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.. index::
single: Nonlinear Complementarity Problems (NCP)
.. contents::
.. _ncp_problem:
Nonlinear Complementarity Problems (NCP)
****************************************
Problem statement
=================
Given a sufficiently smooth function :math:`{F}\colon {{\mathrm{I\!R}}}^{n} \to {{\mathrm{I\!R}}}^{n}` The Nonlinear Complementarity Problem (NCP) is to find two vectors :math:`(z,w \in {{\mathrm{I\!R}}}^{n})` such that:
.. math::
\begin{align*} w &= F(z) \\ 0 &\le w \perp z \ge 0 \end{align*}
Implementation in numerics
==========================
Structure to define the problem: :class:`NonlinearComplementarityProblem`.
The generic driver for all NCP is :func:`ncp_driver()`.
Solvers list :enum:`NCP_SOLVER`
.. _ncp_solvers:
NCP available solvers
=====================
Newton, Fisher-Burmeister (:enumerator:`SICONOS_NCP_NEWTON_FB_FBLSA`)
---------------------------------------------------------------------
NCP Solver using the FB merit function and a Newton-based method with line-search
driver: :func:`ncp_newton_FBLSA`
parameters:
* iparam[SICONOS_IPARAM_MAX_ITER] = 1000
* iparam[SICONOS_IPARAM_LSA_NONMONOTONE_LS] = 0;
* iparam[SICONOS_IPARAM_LSA_NONMONOTONE_LS_M] = 0;
* iparam[SICONOS_IPARAM_STOPPING_CRITERION] = SICONOS_STOPPING_CRITERION_USER_ROUTINE;
* dparam[SICONOS_DPARAM_TOL] = 1e-12
* dparam[SICONOS_DPARAM_LSA_ALPHA_MIN] = 1e-16;
Newton, min merit function (:enumerator:`SICONOS_NCP_NEWTON_MIN_FBLSA`)
-----------------------------------------------------------------------
NCP Solver using the min merit function (+ the FB as backup) and a Newton-based method with line-search
driver: :func:`ncp_newton_minFBLSA`
parameters: same as :enumerator:`SICONOS_NCP_NEWTON_FB_FBLSA`
Path search algorithm (:enumerator:`SICONOS_NCP_PATHSEARCH`)
------------------------------------------------------------
NCP Solver using a path search algorithm, following the work of D. Ralph.
M. Ferris, and many other collaborators of the latter.
driver: :func:`ncp_newton_pathsearch`
parameters:
* iparam[SICONOS_IPARAM_MAX_ITER] = 100
* iparam[SICONOS_IPARAM_LSA_NONMONOTONE_LS] = NM_LS_MEAN;
* iparam[SICONOS_IPARAM_LSA_NONMONOTONE_LS_M] = 10;
* iparam[SICONOS_IPARAM_PATHSEARCH_STACKSIZE] = 5;
* iparam[SICONOS_IPARAM_NMS_WATCHDOG_TYPE] = LINESEARCH;
* iparam[SICONOS_IPARAM_NMS_PROJECTED_GRADIENT_TYPE] = ARCSEARCH;
* iparam[SICONOS_IPARAM_NMS_N_MAX] = 10;
* iparam[SICONOS_IPARAM_PREALLOC] = 0;
* dparam[SICONOS_DPARAM_TOL] = 1e-12
* dparam[SICONOS_DPARAM_NMS_DELTA] = 20;
* dparam[SICONOS_DPARAM_NMS_DELTA_VAR] = .8;
* dparam[SICONOS_DPARAM_NMS_SIGMA] = .01;
* dparam[SICONOS_DPARAM_NMS_ALPHA_MIN_WATCHDOG] = 1e-12;
* dparam[SICONOS_DPARAM_NMS_ALPHA_MIN_PGRAD] = 1e-12;
* dparam[SICONOS_DPARAM_NMS_MERIT_INCR] = 1.1
PATH (Ferris) solver (:enumerator:`SICONOS_NCP_PATH`)
-----------------------------------------------------
driver: :func:`ncp_path`
parameters:
* dparam[SICONOS_DPARAM_TOL] = 1e-12
* iparam[SICONOS_IPARAM_MAX_ITER] = 10000
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