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24   	
25   	/**@file   prop_nlobbt.h
26   	 * @ingroup PROPAGATORS
27   	 * @brief  nonlinear OBBT propagator
28   	 * @author Benjamin Mueller
29   	 *
30   	 * In Nonlinear Optimization-Based Bound Tightening (NLOBBT), we solve auxiliary NLPs of the form
31   	 * \f[
32   	 *      \min / \max \, x_i \\
33   	 * \f]
34   	 * \f[
35   	 *      s.t. \; g_j(x) \le 0 \, \forall j=1,\ldots,m \\
36   	 * \f]
37   	 * \f[
38   	 *      c'x \le \mathcal{U}
39   	 * \f]
40   	 * \f[
41   	 *      x \in [\ell,u]
42   	 * \f]
43   	 *
44   	 * where each \f$ g_j \f$ is a convex function and \f$ \mathcal{U} \f$ the solution value of the current
45   	 * incumbent. Clearly, the optimal objective value of this nonlinear program provides a valid lower/upper bound on
46   	 * variable \f$ x_i \f$.
47   	 *
48   	 * The propagator sorts all variables w.r.t. their occurrences in convex nonlinear constraints and solves sequentially
49   	 * all convex NLPs. Variables which could be successfully tightened by the propagator will be prioritized in the next
50   	 * call of a new node in the branch-and-bound tree. By default, the propagator requires at least one nonconvex
51   	 * constraints to be executed. For purely convex problems, the benefit of having tighter bounds is negligible.
52   	 *
53   	 * By default, NLOBBT is only applied for non-binary variables. A reason for this can be found <a
54   	 * href="http://dx.doi.org/10.1007/s10898-016-0450-4">here </a>. Variables which do not appear non-linearly in the
55   	 * nonlinear constraints will not be considered even though they might lead to additional tightenings.
56   	 *
57   	 * After solving the NLP to optimize \f$ x_i \f$ we try to exploit the dual information to generate a globally valid
58   	 * inequality, called Generalized Variable Bound (see @ref prop_genvbounds.h). Let \f$ \lambda_j \f$, \f$ \mu \f$, \f$
59   	 * \alpha \f$, and \f$ \beta \f$ be the dual multipliers for the constraints of the NLP where \f$ \alpha \f$ and \f$
60   	 * \beta \f$ correspond to the variable bound constraints. Because of the convexity of \f$ g_j \f$ we know that
61   	 *
62   	 * \f[
63   	 *      g_j(x) \ge g_j(x^*) + \nabla g_j(x^*)(x-x^*)
64   	 * \f]
65   	 *
66   	 * holds for every \f$ x^* \in [\ell,u] \f$. Let \f$ x^* \f$ be the optimal solution after solving the NLP for the case
67   	 * of minimizing \f$ x_i \f$ (similiar for the case of maximizing \f$ x_i \f$). Since the NLP is convex we know that the
68   	 * KKT conditions
69   	 *
70   	 * \f[
71   	 *      e_i + \lambda' \nabla g(x^*) + \mu' c + \alpha - \beta = 0
72   	 * \f]
73   	 * \f[
74   	 *      \lambda_j g_j(x^*) = 0
75   	 * \f]
76   	 *
77   	 * hold. Aggregating the inequalities \f$ x_i \ge x_i \f$ and \f$ g_j(x) \le 0 \f$ leads to the inequality
78   	 *
79   	 * \f[
80   	 *      x_i \ge x_i + \sum_{j} g_j(x_i)
81   	 * \f]
82   	 *
83   	 * Instead of calling the (expensive) propagator during the tree search we can use this inequality to derive further
84   	 * reductions on \f$ x_i \f$. Multiplying the first KKT condition by \f$ (x - x^*) \f$ and using the fact that each
85   	 * \f$ g_j \f$ is convex we can rewrite the previous inequality to
86   	 *
87   	 * \f[
88   	 *      x_i \ge (\beta - \alpha)'x + (e_i + \alpha - \beta) x^* + \mu \mathcal{U}.
89   	 * \f]
90   	 *
91   	 * which is passed to the genvbounds propagator. Note that if \f$ \alpha_i \neq \beta_i \f$ we know that the bound of
92   	 * \f$ x_i \f$ is the proof for optimality and thus no useful genvbound can be found.
93   	 */
94   	
95   	/*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
96   	
97   	#ifndef __SCIP_PROP_NLOBBT_H__
98   	#define __SCIP_PROP_NLOBBT_H__
99   	
100  	#include "scip/def.h"
101  	#include "scip/type_retcode.h"
102  	#include "scip/type_scip.h"
103  	
104  	#ifdef __cplusplus
105  	extern "C" {
106  	#endif
107  	
108  	/** creates the nlobbt propagator and includes it in SCIP
109  	 *
110  	 * @ingroup PropagatorIncludes
111  	 */
112  	SCIP_EXPORT
113  	SCIP_RETCODE SCIPincludePropNlobbt(
114  	   SCIP*                 scip                /**< SCIP data structure */
115  	   );
116  	
117  	#ifdef __cplusplus
118  	}
119  	#endif
120  	
121  	#endif
122