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xternal_cycleclustering.c
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1/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
2/* */
3/* This file is part of the program and library */
4/* SCIP --- Solving Constraint Integer Programs */
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8/* Licensed under the Apache License, Version 2.0 (the "License"); */
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23/* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
24
25/**@file xternal_cycleclustering.c
26 * @brief main document page
27 * @author Leon Eifler
28 */
29
30/*--+----1----+----2----+----3----+----4----+----5----+----6----+----7----+----8----+----9----+----0----+----1----+----2*/
31
32/**@page CYCLECLUSTERING_MAIN CycleClustering
33 * @version 0.1
34 * @author Leon Eifler
35 *
36 * This application can be used to solve the cycle clustering problem as described in
37 *
38 * "Mixed-Integer Programming for Cycle Detection in Non-reversible Markov Processes"@n
39 * by Isabel Beckenbach, Leon Eifler, Konstantin Fackeldey, Ambros Gleixner, Andreas Grever, Marcus Weber, and Jakob Witzig, @n
40 * <em> Multiscale Modeling and Simulation </em> , 2016 (accepted for publication, preprint available as <a href="https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/6035">ZIB-Report 16-39</a> ).
41 *
42 * The input format is an \f$n \times n\f$ - matrix \f$Q\f$ of unconditional transition probabilities with a header of the form
43 * "# p nstates ncluster"; nstates is the size of the matrix, ncluster the desired number of clusters; the name of the file must end with ".spa".
44 *
45 * The cycle clustering problem is the following:
46 *
47 * Consider a set of states \f$ \mathcal B = \{1,\ldots,n\}\f$ and a set of clusters \f$\mathcal{C}=\{1,\ldots,m\}\f$.
48 * Let \f$Q \in \mathbb{R}^{n \times n}\f$ with entries \f$ q_{ij}\f$. Then the problem is given by the MINLP
49 *
50 * \f{align*}{
51 * \max \ \ \ \ \ \sum_{t \in \mathcal{K}}f_t \ + \ &\alpha \cdot \sum_{t \in \mathcal{K}} g_t \notag\\
52 * \text{s.t.} \quad \sum_{t \in \mathcal{K}} x_{it} &= 1 && \text{ for all } i \in \mathcal{S} \\
53 * \sum_{i \in \mathcal{S}} x_{it} &\ge 1 && \text{ for all } t \in \mathcal{K} \label{eq:setcover} \\
54 * g_t &= \sum_{\substack{i,j \in \mathcal{S}\\ i < j}} (q_{ij} + q_{ji}) x_{it} x_{jt} && \text{ for all } t \in \mathcal{K}\\
55 * f_t &= {\sum_{\substack{i,j \in \mathcal{S},\\ i \neq j}} (q_{ij}-q_{ji}) x_{it} x_{j \phi(t)}} && \text{ for all } t \in \mathcal{K} \\
56 * x_{it} &\in \{0,1\} && \text{ for all } t \in \mathcal{K}, i \in \mathcal{S} \notag \\
57 * f_t, g_t &\in \mathbb{R}_{\geq 0} && \text{ for all } t \in \mathcal{K}. \notag
58 * \f}
59 *
60 * Further information about particular modules like heuristics and separation routines
61 * can be found in the documentation of the corresponding files.
62 *
63 * Installation
64 * ------------
65 *
66 * See the @ref INSTALL_APPLICATIONS_EXAMPLES "Install file"
67 *
68 *
69 */