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 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 91 SCIP_RETCODE SCIPincludeSepaConvexproj( 92 SCIP* scip /**< SCIP data structure */ 93 ); 94 95 #ifdef __cplusplus 96 } 97 #endif 98 99 #endif 100