1 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
2 /*                                                                           */
3 /*                  This file is part of the program and library             */
4 /*         SCIP --- Solving Constraint Integer Programs                      */
5 /*                                                                           */
6 /*    Copyright (C) 2002-2021 Konrad-Zuse-Zentrum                            */
7 /*                            fuer Informationstechnik Berlin                */
8 /*                                                                           */
9 /*  SCIP is distributed under the terms of the ZIB Academic License.         */
10 /*                                                                           */
11 /*  You should have received a copy of the ZIB Academic License              */
12 /*  along with SCIP; see the file COPYING. If not visit scipopt.org.         */
13 /*                                                                           */
14 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
15 
16 /**@file   heur_multistart.h
17  * @ingroup PRIMALHEURISTICS
18  * @brief  multistart heuristic for convex and nonconvex MINLPs
19  * @author Benjamin Mueller
20  *
21  * The heuristic applies multiple NLP local searches to a mixed-integer nonlinear program with, probably nonconvex,
22  * constraints of the form \f$g_j(x) \le 0\f$. The algorithm tries to identify clusters which approximate the boundary
23  * of the feasible set of the continuous relaxation by sampling and improving randomly generated points. For each
24  * cluster we use a local search heuristic to find feasible solutions. The algorithm consists of the following four
25  * steps:
26  *
27  * 1. sample points
28  *
29  *    Sample random points \f$ x^1, \ldots, x^n \f$ in the box \f$ [\ell,u] \f$. For an unbounded variable \f$ x_i \f$
30  *    we consider \f$ [\ell_i,\ell_i + \alpha], [u_i - \alpha,u_i], \f$ or \f$ [-\alpha / 2, \alpha / 2]\f$ for an \f$
31  *    \alpha > 0 \f$ depending on which bound is infinite.
32  *
33  * 2. reduce infeasibility
34  *
35  *   For each point \f$ x^i \f$ we use a gradient descent method to reduce the maximum infeasibility. We first compute
36  *
37  *    \f[
38  *        d_j = -\frac{g_j(x^i)}{||\nabla g_j(x^i)||^2} \nabla g_j(x^i)
39  *    \f]
40  *
41  *    and update the current point \f$ x^i \f$ with
42  *
43  *    \f[
44  *        x^i := x^i + \frac{1}{n_j} \sum_{j} d_j
45  *    \f]
46  *
47  *    where \f$ n_j \f$ is the number of strictly positive \f$ d_j \f$. The algorithm is called Constraint Consensus
48  *    Method and has been introduced by <a
49  *    href="http://www.sce.carleton.ca/faculty/chinneck/docs/ConstraintConsensusJoC.pdf">here </a>.
50  *
51  * 3. cluster points
52  *
53  *    We use a greedy algorithm to all of the resulting points of step 3. to find clusters which (hopefully) approximate
54  *    the boundary of the feasible set locally. Points with a too large violations will be filtered.
55  *
56  * 4. solve sub-problems
57  *
58  *    Depending on the current setting, we solve a sub-problem for each identified cluster. The default strategy is to
59  *    compute a starting point for the sub-NLP heuristic (see @ref heur_subnlp.h) by using a linear combination of the
60  *    points in a cluster \f$ C \f$, i.e.,
61  *
62  *    \f[
63  *        s := \sum_{x \in C} \lambda_x x
64  *    \f]
65  *
66  *    Since the sub-NLP heuristic requires a starting point which is integer feasible we round each fractional
67  *    value \f$ s_i \f$ to its closest integer.
68  */
69 
70 
71 /*---+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
72 
73 #ifndef __SCIP_HEUR_MULTISTART_H__
74 #define __SCIP_HEUR_MULTISTART_H__
75 
76 #include "scip/def.h"
77 #include "scip/type_retcode.h"
78 #include "scip/type_scip.h"
79 
80 #ifdef __cplusplus
81 extern "C" {
82 #endif
83 
84 /** creates the multistart primal heuristic and includes it in SCIP
85  *
86  *  @ingroup PrimalHeuristicIncludes
87  */
88 SCIP_EXPORT
89 SCIP_RETCODE SCIPincludeHeurMultistart(
90    SCIP*                 scip                /**< SCIP data structure */
91    );
92 
93 #ifdef __cplusplus
94 }
95 #endif
96 
97 #endif
98