..
   This file is part of ManiSolve https://github.com/baharev/ManiSolve, 
   and is Copyright (C) 2015-2018 the University of Vienna.
   It is licensed under the three-clause BSD license; see LICENSE.
   Contact: ali.baharev@gmail.com
   
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*********
Citations
*********

Software Citation
=================

If you use the ManiSolve software, please cite our paper:

`A manifold-based approach to sparse global constraint satisfaction problems <https://doi.org/10.1007/s10898-019-00805-x>`_

.. code-block:: tex

    @Article{Baharev2019,
        author="Baharev, Ali
        and Neumaier, Arnold
        and Schichl, Hermann",
        title="A manifold-based approach to sparse global constraint satisfaction problems",
        journal="Journal of Global Optimization",
        year="2019",
        month="Dec",
        day="01",
        volume="75",
        number="4",
        pages="949--971",
        abstract="We consider square, sparse nonlinear systems of 
        equations whose Jacobian is structurally nonsingular, with reasonable 
        bound constraints on all variables. We propose an algorithm for 
        finding good approximations to all well-separated solutions of such 
        systems. We assume that the input system is ordered such that its 
        Jacobian is in bordered block lower triangular form with small 
        diagonal blocks and with small border width; this can be performed 
        fully automatically with off-the-shelf decomposition methods. Five 
        decades of numerical experience show that models of technical systems 
        tend to decompose favorably in practice. Once the block decomposition 
        is available, we reduce the task of solving the large nonlinear system 
        of equations to that of solving a sequence of low-dimensional ones. 
        The most serious weakness of this approach is well-known: It may 
        suffer from severe numerical instability. The proposed method resolves 
        this issue with the novel backsolve step. We study the effect of the 
        decomposition on a sequence of challenging problems. Beyond a certain 
        problem size, the computational effort of multistart (no 
        decomposition) grows exponentially. In contrast, thanks to the 
        decomposition, for the proposed method the computational effort grows 
        only linearly with the problem size. It depends on the problem size 
        and on the hyperparameter settings whether the decomposition and the 
        more sophisticated algorithm pay off. Although there is no theoretical 
        guarantee that all solutions will be found in the general case, 
        increasing the so-called sample size hyperparameter improves the 
        robustness of the proposed method.",
        issn="1573-2916",
        doi="10.1007/s10898-019-00805-x",
        url="https://doi.org/10.1007/s10898-019-00805-x"
    }