Drake
Drake C++ Documentation
LorentzConeConstraint Class Reference

## Detailed Description

Constraining the linear expression $$z=Ax+b$$ lies within the Lorentz cone.

A vector z ∈ ℝ ⁿ lies within Lorentz cone if

$z_0 \ge \sqrt{z_1^2+...+z_{n-1}^2}$

where A ∈ ℝ ⁿˣᵐ, b ∈ ℝ ⁿ are given matrices. Ideally this constraint should be handled by a second-order cone solver. In case the user wants to enforce this constraint through general nonlinear optimization, we provide three different formulations on the Lorentz cone constraint

1. [kConvex] g(z) = z₀ - sqrt(z₁² + ... + zₙ₋₁²) ≥ 0 This formulation is not differentiable at z₁=...=zₙ₋₁=0
2. [kConvexSmooth] g(z) = z₀ - sqrt(z₁² + ... + zₙ₋₁²) ≥ 0 but the gradient of g(z) is approximated as ∂g(z)/∂z = [1, -z₁/sqrt(z₁² + ... zₙ₋₁² + ε), ..., -zₙ₋₁/sqrt(z₁²+...+zₙ₋₁²+ε)] where ε is a small positive number.
3. [kNonconvex] z₀²-(z₁²+...+zₙ₋₁²) ≥ 0 z₀ ≥ 0 This constraint is differentiable everywhere, but z₀²-(z₁²+...+zₙ₋₁²) ≥ 0 is non-convex. For more information and visualization, please refer to https://www.epfl.ch/labs/disopt/wp-content/uploads/2018/09/7.pdf and https://docs.mosek.com/modeling-cookbook/cqo.html (Fig 3.1)

#include <drake/solvers/constraint.h>

## Public Types

enum  EvalType { kConvex, kConvexSmooth, kNonconvex }
We provide three possible Eval functions to represent the Lorentz cone constraint z₀ ≥ sqrt(z₁² + ... More...

## Public Member Functions

LorentzConeConstraint (const Eigen::Ref< const Eigen::MatrixXd > &A, const Eigen::Ref< const Eigen::VectorXd > &b, EvalType eval_type=EvalType::kConvexSmooth)

~LorentzConeConstraint () override

const Eigen::SparseMatrix< double > & A () const
Getter for A. More...

const Eigen::MatrixXd & A_dense () const
Getter for dense version of A. More...

const Eigen::VectorXd & b () const
Getter for b. More...

EvalType eval_type () const
Getter for eval type. More...

void UpdateCoefficients (const Eigen::Ref< const Eigen::MatrixXd > &new_A, const Eigen::Ref< const Eigen::VectorXd > &new_b)
Updates the coefficients, the updated constraint is z=new_A * x + new_b in the Lorentz cone. More...

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

LorentzConeConstraintoperator= (const LorentzConeConstraint &)=delete

LorentzConeConstraint (LorentzConeConstraint &&)=delete

LorentzConeConstraintoperator= (LorentzConeConstraint &&)=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. More...

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. More...

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. More...

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. More...

Constraint (const Constraint &)=delete

Constraintoperator= (const Constraint &)=delete

Constraint (Constraint &&)=delete

Constraintoperator= (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. More...

void Eval (const Eigen::Ref< const AutoDiffVecXd > &x, AutoDiffVecXd *y) const
Evaluates the expression. More...

void Eval (const Eigen::Ref< const VectorX< symbolic::Variable >> &x, VectorX< symbolic::Expression > *y) const
Evaluates the expression. More...

void set_description (const std::string &description)
Set a human-friendly description for the evaluator. More...

const std::string & get_description () const
Getter for a human-friendly description for the evaluator. More...

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. More...

std::ostream & Display (std::ostream &os) const
Formats this evaluator into the given stream, without displaying the decision variables it is bound to. More...

std::string ToLatex (const VectorX< symbolic::Variable > &vars, int precision=3) const
Returns a LaTeX string describing this evaluator. More...

int num_vars () const
Getter for the number of variables, namely the number of rows in x, as used in Eval(x, y). More...

int num_outputs () const
Getter for the number of outputs, namely the number of rows in y, as used in Eval(x, y). More...

Set the sparsity pattern of the gradient matrix ∂y/∂x (the gradient of y value in Eval, w.r.t x in Eval) . More...

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. More...

EvaluatorBase (const EvaluatorBase &)=delete

EvaluatorBaseoperator= (const EvaluatorBase &)=delete

EvaluatorBase (EvaluatorBase &&)=delete

EvaluatorBaseoperator= (EvaluatorBase &&)=delete

Protected Member Functions inherited from Constraint
void UpdateLowerBound (const Eigen::Ref< const Eigen::VectorXd > &new_lb)

void UpdateUpperBound (const Eigen::Ref< const Eigen::VectorXd > &new_ub)

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. More...

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. More...

void set_num_outputs (int num_outputs)

## ◆ EvalType

 enum EvalType
strong

We provide three possible Eval functions to represent the Lorentz cone constraint z₀ ≥ sqrt(z₁² + ...

• zₙ₋₁²). For more explanation on the three formulations, refer to LorentzConeConstraint documentation.
Enumerator
kConvex

The constraint is g(z) = z₀ - sqrt(z₁² + ...

• zₙ₋₁²) ≥ 0. Note this formulation is non-differentiable at z₁= ...= zₙ₋₁=0
kConvexSmooth

Same as kConvex, but with approximated gradient that exists everywhere.

kNonconvex

Nonconvex constraint z₀²-(z₁²+...+zₙ₋₁²) ≥ 0 and z₀ ≥ 0.

Note this formulation is differentiable, but at z₁= ...= zₙ₋₁=0 the gradient is also 0, so a gradient-based nonlinear solver can get stuck.

## ◆ LorentzConeConstraint() [1/3]

 LorentzConeConstraint ( const LorentzConeConstraint & )
delete

## ◆ LorentzConeConstraint() [2/3]

 LorentzConeConstraint ( LorentzConeConstraint && )
delete

## ◆ LorentzConeConstraint() [3/3]

 LorentzConeConstraint ( const Eigen::Ref< const Eigen::MatrixXd > & A, const Eigen::Ref< const Eigen::VectorXd > & b, EvalType eval_type = EvalType::kConvexSmooth )

## ◆ ~LorentzConeConstraint()

 ~LorentzConeConstraint ( )
override

## ◆ A()

 const Eigen::SparseMatrix& A ( ) const

Getter for A.

## ◆ A_dense()

 const Eigen::MatrixXd& A_dense ( ) const

Getter for dense version of A.

## ◆ b()

 const Eigen::VectorXd& b ( ) const

Getter for b.

## ◆ eval_type()

 EvalType eval_type ( ) const

Getter for eval type.

## ◆ operator=() [1/2]

 LorentzConeConstraint& operator= ( LorentzConeConstraint && )
delete

## ◆ operator=() [2/2]

 LorentzConeConstraint& operator= ( const LorentzConeConstraint & )
delete

## ◆ UpdateCoefficients()

 void UpdateCoefficients ( const Eigen::Ref< const Eigen::MatrixXd > & new_A, const Eigen::Ref< const Eigen::VectorXd > & new_b )

Updates the coefficients, the updated constraint is z=new_A * x + new_b in the Lorentz cone.

Exceptions
 std::exception if the new_A.cols() != A.cols(), namely the variable size should not change.
Precondition
new_A has to have at least 2 rows and new_A.rows() == new_b.rows().

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