  # SCIP

Solving Constraint Integer Programs

prop_nlobbt.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 prop_nlobbt.h
17  * @ingroup PROPAGATORS
18  * @brief nonlinear OBBT propagator
19  * @author Benjamin Mueller
20  *
21  * In Nonlinear Optimization-Based Bound Tightening (NLOBBT), we solve auxiliary NLPs of the form
22  * \f[
23  * \min / \max \, x_i \\
24  * \f]
25  * \f[
26  * s.t. \; g_j(x) \le 0 \, \forall j=1,\ldots,m \\
27  * \f]
28  * \f[
29  * c'x \le \mathcal{U}
30  * \f]
31  * \f[
32  * x \in [\ell,u]
33  * \f]
34  *
35  * where each \f$g_j \f$ is a convex function and \f$\mathcal{U} \f$ the solution value of the current
36  * incumbent. Clearly, the optimal objective value of this nonlinear program provides a valid lower/upper bound on
37  * variable \f$x_i \f$.
38  *
39  * The propagator sorts all variables w.r.t. their occurrences in convex nonlinear constraints and solves sequentially
40  * all convex NLPs. Variables which could be successfully tightened by the propagator will be prioritized in the next
41  * call of a new node in the branch-and-bound tree. By default, the propagator requires at least one nonconvex
42  * constraints to be executed. For purely convex problems, the benefit of having tighter bounds is negligible.
43  *
44  * By default, NLOBBT is only applied for non-binary variables. A reason for this can be found <a
45  * href="http://dx.doi.org/10.1007/s10898-016-0450-4">here </a>. Variables which do not appear non-linearly in the
46  * nonlinear constraints will not be considered even though they might lead to additional tightenings.
47  *
48  * After solving the NLP to optimize \f$x_i \f$ we try to exploit the dual information to generate a globally valid
49  * inequality, called Generalized Variable Bound (see @ref prop_genvbounds.h). Let \f$\lambda_j \f$, \f$\mu \f$, \f$50 * \alpha \f$, and \f$\beta \f$ be the dual multipliers for the constraints of the NLP where \f$\alpha \f$ and \f$51 * \beta \f$ correspond to the variable bound constraints. Because of the convexity of \f$g_j \f$ we know that
52  *
53  * \f[
54  * g_j(x) \ge g_j(x^*) + \nabla g_j(x^*)(x-x^*)
55  * \f]
56  *
57  * holds for every \f$x^* \in [\ell,u] \f$. Let \f$x^* \f$ be the optimal solution after solving the NLP for the case
58  * of minimizing \f$x_i \f$ (similiar for the case of maximizing \f$x_i \f$). Since the NLP is convex we know that the
59  * KKT conditions
60  *
61  * \f[
62  * e_i + \lambda' \nabla g(x^*) + \mu' c + \alpha - \beta = 0
63  * \f]
64  * \f[
65  * \lambda_j g_j(x^*) = 0
66  * \f]
67  *
68  * hold. Aggregating the inequalities \f$x_i \ge x_i \f$ and \f$g_j(x) \le 0 \f$ leads to the inequality
69  *
70  * \f[
71  * x_i \ge x_i + \sum_{j} g_j(x_i)
72  * \f]
73  *
74  * Instead of calling the (expensive) propagator during the tree search we can use this inequality to derive further
75  * reductions on \f$x_i \f$. Multiplying the first KKT condition by \f$(x - x^*) \f$ and using the fact that each
76  * \f$g_j \f$ is convex we can rewrite the previous inequality to
77  *
78  * \f[
79  * x_i \ge (\beta - \alpha)'x + (e_i + \alpha - \beta) x^* + \mu \mathcal{U}.
80  * \f]
81  *
82  * which is passed to the genvbounds propagator. Note that if \f$\alpha_i \neq \beta_i \f$ we know that the bound of
83  * \f$x_i \f$ is the proof for optimality and thus no useful genvbound can be found.
84  */
85
86 /*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
87
88 #ifndef __SCIP_PROP_NLOBBT_H__
89 #define __SCIP_PROP_NLOBBT_H__
90
91 #include "scip/def.h"
92 #include "scip/type_retcode.h"
93 #include "scip/type_scip.h"
94
95 #ifdef __cplusplus
96 extern "C" {
97 #endif
98
99 /** creates the nlobbt propagator and includes it in SCIP
100  *
101  * @ingroup PropagatorIncludes
102  */
105  SCIP* scip /**< SCIP data structure */
106  );
107
108 #ifdef __cplusplus
109 }
110 #endif
111
112 #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 SCIPincludePropNlobbt(SCIP *scip)
Definition: prop_nlobbt.c:724
type definitions for SCIP&#39;s main datastructure
common defines and data types used in all packages of SCIP