PositiveSemidefiniteConstraint Class Reference

Detailed Description

Implements a positive semidefinite constraint on a symmetric matrix S.

\[\text{ S is p.s.d }\]

namely, all eigen values of S are non-negative.

Note

if the matix S has 1 row, then it is better to impose a linear inequality constraints; if it has 2 rows, then it is better to impose a rotated Lorentz cone constraint, since a 2 x 2 matrix S being p.s.d is equivalent to the constraint [S(0, 0), S(1, 1), S(0, 1)] in the rotated Lorentz cone.

#include <drake/solvers/constraint.h>

Public Member Functions
	PositiveSemidefiniteConstraint (int rows)
	Impose the constraint that a symmetric matrix with size rows x rows is positive semidefinite.
	~PositiveSemidefiniteConstraint () override
int	matrix_rows () const
void	WarnOnSmallMatrixSize () const
	Throw a warning if the matrix size is either 1x1 or 2x2.
Does not allow copy, move, or assignment
	PositiveSemidefiniteConstraint (const PositiveSemidefiniteConstraint &)=delete
PositiveSemidefiniteConstraint &	operator= (const PositiveSemidefiniteConstraint &)=delete
	PositiveSemidefiniteConstraint (PositiveSemidefiniteConstraint &&)=delete
PositiveSemidefiniteConstraint &	operator= (PositiveSemidefiniteConstraint &&)=delete
Public Member Functions inherited from Constraint
template<typename DerivedLB, typename DerivedUB>
	Constraint (int num_constraints, int num_vars, const Eigen::MatrixBase< DerivedLB > &lb, const Eigen::MatrixBase< DerivedUB > &ub, const std::string &description="")
	Constructs a constraint which has num_constraints rows, with an input num_vars x 1 vector.
	Constraint (int num_constraints, int num_vars)
	Constructs a constraint which has num_constraints rows, with an input num_vars x 1 vector, with no bounds.
bool	CheckSatisfied (const Eigen::Ref< const Eigen::VectorXd > &x, double tol=1E-6) const
	Return whether this constraint is satisfied by the given value, x.
bool	CheckSatisfied (const Eigen::Ref< const AutoDiffVecXd > &x, double tol=1E-6) const
symbolic::Formula	CheckSatisfied (const Eigen::Ref< const VectorX< symbolic::Variable > > &x) const
const Eigen::VectorXd &	lower_bound () const
const Eigen::VectorXd &	upper_bound () const
int	num_constraints () const
	Number of rows in the output constraint.
	Constraint (const Constraint &)=delete
Constraint &	operator= (const Constraint &)=delete
	Constraint (Constraint &&)=delete
Constraint &	operator= (Constraint &&)=delete
Public Member Functions inherited from EvaluatorBase
virtual	~EvaluatorBase ()
void	Eval (const Eigen::Ref< const Eigen::VectorXd > &x, Eigen::VectorXd *y) const
	Evaluates the expression.
void	Eval (const Eigen::Ref< const AutoDiffVecXd > &x, AutoDiffVecXd *y) const
	Evaluates the expression.
void	Eval (const Eigen::Ref< const VectorX< symbolic::Variable > > &x, VectorX< symbolic::Expression > *y) const
	Evaluates the expression.
void	set_description (const std::string &description)
	Set a human-friendly description for the evaluator.
const std::string &	get_description () const
	Getter for a human-friendly description for the evaluator.
std::ostream &	Display (std::ostream &os, const VectorX< symbolic::Variable > &vars) const
	Formats this evaluator into the given stream using vars for the bound decision variable names.
std::ostream &	Display (std::ostream &os) const
	Formats this evaluator into the given stream, without displaying the decision variables it is bound to.
std::string	ToLatex (const VectorX< symbolic::Variable > &vars, int precision=3) const
	Returns a LaTeX string describing this evaluator.
int	num_vars () const
	Getter for the number of variables, namely the number of rows in x, as used in Eval(x, y).
int	num_outputs () const
	Getter for the number of outputs, namely the number of rows in y, as used in Eval(x, y).
void	SetGradientSparsityPattern (const std::vector< std::pair< int, int > > &gradient_sparsity_pattern)
	Set the sparsity pattern of the gradient matrix ∂y/∂x (the gradient of y value in Eval, w.r.t x in Eval) .
const std::optional< std::vector< std::pair< int, int > > > &	gradient_sparsity_pattern () const
	Returns the vector of (row_index, col_index) that contains all the entries in the gradient of Eval function (∂y/∂x) whose value could be non-zero, namely if ∂yᵢ/∂xⱼ could be non-zero, then the pair (i, j) is in gradient_sparsity_pattern.
bool	is_thread_safe () const
	Returns whether it is safe to call Eval in parallel.
	EvaluatorBase (const EvaluatorBase &)=delete
EvaluatorBase &	operator= (const EvaluatorBase &)=delete
	EvaluatorBase (EvaluatorBase &&)=delete
EvaluatorBase &	operator= (EvaluatorBase &&)=delete

Protected Member Functions
void	DoEval (const Eigen::Ref< const Eigen::VectorXd > &x, Eigen::VectorXd *y) const override
	Evaluate the eigen values of the symmetric matrix.
void	DoEval (const Eigen::Ref< const AutoDiffVecXd > &x, AutoDiffVecXd *y) const override
void	DoEval (const Eigen::Ref< const VectorX< symbolic::Variable > > &x, VectorX< symbolic::Expression > *y) const override
std::string	DoToLatex (const VectorX< symbolic::Variable > &, int) const override
Protected Member Functions inherited from Constraint
void	UpdateLowerBound (const Eigen::Ref< const Eigen::VectorXd > &new_lb)
	Updates the lower bound.
void	UpdateUpperBound (const Eigen::Ref< const Eigen::VectorXd > &new_ub)
	Updates the upper bound.
void	set_bounds (const Eigen::Ref< const Eigen::VectorXd > &new_lb, const Eigen::Ref< const Eigen::VectorXd > &new_ub)
	Set the upper and lower bounds of the constraint.
virtual bool	DoCheckSatisfied (const Eigen::Ref< const Eigen::VectorXd > &x, const double tol) const
virtual bool	DoCheckSatisfied (const Eigen::Ref< const AutoDiffVecXd > &x, const double tol) const
virtual symbolic::Formula	DoCheckSatisfied (const Eigen::Ref< const VectorX< symbolic::Variable > > &x) const
Protected Member Functions inherited from EvaluatorBase
	EvaluatorBase (int num_outputs, int num_vars, const std::string &description="")
	Constructs a evaluator.
virtual std::ostream &	DoDisplay (std::ostream &os, const VectorX< symbolic::Variable > &vars) const
	NVI implementation of Display.
void	set_num_outputs (int num_outputs)
void	set_is_thread_safe (bool is_thread_safe)

Constructor & Destructor Documentation

PositiveSemidefiniteConstraint() [1/3]

PositiveSemidefiniteConstraint ( const PositiveSemidefiniteConstraint & )

delete

PositiveSemidefiniteConstraint() [2/3]

PositiveSemidefiniteConstraint ( PositiveSemidefiniteConstraint && )

delete

PositiveSemidefiniteConstraint() [3/3]

PositiveSemidefiniteConstraint ( int rows )

explicit

Impose the constraint that a symmetric matrix with size rows x rows is positive semidefinite.

See also

MathematicalProgram::AddPositiveSemidefiniteConstraint() for how to use this constraint on some decision variables. We currently use this constraint as a place holder in MathematicalProgram, to indicate the positive semidefiniteness of some decision variables.

Parameters

rows	The number of rows (and columns) of the symmetric matrix.

Note

rows should be a positive integer. If rows==1 or rows==2, then consider imposing a linear inequality or rotated Lorentz cone constraint respectively.

Example:

// Create a MathematicalProgram object.
auto prog = MathematicalProgram();
 
// Add a 3 x 3 symmetric matrix S to optimization program as new decision
// variables.
auto S = prog.NewSymmetricContinuousVariables<3>("S");
 
// Impose a positive semidefinite constraint on S.
std::shared_ptr<PositiveSemidefiniteConstraint> psd_constraint =
    prog.AddPositiveSemidefiniteConstraint(S);
 
/////////////////////////////////////////////////////////////
// Add more constraints to make the program more interesting,
// but this is not needed.
 
// Add the constraint that S(1, 0) = 1.
prog.AddBoundingBoxConstraint(1, 1, S(1, 0));
 
// Minimize S(0, 0) + S(1, 1) + S(2, 2).
prog.AddLinearCost(Eigen::RowVector3d(1, 1, 1), {S.diagonal()});
 
/////////////////////////////////////////////////////////////
 
// Now solve the program.
auto result = Solve(prog);
 
// Retrieve the solution of matrix S.
auto S_value = GetSolution(S, result);
 
// Compute the eigen values of the solution, to see if they are
// all non-negative.
Vector6d S_stacked;
S_stacked << S_value.col(0), S_value.col(1), S_value.col(2);
 
Eigen::VectorXd S_eigen_values;
psd_constraint->Eval(S_stacked, S_eigen_values);
 
std::cout<<"S solution is: " << S << std::endl;
std::cout<<"The eigen value of S is " << S_eigen_values << std::endl;

~PositiveSemidefiniteConstraint()

~PositiveSemidefiniteConstraint ( )

override

Member Function Documentation

DoEval() [1/3]

void DoEval ( const Eigen::Ref< const AutoDiffVecXd > & x, AutoDiffVecXd * y ) const

overrideprotectedvirtual

Parameters

x	The stacked columns of the symmetric matrix. This function is not supported yet, since Eigen's eigen value solver does not accept AutoDiffXd.

Implements EvaluatorBase.

DoEval() [2/3]

void DoEval ( const Eigen::Ref< const Eigen::VectorXd > & x, Eigen::VectorXd * y ) const

overrideprotectedvirtual

Evaluate the eigen values of the symmetric matrix.

Parameters

x	The stacked columns of the symmetric matrix.

Implements EvaluatorBase.

DoEval() [3/3]

void DoEval ( const Eigen::Ref< const VectorX< symbolic::Variable > > & x, VectorX< symbolic::Expression > * y ) const

overrideprotectedvirtual

Parameters

x	The stacked columns of the symmetric matrix. This function is not supported, since Eigen's eigen value solver does not accept symbolic::Expression.

Implements EvaluatorBase.

DoToLatex()

std::string DoToLatex ( const VectorX< symbolic::Variable > & , int  ) const

overrideprotectedvirtual

Reimplemented from EvaluatorBase.

matrix_rows()

int matrix_rows

(

)

const

operator=() [1/2]

PositiveSemidefiniteConstraint & operator= ( const PositiveSemidefiniteConstraint & )

delete

operator=() [2/2]

PositiveSemidefiniteConstraint & operator= ( PositiveSemidefiniteConstraint && )

delete

WarnOnSmallMatrixSize()

void WarnOnSmallMatrixSize

(

)

const

Throw a warning if the matrix size is either 1x1 or 2x2.

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

• drake/solvers/constraint.h