Drake

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...
 
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...
 
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...
 
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) . 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
 

Additional Inherited Members

- Protected Member Functions inherited from Constraint
void UpdateLowerBound (const Eigen::Ref< const Eigen::VectorXd > &new_lb)
 Updates the lower bound. More...
 
void UpdateUpperBound (const Eigen::Ref< const Eigen::VectorXd > &new_ub)
 Updates the upper bound. More...
 
void set_bounds (const Eigen::Ref< const Eigen::VectorXd > &lower_bound, const Eigen::Ref< const Eigen::VectorXd > &upper_bound)
 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)
 

Member Enumeration Documentation

◆ 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.

Constructor & Destructor Documentation

◆ LorentzConeConstraint() [1/3]

◆ LorentzConeConstraint() [2/3]

◆ 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

Member Function Documentation

◆ A()

const Eigen::SparseMatrix<double>& 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

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