# 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 /* */
7 /* fuer Informationstechnik Berlin */
8 /* */
10 /* */
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 separator 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{align}{
30  * \min \; & || x - x_0 ||^2 \\
31  * s.t. \; & g_j(x) \le 0 & \forall j=1,\ldots,m.
32  * \f}
33  *
34  * By default, if enabled, the separator runs only if the convex relaxation has at least one nonlinear convex function.
35  *
36  * The separator generates cuts for constraints which were violated by the solution we want to separate and active
37  * at the projection. If the projection problem is not solved to optimality, it still tries to add a cut at the
38  * best solution found. In case that the projection problem is solved to optimality, it is guaranteed that a cut
39  * separates the point. To see this, remember that \f$z \f$ is the projection if and only if
40  * \f[
41  * \langle x - z, z - x_0 \rangle \ge 0 \, \forall x \in C \\
42  * \f]
43  * This inequality is violated for \f$x = x_0 \f$. On the other hand, one of the optimality conditions of the
44  * projection problem at the optimum looks like
45  * \f[
46  * 2 (z - x_0) + \sum_j \lambda_j \nabla g_j(z) = 0.
47  * \f]
48  * Now suppose that the no gradient cut at \f$z \f$ separates \f$x_0 \f$, i.e.,
49  * \f[
50  * g_j(z) + \langle \nabla g_j(z), x_0 - z \rangle \le 0.
51  * \f]
52  * Multiplying each inequality with \f$\lambda_j \ge 0 \f$ and summing up, we get the following contradiction:
53  * \f[
54  * \langle -2(z - x_0), x_0 - z \rangle \le 0.
55  * \f]
56  *
57  * This separator is currently disabled by default. It requires additional
58  * tuning to be enabled by default. However, it may be useful to enable
59  * it on instances with convex nonlinear constraints if SCIP spends
60  * many iterations in the separation loop without doing sufficient progress.
61  */
62
63 /*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
64
65 #ifndef __SCIP_SEPA_CONVEXPROJ_H__
66 #define __SCIP_SEPA_CONVEXPROJ_H__
67
68
69 #include "scip/def.h"
70 #include "scip/type_retcode.h"
71 #include "scip/type_scip.h"
72
73 #ifdef __cplusplus
74 extern "C" {
75 #endif
76
77 /** creates the convexproj separator and includes it in SCIP
78  *
79  * @ingroup SeparatorIncludes
80  */
81 SCIP_EXPORT
83  SCIP* scip /**< SCIP data structure */
84  );
85
86 #ifdef __cplusplus
87 }
88 #endif
89
90 #endif
enum SCIP_Retcode SCIP_RETCODE
Definition: type_retcode.h:54
type definitions for return codes for SCIP methods
SCIP_RETCODE SCIPincludeSepaConvexproj(SCIP *scip)
type definitions for SCIP&#39;s main datastructure
common defines and data types used in all packages of SCIP