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) 2002-2024 Zuse Institute Berlin (ZIB) */
7 /* */
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23 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
24
25 /**@file sepa_convexproj.h
26  * @ingroup SEPARATORS
27  * @brief convexproj separator
28  * @author Felipe Serrano
29  *
30  * This separator receives a point \f$x_0 \f$ to separate, projects it onto a convex relaxation
31  * of the current problem and then generates gradient cuts at the projection.
32  *
33  * In more detail, the separator builds and stores a convex relaxation of the problem
34  * \f[
35  * C = \{ x \colon g_j(x) \le 0 \, \forall j=1,\ldots,m \}
36  * \f]
37  * where each \f$g_j \f$ is a convex function and computes the projection by solving
38  * \f{align}{
39  * \min \; & || x - x_0 ||^2 \\
40  * s.t. \; & g_j(x) \le 0 & \forall j=1,\ldots,m.
41  * \f}
42  *
43  * By default, if enabled, the separator runs only if the convex relaxation has at least one nonlinear convex function.
44  *
45  * The separator generates cuts for constraints which were violated by the solution we want to separate and active
46  * at the projection. If the projection problem is not solved to optimality, it still tries to add a cut at the
47  * best solution found. In case that the projection problem is solved to optimality, it is guaranteed that a cut
48  * separates the point. To see this, remember that \f$z \f$ is the projection if and only if
49  * \f[
50  * \langle x - z, z - x_0 \rangle \ge 0 \, \forall x \in C \\
51  * \f]
52  * This inequality is violated for \f$x = x_0 \f$. On the other hand, one of the optimality conditions of the
53  * projection problem at the optimum looks like
54  * \f[
55  * 2 (z - x_0) + \sum_j \lambda_j \nabla g_j(z) = 0.
56  * \f]
57  * Now suppose that the no gradient cut at \f$z \f$ separates \f$x_0 \f$, i.e.,
58  * \f[
59  * g_j(z) + \langle \nabla g_j(z), x_0 - z \rangle \le 0.
60  * \f]
61  * Multiplying each inequality with \f$\lambda_j \ge 0 \f$ and summing up, we get the following contradiction:
62  * \f[
63  * \langle -2(z - x_0), x_0 - z \rangle \le 0.
64  * \f]
65  *
66  * This separator is currently disabled by default. It requires additional
67  * tuning to be enabled by default. However, it may be useful to enable
68  * it on instances with convex nonlinear constraints if SCIP spends
69  * many iterations in the separation loop without doing sufficient progress.
70  */
71
72 /*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
73
74 #ifndef __SCIP_SEPA_CONVEXPROJ_H__
75 #define __SCIP_SEPA_CONVEXPROJ_H__
76
77
78 #include "scip/def.h"
79 #include "scip/type_retcode.h"
80 #include "scip/type_scip.h"
81
82 #ifdef __cplusplus
83 extern "C" {
84 #endif
85
86 /** creates the convexproj separator and includes it in SCIP
87  *
88  * @ingroup SeparatorIncludes
89  */
90 SCIP_EXPORT
92  SCIP* scip /**< SCIP data structure */
93  );
94
95 #ifdef __cplusplus
96 }
97 #endif
98
99 #endif
enum SCIP_Retcode SCIP_RETCODE
Definition: type_retcode.h:63
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