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.. index::
single: Relay or box-constrained AVI problems
.. contents::
.. _lcp_problem:
Relay or box-constrained AVI problems
*************************************
Problem statement
=================
Find :math:`(z,w)` such that:
.. math::
\begin{equation*}
\left\lbrace \begin{array}{l}
w = M z + q\\
-w \in \mathcal{N}_{K}(z)\\
\end{array}, \right.
\end{equation*}
where M is an ( :math:`n \times n` )-matrix, q, z and w are n-dimensional vectors, K is the box defined by :math:`K=\{x\in\mathbb{R}^n\mid lb_i\leq x_i\leq ub_i,i=1,...,n\}` and :math:`\mathcal{N}_K(z)` is the normal cone to :math:`K` at :math:`z` .
Implementation in numerics
==========================
Structure to define the problem: :class:`RelayProblem`.
The generic driver for all Relay problems is :func:`relay_driver()`.
Solvers list :enum:`RELAY_SOLVER`
.. _relay_solvers:
Relay available solvers
=======================
Direct solvers
--------------
Enumerative solver (:enumerator:`SICONOS_RELAY_ENUM`)
"""""""""""""""""""""""""""""""""""""""""""""""""""""
The relay problem is reformulated as a LCP and solved with enum method, see :ref:`lcp_solvers`.
driver: :func:`relay_enum()`
parameters: same as :enumerator:`SICONOS_LCP_ENUM`.
Lemke solver (:enumerator:`SICONOS_RELAY_LEMKE`)
""""""""""""""""""""""""""""""""""""""""""""""""
The relay problem is reformulated as a LCP and solved with Lemke's method, see :ref:`lcp_solvers`.
driver: :func:`relay_lexicolemke()`
parameters: same as :enumerator:`SICONOS_LCP_ENUM`.
PATH (:enumerator:`SICONOS_RELAY_PATH`)
"""""""""""""""""""""""""""""""""""""""
The relay problem is reformulated as a LCP and solved with the PATH solver
*Works only if Siconos has been built with path support (if PathFerris or PathVI has been found, see :ref:`siconos_install_guide`)**
driver: :func:`relay_path()`
parameters : none.
AVI reformulation
-----------------
AVI, Cao/Ferris solver (:enumerator:`SICONOS_RELAY_AVI_CAOFERRIS`)
""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""
Based on an algorithm by Cao and Ferris for AVI with a polytopic set :math:`K` .
driver: :func:`relay_avi_caoferris()`
parameters: same as :enumerator:`SICONOS_AVI_CAOFERRIS`, see :ref:`avi_solvers`.
There also exists a test version :enumerator:`SICONOS_RELAY_AVI_CAOFERRIS_TEST` with
driver: :func:`relay_avi_caoferris_test()`
parameters: same as :enumerator:`SICONOS_AVI_CAOFERRIS`, see :ref:`avi_solvers`.
Iterative solvers
-----------------
Projected Gauss-Seidel (:enumerator:`SICONOS_RELAY_PGS`)
""""""""""""""""""""""""""""""""""""""""""""""""""""""""
driver: :func:`relay_pgs()`
parameters:
* iparam[SICONOS_IPARAM_MAX_ITER] = 1000
* dparam[SICONOS_DPARAM_TOL] = 1e-6
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