prop_nlobbt.h
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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
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.
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.
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
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$). Since the NLP is convex we know that the
68 * hold. Aggregating the inequalities \f$ x_i \ge x_i \f$ and \f$ g_j(x) \le 0 \f$ leads to the inequality
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
82 * which is passed to the genvbounds propagator. Note that if \f$ \alpha_i \neq \beta_i \f$ we know that the bound of
86 /*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
Definition: struct_scip.h:59
type definitions for return codes for SCIP methods
SCIP_EXPORT SCIP_RETCODE SCIPincludePropNlobbt(SCIP *scip)
Definition: prop_nlobbt.c:724
type definitions for SCIP's main datastructure
common defines and data types used in all packages of SCIP
Definition: objbenders.h:33