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.. index::
single: Second Order Cone Linear Complementarity Problem (SOCLCP)
.. contents::
.. _soclcp_problem:
Second Order Cone Linear Complementarity Problem (SOCLCP)
*********************************************************
Problem statement
=================
Given
* a symmetric positive semi definite matrix :math:`{M} \in {{\mathrm{I\!R}}}^{n \times n}`
* a vector :math:`{q} \in {{\mathrm{I\!R}}}^n`
* a vector of coefficients :math:`\mu \in{{\mathrm{I\!R}}}^{n_c}`
the second order cone linear complementarity problem (SOCLCP) is to find two vectors :math:`u\in{{\mathrm{I\!R}}}^n` , and :math:`r\in {{\mathrm{I\!R}}}^n` , denoted by :math:`\mathrm{SOCCLP}(M,q,\mu)` such that
.. math::
\begin{eqnarray*} \begin{cases}
u = M r + q \\
\ C^\star_{\mu} \ni {u} \perp r \in C_{\mu}
\end{cases} \end{eqnarray*}
and the set :math:`C^{\alpha,\star}_{\mu^\alpha}` is its dual.
The set C is the second order cone given by
.. math::
\begin{eqnarray}
C_{\mu} = \{ r \} = \prod_{\alpha =1}^{n_c} C^\alpha_{\mu}
\end{eqnarray}
with
.. math::
\begin{eqnarray}
C^\alpha_{\mu} = \{ r \mid \|[r_1, \ldots, r_{n^\alpha}]\| \leq \mu^\alpha * r_0 \} \subset {\mathrm{I\!R}}^{n^\alpha}
\end{eqnarray}
Implementation in numerics
==========================
Structure to define the problem: :class:`SecondOrderConeLinearComplementarityProblem`.
The generic driver for all SOCLCP problems is :func:`soclcp_driver()`.
Solvers list :enum:`SOCLCP_SOLVER`
.. _soclcp_solvers:
SOCLCP available solvers
========================
Gauss-Seidel (:enumerator:`SICONOS_SOCLCP_NSGS`)
""""""""""""""""""""""""""""""""""""""""""""""""
PSOR (Gauss-Seidel with overrelaxation) solver.
driver: :func:`soclcp_nsgs()`
parameters:
* iparam[SICONOS_IPARAM_MAX_ITER] = 1000;
* iparam[SICONOS_IPARAM_ERROR_EVALUATION] : error computation method,
* SICONOS_ERROR_FULL_EVALUATION Complete error computation with v computation (Default)
* SICONOS_ERROR_LIGHT_EVALUATION for Light error computation with incremental values on r verification of absolute error at the end
* SICONOS_ERROR_LIGHT_EVALUATION_NO_UPDATE for light error computation, without update for v
* iparam[SICONOS_IPARAM_SOCLCP_NSGS_WITH_RELAXATION] = 0;
* iparam[SICONOS_IPARAM_SOCLCP_NSGS_WITH_RELAXATION] : method uses overrelaxation
* iparam[SICONOS_IPARAM_NSGS_SHUFFLE] : if 1, shuffle the contact indices in the loop
* dparam[SICONOS_DPARAM_TOL] = 1e-4;
* dparam[SICONOS_DPARAM_SOCLCP_NSGS_RELAXATION] = 1., relaxation parameter value
internal solver: :enumerator:`SICONOS_SOCLCP_ProjectionOnConeWithLocalIteration`.
VI, fixed-point (:enumerator:`SICONOS_SOCLCP_VI_FPP`)
"""""""""""""""""""""""""""""""""""""""""""""""""""""
VI formulation and fixed point projection.
driver: :func:`soclcp_VI_FixedPointProjection()`
parameters: same as :enumerator:`SICONO_VI_FPP`, see :ref:`vi_solvers`.
VI, Extra-gradient (:enumerator:`SICONOS_SOCLCP_VI_EG`)
"""""""""""""""""""""""""""""""""""""""""""""""""""""""
VI formulation and extra-gradient solver.
driver: :func:`soclcp_VI_ExtraGradient()`
parameters: same as :enumerator:`SICONO_VI_EG`, see :ref:`vi_solvers`.
VI, Extra-gradient (:enumerator:`SICONOS_SOCLCP_VI_EG`)
"""""""""""""""""""""""""""""""""""""""""""""""""""""""
VI formulation and extra-gradient solver.
driver: :func:`soclcp_VI_ExtraGradient()`
parameters: same as :enumerator:`SICONO_VI_EG`, see :ref:`vi_solvers`.
Projections
"""""""""""
Used as internal solver for :enumerator:`SICONOS_SOCLCP_NSGS`.
ids: :enumerator:`SICONOS_SOCLCP_ProjectionOnConeWithLocalIteration`,
:enumerator:`SICONOS_SOCLCP_ProjectionOnCone`,
:enumerator:`SICONOS_SOCLCP_ProjectionOnConeWithRegularization`.
drivers:
* :func:`soclcp_projectionOnCone_solve` for ProjectionOnCone and ProjectionOnConeWithRegularization,
* :func:`soclcp_projectionOnConeWithLocalIteration` for ProjectionOnConeWithLocalIteration.
parameters:
* iparam[SICONOS_IPARAM_SOCLCP_PROJECTION_CONE_INDEX] (set by soclcp_nsgs)
* dparam[SICONOS_DPARAM_SOCLCP_PROJECTION_RHO] = 0.
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