1 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */ 2 /* */ 3 /* This file is part of the program and library */ 4 /* SCIP --- Solving Constraint Integer Programs */ 5 /* */ 6 /* Copyright (c) 2002-2023 Zuse Institute Berlin (ZIB) */ 7 /* */ 8 /* Licensed under the Apache License, Version 2.0 (the "License"); */ 9 /* you may not use this file except in compliance with the License. */ 10 /* You may obtain a copy of the License at */ 11 /* */ 12 /* http://www.apache.org/licenses/LICENSE-2.0 */ 13 /* */ 14 /* Unless required by applicable law or agreed to in writing, software */ 15 /* distributed under the License is distributed on an "AS IS" BASIS, */ 16 /* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. */ 17 /* See the License for the specific language governing permissions and */ 18 /* limitations under the License. */ 19 /* */ 20 /* You should have received a copy of the Apache-2.0 license */ 21 /* along with SCIP; see the file LICENSE. If not visit scipopt.org. */ 22 /* */ 23 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */ 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