Mixed (Non Linear) Complementarity problem (MCP)

Problem statement

Given a sufficiently smooth function \({F} \colon {{\mathrm{I\!R}}}^{n+m} \to {{\mathrm{I\!R}}}^{n+m}\) , the Mixed Complementarity problem (MCP) is to find two vectors \((z,w \in {{\mathrm{I\!R}}}^{n+m})\) such that:

\[\begin{split}\begin{align*} w &= \begin{pmatrix}w_e\\w_i\end{pmatrix} = F(z) \\ w_e &=0 \\ 0 &\le w_i \perp z_i \ge 0, \end{align*}\end{split}\]

where “i” (resp. “e”) stands for inequalities (resp. equalities). The vector \(z\) is splitted like \(w\) :

\[\begin{split}\begin{equation*}z =\begin{pmatrix}z_e\\z_i\end{pmatrix}.\end{equation*}\end{split}\]

The vectors \(z_i,w_i\) are of size sizeEqualities , the vectors \(z_e,w_e\) are of size sizeInequalities and \(F\) is a non linear function that must be user-defined.

A Mixed Complementarity problem (MCP) is a Nonlinear Complementarity Problems (NCP) “augmented” with equality constraints.

Implementation in numerics

Structure to define the problem: MixedLinearComplementarityProblem.

The generic driver for all MCP is mcp_driver().

Solvers list MCP_SOLVER

MCP available solvers

Newton, Fisher-Burmeister (SICONOS_MCP_NEWTON_FB_FBLSA)

Solver based on Fischer-Burmeister reformulation and line search (VFBLSA in Facchinei–Pang 2003 p. 834)

driver: mcp_newton_FB_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-10

  • dparam[SICONOS_DPARAM_LSA_ALPHA_MIN] = 1e-16;

Newton (min), Fisher-Burmeister (SICONOS_MCP_NEWTON_MIN_FB_FBLSA)

Solver based on Fischer-Burmeister reformulation and line search. The descent direction is found using a min reformulation (minFBLSA in Facchinei–Pang 2003 p. 855)

driver: mcp_newton_min_FB_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-10

  • dparam[SICONOS_DPARAM_LSA_ALPHA_MIN] = 1e-16;

Newton, Fisher-Burmeister (SICONOS_MCP_OLD_FB)

Solver based on Fischer-Burmeister reformulation, old (outdated) version, for the records.

driver: mcp_old_FischerBurmeister()

parameters:

  • iparam[SICONOS_IPARAM_MAX_ITER] = 10

  • iparam[SICONOS_IPARAM_STOPPING_CRITERION] = SICONOS_STOPPING_CRITERION_USER_ROUTINE;

  • dparam[SICONOS_DPARAM_TOL] = 1e-7