# File numerics/src/LCP/LCP_Solvers.h¶

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Subroutines for the resolution of Linear Complementarity Problems.

See detailed documentation in LCP available solvers

Functions

void lcp_qp(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_qp uses a quadratic programm formulation for solving a LCP

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence / minimization sucessfull 1 : Too Many iterations 2 : Accuracy insuficient to satisfy convergence criterion 5 : Length of working array insufficient Other : The constraints are inconstent

• options[inout] structure used to define the solver and its parameters.

void lcp_cpg(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_cpg is a CPG (Conjugated Projected Gradient) solver for LCP based on quadratic minimization.

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0: convergence 1: iter = itermax 2: negative diagonal term 3: pWp nul

• options[inout] structure used to define the solver and its parameters.

void lcp_pgs(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_pgs (Projected Gauss-Seidel) is a basic Projected Gauss-Seidel solver for LCP.

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence 1 : iter = itermax 2 : negative diagonal term

• options[inout] structure used to define the solver and its parameters.

void lcp_rpgs(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_rpgs (Regularized Projected Gauss-Seidel ) is a solver for LCP, able to handle matrices with null diagonal terms.

Todo:

Sizing the regularization paramter and apply it only on null diagnal term

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence 1 : iter = itermax 2 : negative diagonal term

• options[inout] structure used to define the solver and its parameters.

void lcp_psor(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_psor Projected Succesive over relaxation solver for LCP.

See cottle, Pang Stone Chap 5

Todo:

use the relax parameter

add a vector of relaxation parameter wtith an auto sizing (see SOR algorithm for linear solver.)

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence 1 : iter = itermax 2 : negative diagonal term

• options[inout] structure used to define the solver and its parameters.

void lcp_nsqp(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_nsqp use a quadratic programm formulation for solving an non symmetric LCP

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence / minimization sucessfull 1 : Too Many iterations 2 : Accuracy insuficient to satisfy convergence criterion 5 : Length of working array insufficient Other : The constraints are inconstent

• options[inout] structure used to define the solver and its parameters.

void lcp_latin(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_latin (LArge Time INcrements) is a basic latin solver for LCP.

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence 1 : iter = itermax 2 : Cholesky Factorization failed 3 : nul diagonal term

• options[inout] structure used to define the solver and its parameters.

void lcp_latin_w(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_latin_w (LArge Time INcrements) is a basic latin solver with relaxation for LCP.

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence 1 : iter = itermax 2 : Cholesky Factorization failed 3 : nul diagonal term

• options[inout] structure used to define the solver and its parameters.

void lcp_lexicolemke(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_lexicolemke is a direct solver for LCP based on pivoting method principle for degenerate problem Choice of pivot variable is performed via lexicographic ordering Ref: “The Linear Complementarity Problem” Cottle, Pang, Stone (1992)

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence 1 : iter = itermax 2 : negative diagonal term

• options[inout] structure used to define the solver and its parameters.

void lcp_newton_min(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_newton_min uses a nonsmooth Newton method based on the min formulation (or max formulation) of the LCP

Todo:

Optimizing the memory allocation (Try to avoid the copy of JacH into A)

Add rules for the computation of the penalization rho

Add a globalization strategy based on a decrease of a merit function. (Nonmonotone LCP) Reference in Ferris Kanzow 2002

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence / minimization sucessfull 1 : Too Many iterations 2 : Accuracy insuficient to satisfy convergence criterion 5 : Length of working array insufficient Other : The constraints are inconstent

• options[inout] structure used to define the solver and its parameters.

void lcp_newton_FB(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_newton_FB use a nonsmooth newton method based on the Fischer-Bursmeister convex function

Todo:

Optimizing the memory allocation (Try to avoid the copy of JacH into A)

Add rules for the computation of the penalization rho

Add a globalization strategy based on a decrease of a merit function. (Nonmonotone LCP) Reference in Ferris Kanzow 2002

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 - convergence 1 - iter = itermax 2 - failure in the descent direction search (in LAPACK)

• options[inout] structure used to define the solver and its parameters.

void lcp_newton_minFB(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_newton_minFB use a nonsmooth newton method based on both a min and Fischer-Bursmeister reformulation References: Facchinei&#8212;Pang (2003)

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 - convergence 1 - iter = itermax 2 - failure in the descent direction search (in LAPACK)

• options[inout] structure used to define the solver and its parameters.

void lcp_path(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

path solver

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence 1 : iter = itermax 2 : negative diagonal term

• options[inout] structure used to define the solver and its parameters.

void lcp_enum(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

enumerative solver

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : success 1 : failed

• options[inout] structure used to define the solver and its parameters.

void lcp_enum_init(LinearComplementarityProblem *problem, SolverOptions *options, int withMemAlloc)

Proceed with initialisation required before any call of the enum solver.

Parameters
• problem[in] structure that represents the LCP (M, q…)

• options[inout] structure used to define the solver and its parameters.

• withMemAlloc[in] If it is not 0, then the necessary work memory is allocated.

void lcp_enum_reset(LinearComplementarityProblem *problem, SolverOptions *options, int withMemAlloc)

Reset state for enum solver parameters.

Parameters
• problem[in] structure that represents the LCP (M, q…)

• options[inout] structure used to define the solver and its parameters.

• withMemAlloc[in] If it is not 0, then the work memory is free.

void lcp_avi_caoferris(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_avi_caoferris is a direct solver for LCP based on an Affine Variational Inequalities (AVI) reformulation The AVI solver is here the one from Cao and Ferris Ref: “A Pivotal Method for Affine Variational Inequalities” Menglin Cao et Michael Ferris (1996)

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence 1 : iter = itermax

• options[inout] structure used to define the solver and its parameters.

void lcp_pivot(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_pivot is a direct solver for LCP based on a pivoting method It can currently use Bard, Murty’s least-index or Lemke rule for choosing the pivot.

The default one is Lemke and it can be changed by setting iparam[SICONOS_LCP_IPARAM_PIVOTING_METHOD_TYPE]. The list of choices are in the enum LCP_PIVOT (see lcp_cst.h).

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence 1 : iter = itermax

• options[inout] structure used to define the solver and its parameters.

void lcp_pivot_covering_vector(LinearComplementarityProblem *problem, double *u, double *s, int *info, SolverOptions *options, double *cov_vec)
void lcp_pivot_lumod(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)
void lcp_pivot_lumod_covering_vector(LinearComplementarityProblem *problem, double *u, double *s, int *info, SolverOptions *options, double *cov_vec)
void lcp_pathsearch(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_pathsearch is a direct solver for LCP based on the pathsearch algorithm

Warning

this solver is available for testing purposes only! consider using lcp_pivot() if you are looking for simular solvers

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence 1 : iter = itermax

• options[inout] structure used to define the solver and its parameters.

void lcp_gams(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

lcp_gams uses the solver provided by GAMS

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info[out] an integer which returns the termination value: 0 : convergence 1 : iter = itermax

• options[inout] structure used to define the solver and its parameters.

void lcp_nsgs_SBM(LinearComplementarityProblem *problem, double *z, double *w, int *info, SolverOptions *options)

generic interface used to call any LCP solver applied on a Sparse-Block structured matrix M, with a Gauss-Seidel process to solve the global problem (formulation/solving of local problems for each row of blocks)

Parameters
• problem[in] structure that represents the LCP (M, q…). M must be a SparseBlockStructuredMatrix

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• info – an integer which returns the termination value: 0 : convergence >0 : failed, depends on local solver

• options[inout] structure used to define the solver and its parameters.

void lcp_nsgs_SBM_buildLocalProblem(int rowNumber, SparseBlockStructuredMatrix *const blmat, LinearComplementarityProblem *local_problem, double *q, double *z)

Construct local problem from a “global” one.

Parameters
• rowNumber – index of the local problem

• blmat – matrix containing the problem

• local_problem – problem to fill

• q – big q

• z – big z

int lcp_compute_error(LinearComplementarityProblem *problem, double *z, double *w, double tolerance, double *error)

Computes error criterion and update $$w = Mz + q$$.

Parameters
• problem[in] structure that represents the LCP (M, q…)

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• tolerance[in] threshold used to validate the solution: if the error is less than this value, the solution is accepted

• error[out] the actual error of the solution with respect to the problem

Returns

status: 0 : convergence, 1: error > tolerance

void lcp_compute_error_only(unsigned int n, double *z, double *w, double *error)

Computes error criterion.

Parameters
• n[in] size of the LCP

• z[inout] a n-vector of doubles which contains the initial solution and returns the solution of the problem.

• w[inout] a n-vector of doubles which returns the solution of the problem.

• error[out] the result of the computation

int lcp_driver_DenseMatrix(LinearComplementarityProblem *problem, double *z, double *w, SolverOptions *options)

Interface to solvers for Linear Complementarity Problems, dedicated to dense matrix storage.

Parameters
• problem[in] the LinearComplementarityProblem structure which handles the problem (M,q)

• z[inout] a n-vector of doubles which contains the solution of the problem.

• w[inout] a n-vector of doubles which contains the solution of the problem.

• options[inout] structure used to define the solver(s) and their parameters

Returns

info termination value

• 0 : successful