# SCIP

Solving Constraint Integer Programs

sepa_convexproj.h
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1 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
2 /* */
3 /* This file is part of the program and library */
4 /* SCIP --- Solving Constraint Integer Programs */
5 /* */
6 /* Copyright (C) 2002-2020 Konrad-Zuse-Zentrum */
7 /* fuer Informationstechnik Berlin */
8 /* */
9 /* SCIP is distributed under the terms of the ZIB Academic License. */
10 /* */
11 /* You should have received a copy of the ZIB Academic License */
12 /* along with SCIP; see the file COPYING. If not visit scipopt.org. */
13 /* */
14 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
15
16 /**@file sepa_convexproj.h
17  * @ingroup SEPARATORS
18  * @brief convexproj separator
19  * @author Felipe Serrano
20  *
21  * This separator receives a point \f$x_0 \f$ to separate, projects it onto a convex relaxation
22  * of the current problem and then generates gradient cuts at the projection.
23  *
24  * In more detail, the separators builds and stores a convex relaxation of the problem
25  * \f[
26  * C = \{ x \colon g_j(x) \le 0 \, \forall j=1,\ldots,m \}
27  * \f]
28  * where each \f$g_j \f$ is a convex function and computes the projection by solving
29  * \f[
30  * \min || x - x_0 ||^2 \\
31  * \f]
32  * \f[
33  * s.t. \; g_j(x) \le 0 \, \forall j=1,\ldots,m \\
34  * \f]
35  *
36  * By default, the separator runs only if the convex relaxation has at least one nonlinear convex function
37  *
38  * The separator generates cuts for constraints which were violated by the solution we want to separate and active
39  * at the projection. If the projection problem is not solved to optimality, it still tries to add a cut at the
40  * best solution found. In case that the projection problem is solved to optimality, it is guaranteed that a cut
41  * separates the point. To see this, remember that \f$z \f$ is the projection if and only if
42  * \f[
43  * \langle x - z, z - x_0 \rangle \ge 0 \, \forall x \in C \\
44  * \f]
45  * This inequality is violated by for \f$x = x_0 \f$. On the other hand, one of the optimality conditions of the
46  * projection problem at the optimum looks like
47  * \f[
48  * 2 (z - x_0) + \sum_j \lambda_j \nabla g_j(z) = 0
49  * \f]
50  * Now suppose that the no gradient cut at \f$z \f$ separates \f$x_0 \f$, i.e.,
51  * \f[
52  * g_j(z) + \langle \nabla g_j(z), x_0 - z \rangle \le 0
53  * \f]
54  * Multiplying each inequality with \f$\lambda_j \ge 0 \f$ and summing up, we get the following contradiction:
55  * \f[
56  * \langle -2(z - x_0), x_0 - z \rangle \le 0
57  * \f]
58  */
59
60 /*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
61
62 #ifndef __SCIP_SEPA_CONVEXPROJ_H__
63 #define __SCIP_SEPA_CONVEXPROJ_H__
64
65
66 #include "scip/def.h"
67 #include "scip/type_retcode.h"
68 #include "scip/type_scip.h"
69
70 #ifdef __cplusplus
71 extern "C" {
72 #endif
73
74 /** creates the convexproj separator and includes it in SCIP
75  *
76  * @ingroup SeparatorIncludes
77  */
80  SCIP* scip /**< SCIP data structure */
81  );
82
83 #ifdef __cplusplus
84 }
85 #endif
86
87 #endif
#define SCIP_EXPORT
Definition: def.h:100
enum SCIP_Retcode SCIP_RETCODE
Definition: type_retcode.h:54
type definitions for return codes for SCIP methods
SCIP_EXPORT SCIP_RETCODE SCIPincludeSepaConvexproj(SCIP *scip)
type definitions for SCIP&#39;s main datastructure
common defines and data types used in all packages of SCIP