
98   	#ifdef __cplusplus
99   	}
100  	#endif
101  	
102  	#endif
103