Drake
MobyLCPSolver< T > Class Template Reference

A class for solving Linear Complementarity Problems (LCPs). More...

#include <drake/solvers/moby_lcp_solver.h>

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## Public Member Functions

MobyLCPSolver ()=default

~MobyLCPSolver () override=default

void SetLoggingEnabled (bool enabled)

bool SolveLcpFast (const MatrixX< T > &M, const VectorX< T > &q, VectorX< T > *z, const T &zero_tol=T(-1)) const
Fast pivoting algorithm for LCPs of the form M = PAPᵀ, q = Pb, where b ∈ ℝᵐ, P ∈ ℝⁿˣᵐ, and A ∈ ℝᵐˣᵐ (where A is positive definite). More...

bool SolveLcpFastRegularized (const MatrixX< T > &M, const VectorX< T > &q, VectorX< T > *z, int min_exp=-20, unsigned step_exp=4, int max_exp=20, const T &zero_tol=T(-1)) const
Regularized version of the fast pivoting algorithm for LCPs of the form M = PAPᵀ, q = Pb, where b ∈ ℝᵐ, P ∈ ℝⁿˣᵐ, and A ∈ ℝᵐˣᵐ (where A is positive definite). More...

bool SolveLcpLemke (const MatrixX< T > &M, const VectorX< T > &q, VectorX< T > *z, const T &piv_tol=T(-1), const T &zero_tol=T(-1)) const
Lemke's Algorithm for solving LCPs in the matrix class E, which contains all strictly semimonotone matrices, all P-matrices, and all strictly copositive matrices. More...

bool SolveLcpLemkeRegularized (const MatrixX< T > &M, const VectorX< T > &q, VectorX< T > *z, int min_exp=-20, unsigned step_exp=1, int max_exp=1, const T &piv_tol=T(-1), const T &zero_tol=T(-1)) const
Lemke's Algorithm for solving LCPs in the matrix class E, which contains all strictly semimonotone matrices, all P-matrices, and all strictly copositive matrices. More...

bool SolveLcpLemke (const Eigen::SparseMatrix< double > &M, const Eigen::VectorXd &q, Eigen::VectorXd *z, double piv_tol=-1.0, double zero_tol=-1.0) const
Lemke's Algorithm for solving LCPs in the matrix class E, which contains all strictly semimonotone matrices, all P-matrices, and all strictly copositive matrices, for the special case of sparse matrices. More...

bool SolveLcpLemkeRegularized (const Eigen::SparseMatrix< double > &M, const Eigen::VectorXd &q, Eigen::VectorXd *z, int min_exp=-20, unsigned step_exp=4, int max_exp=20, double piv_tol=-1.0, double zero_tol=-1.0) const
Regularized wrapper around Lemke's Algorithm for solving LCPs in the matrix class E. More...

bool available () const override
Returns true iff this solver was enabled at compile-time. More...

SolutionResult Solve (MathematicalProgram &prog) const override
Sets values for the decision variables on the given MathematicalProgram prog, or: More...

SolverId solver_id () const override
Returns the identifier of this solver. More...

int get_num_pivots () const
Returns the number of pivoting operations made by the last LCP solve. More...

void reset_num_pivots ()
Resets the number of pivoting operations made by the last LCP solver to zero. More...

template<>
SolutionResult Solve (MathematicalProgram &) const
Sets values for the decision variables on the given MathematicalProgram prog, or: More...

Does not allow copy, move, or assignment
MobyLCPSolver (const MobyLCPSolver &)=delete

MobyLCPSolveroperator= (const MobyLCPSolver &)=delete

MobyLCPSolver (MobyLCPSolver &&)=delete

MobyLCPSolveroperator= (MobyLCPSolver &&)=delete

Public Member Functions inherited from MathematicalProgramSolverInterface
MathematicalProgramSolverInterface ()=default

virtual ~MathematicalProgramSolverInterface ()=default

MathematicalProgramSolverInterface (const MathematicalProgramSolverInterface &)=delete

MathematicalProgramSolverInterfaceoperator= (const MathematicalProgramSolverInterface &)=delete

MathematicalProgramSolverInterface (MathematicalProgramSolverInterface &&)=delete

MathematicalProgramSolverInterfaceoperator= (MathematicalProgramSolverInterface &&)=delete

## Static Public Member Functions

template<class U >
static U ComputeZeroTolerance (const MatrixX< U > &M)
Calculates the zero tolerance that the solver would compute if the user does not specify a tolerance. More...

## Detailed Description

### template<class T> class drake::solvers::MobyLCPSolver< T >

A class for solving Linear Complementarity Problems (LCPs).

Solving a LCP requires finding a solution to the problem:

Mz + q = w
z ≥ 0
w ≥ 0
zᵀw = 0


(where M ∈ ℝⁿˣⁿ and q ∈ ℝⁿ are problem inputs and z ∈ ℝⁿ and w ∈ ℝⁿ are unknown vectors) or correctly reporting that such a solution does not exist. In spite of their linear structure, solving LCPs is NP-Hard [Cottle 1992]. However, some LCPs are significantly easier to solve. For instance, it can be seen that the LCP is solvable in worst-case polynomial time for the case of symmetric positive-semi-definite M by formulating it as the following convex quadratic program:

minimize:   f(z) = zᵀw = zᵀ(Mz + q)
subject to: z ≥ 0
Mz + q ≥ 0


Note that this quadratic program's (QP) objective function at the minimum z cannot be less than zero, and the LCP is only solved if the objective function at the minimum is equal to zero. Since the seminal result of Karmarkar, it has been known that convex QPs are solvable in polynomial time [Karmarkar 1984].

The difficulty of solving an LCP is characterized by the properties of its particular matrix, namely the class of matrices it belongs to. Classes include, for example, Q, Q₀, P, P₀, copositive, and Z matrices. [Cottle 1992] and Murty 1998 describe relevant matrix classes in more detail.

• [Cottle 1992] R. Cottle, J.-S. Pang, and R. Stone. The Linear Complementarity Problem. Academic Press, 1992.
• [Karmarkar 1984] N. Karmarkar. A New Polynomial-Time Algorithm for Linear Programming. Combinatorica, 4(4), pp. 373-395.
• [Murty 1988] K. Murty. Linear Complementarity, Linear and Nonlinear Programming. Heldermann Verlag, 1988.

## Constructor & Destructor Documentation

 MobyLCPSolver ( const MobyLCPSolver< T > & )
delete
 MobyLCPSolver ( MobyLCPSolver< T > && )
delete
 MobyLCPSolver ( )
default
 ~MobyLCPSolver ( )
overridedefault

## Member Function Documentation

 bool available ( ) const
inlineoverridevirtual

Returns true iff this solver was enabled at compile-time.

Implements MathematicalProgramSolverInterface.

 static U ComputeZeroTolerance ( const MatrixX< U > & M )
inlinestatic

Calculates the zero tolerance that the solver would compute if the user does not specify a tolerance.

 int get_num_pivots ( ) const
inline

Returns the number of pivoting operations made by the last LCP solve.

 MobyLCPSolver& operator= ( MobyLCPSolver< T > && )
delete
 MobyLCPSolver& operator= ( const MobyLCPSolver< T > & )
delete
 void reset_num_pivots ( )
inline

Resets the number of pivoting operations made by the last LCP solver to zero.

 void SetLoggingEnabled ( bool enabled )
 SolutionResult Solve ( MathematicalProgram & prog ) const
virtual

Sets values for the decision variables on the given MathematicalProgram prog, or:

• If no solver is available, throws std::runtime_error
• If the solver returns an error, returns a nonzero SolutionResult.

Implements MathematicalProgramSolverInterface.

 SolutionResult Solve ( MathematicalProgram & prog ) const
overridevirtual

Sets values for the decision variables on the given MathematicalProgram prog, or:

• If no solver is available, throws std::runtime_error
• If the solver returns an error, returns a nonzero SolutionResult.

Implements MathematicalProgramSolverInterface.

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 bool SolveLcpFast ( const MatrixX< T > & M, const VectorX< T > & q, VectorX< T > * z, const T & zero_tol = T(-1) ) const

Fast pivoting algorithm for LCPs of the form M = PAPᵀ, q = Pb, where b ∈ ℝᵐ, P ∈ ℝⁿˣᵐ, and A ∈ ℝᵐˣᵐ (where A is positive definite).

Therefore, q is in the range of P and M is positive semi-definite. An LCP of this form is also guaranteed to have a solution [Cottle 1992].

This particular implementation focuses on the case where the solution requires few nonzero nonbasic variables, meaning that few z variables need be nonzero to find a solution to Mz + q = w. This algorithm, which is based off of Dantzig's Principle Pivoting Method I [Cottle 1992] is described in [Drumwright 2015]. This algorithm is able to use "warm" starting- a solution to a "nearby" LCP can be used to find the solution to a given LCP more quickly.

Although this solver is theoretically guaranteed to give a solution to the LCPs described above, accumulated floating point error from pivoting operations could cause the solver to fail. Additionally, the solver can be applied with some success to problems outside of its guaranteed matrix class. For these reasons, the solver returns a flag indicating success/failure.

Parameters
 [in] M the LCP matrix. [in] q the LCP vector. [in,out] z the solution to the LCP on return (if the solver succeeds). If the length of z is equal to the length of q, the solver will attempt to use z's value as a starting solution. If the solver fails (returns false), z will be set to the zero vector. [in] zero_tol The tolerance for testing against zero. If the tolerance is negative (default) the solver will determine a generally reasonable tolerance.
Exceptions
 std::logic_error if M is non-square or M's dimensions do not equal q's dimension.
Returns
true if the solver succeeded and false otherwise.
• [Cottle 1992] R. Cottle, J.-S. Pang, and R. Stone. The Linear Complementarity Problem. Academic Press, 1992.
• [Drumwright 2015] E. Drumwright. Rapidly computable viscous friction and no-slip rigid contact models. arXiv: 1504.00719v1. 2015.

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 bool SolveLcpFastRegularized ( const MatrixX< T > & M, const VectorX< T > & q, VectorX< T > * z, int min_exp = -20, unsigned step_exp = 4, int max_exp = 20, const T & zero_tol = T(-1) ) const

Regularized version of the fast pivoting algorithm for LCPs of the form M = PAPᵀ, q = Pb, where b ∈ ℝᵐ, P ∈ ℝⁿˣᵐ, and A ∈ ℝᵐˣᵐ (where A is positive definite).

Therefore, q is in the range of P and M is positive semi-definite. Please see SolveLcpFast() for more documentation about the particular algorithm.

This implementation wraps that algorithm with a Tikhonov-type regularization approach. Specifically, this implementation repeatedly attempts to solve the LCP:

(M + Iα)z + q = w
z ≥ 0
w ≥ 0
zᵀw = 0


where I is the identity matrix and α ≪ 1, using geometrically increasing values of α, until the LCP is solved. Cottle et al. describe how, for sufficiently large α, the LCP will always be solvable [Cottle 1992], p. 493.

Although this solver is theoretically guaranteed to give a solution to the LCPs described above, accumulated floating point error from pivoting operations could cause the solver to fail. Additionally, the solver can be applied with some success to problems outside of its guaranteed matrix class. For these reasons, the solver returns a flag indicating success/failure.

Parameters
 [in] M the LCP matrix. [in] q the LCP vector. [in,out] z the solution to the LCP on return (if the solver succeeds). If the length of z is equal to the length of q, the solver will attempt to use z's value as a starting solution. [in] min_exp The minimum exponent for computing α over [10ᵝ, 10ᵞ] in steps of 10ᵟ, where β is the minimum exponent, γ is the maximum exponent, and δ is the stepping exponent. [in] step_exp The stepping exponent for computing α over [10ᵝ, 10ᵞ] in steps of 10ᵟ, where β is the minimum exponent, γ is the maximum exponent, and δ is the stepping exponent. [in] max_exp The maximum exponent for computing α over [10ᵝ, 10ᵞ] in steps of 10ᵟ, where β is the minimum exponent, γ is the maximum exponent, and δ is the stepping exponent. [in] zero_tol The tolerance for testing against zero. If the tolerance is negative (default) the solver will determine a generally reasonable tolerance.
Exceptions
 std::logic_error if M is non-square or M's dimensions do not equal q's dimension.
Returns
true if the solver succeeded and false if the solver did not find a solution for α = 10ᵞ.
SolveLcpFast()
• [Cottle, 1992] R. Cottle, J.-S. Pang, and R. Stone. The Linear Complementarity Problem. Academic Press, 1992.

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 bool SolveLcpLemke ( const MatrixX< T > & M, const VectorX< T > & q, VectorX< T > * z, const T & piv_tol = T(-1), const T & zero_tol = T(-1) ) const

Lemke's Algorithm for solving LCPs in the matrix class E, which contains all strictly semimonotone matrices, all P-matrices, and all strictly copositive matrices.

Lemke's Algorithm is described in [Cottle 1992], Section 4.4. This implementation was adapted from the LEMKE Library [LEMKE] for Matlab; this particular implementation fixes a bug in LEMKE that could occur when multiple indices passed the minimum ratio test.

Although this solver is theoretically guaranteed to give a solution to the LCPs described above, accumulated floating point error from pivoting operations could cause the solver to fail. Additionally, the solver can be applied with some success to problems outside of its guaranteed matrix classes. For these reasons, the solver returns a flag indicating success/failure.

Parameters
 [in] M the LCP matrix. [in] q the LCP vector. [in,out] z the solution to the LCP on return (if the solver succeeds). If the length of z is equal to the length of q, the solver will attempt to use z's value as a starting solution. This warmstarting is generally not recommended: it has a predisposition to lead to a failing pivoting sequence. If the solver fails (returns false), z will be set to the zero vector. [in] zero_tol The tolerance for testing against zero. If the tolerance is negative (default) the solver will determine a generally reasonable tolerance. [in] piv_tol The tolerance for testing against zero, specifically used for the purpose of finding variables for pivoting. If the tolerance is negative (default) the solver will determine a generally reasonable tolerance.
Returns
true if the solver believes it has computed a solution (which it determines by the ability to "pivot out" the "artificial" variable (see [Cottle 1992]) and false otherwise.
Warning
The caller should verify that the algorithm has solved the LCP to the desired tolerances on returns indicating success.
Exceptions
 std::logic_error if M is not square or the dimensions of M do not match the length of q.

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 bool SolveLcpLemke ( const Eigen::SparseMatrix< double > & M, const Eigen::VectorXd & q, Eigen::VectorXd * z, double piv_tol = -1.0, double zero_tol = -1.0 ) const

Lemke's Algorithm for solving LCPs in the matrix class E, which contains all strictly semimonotone matrices, all P-matrices, and all strictly copositive matrices, for the special case of sparse matrices.

See the non-sparse version of SolveLcpLemke() for descriptions of the calling and return parameters.

Note
This function is not templatized because the pivoting operations make single-precision floating point solves untenable and because the underlying sparse linear system solver does not support AutoDiff.

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 bool SolveLcpLemkeRegularized ( const MatrixX< T > & M, const VectorX< T > & q, VectorX< T > * z, int min_exp = -20, unsigned step_exp = 1, int max_exp = 1, const T & piv_tol = T(-1), const T & zero_tol = T(-1) ) const

Lemke's Algorithm for solving LCPs in the matrix class E, which contains all strictly semimonotone matrices, all P-matrices, and all strictly copositive matrices.

Lemke's Algorithm is described in [Cottle 1992], Section 4.4.

This implementation wraps that algorithm with a Tikhonov-type regularization approach. Specifically, this implementation repeatedly attempts to solve the LCP:

(M + Iα)z + q = w
z ≥ 0
w ≥ 0
zᵀw = 0


where I is the identity matrix and α ≪ 1, using geometrically increasing values of α, until the LCP is solved. See SolveLcpFastRegularized() for description of the regularization process and the function parameters, which are identical. See SolveLcpLemke() for a description of Lemke's Algorithm. See SolveLcpFastRegularized() for a description of all calling parameters other than z, which apply equally well to this function.

Parameters
 [in,out] z the solution to the LCP on return (if the solver succeeds). If the length of z is equal to the length of q, the solver will attempt to use z's value as a starting solution. This warmstarting is generally not recommended: it has a predisposition to lead to a failing pivoting sequence.
SolveLcpFastRegularized()
SolveLcpLemke()
• [Cottle 1992] R. Cottle, J.-S. Pang, and R. Stone. The Linear Complementarity Problem. Academic Press, 1992.

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 bool SolveLcpLemkeRegularized ( const Eigen::SparseMatrix< double > & M, const Eigen::VectorXd & q, Eigen::VectorXd * z, int min_exp = -20, unsigned step_exp = 4, int max_exp = 20, double piv_tol = -1.0, double zero_tol = -1.0 ) const

Regularized wrapper around Lemke's Algorithm for solving LCPs in the matrix class E.

See the non-sparse version of SolveLcpLemkeRegularized() for descriptions of the calling and return parameters.

Note
This function is not templatized because the pivoting operations make single-precision floating point solves untenable and because the underlying sparse linear system solver does not support AutoDiff.

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 SolverId solver_id ( ) const
overridevirtual

Returns the identifier of this solver.

Implements MathematicalProgramSolverInterface.

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The documentation for this class was generated from the following files: