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
- [kConvex] g(z) = z₀ - sqrt(z₁² + ... + zₙ₋₁²) ≥ 0 This formulation is not differentiable at z₁=...=zₙ₋₁=0
- [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.
- [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)
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| | 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.
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| const Eigen::MatrixXd & | A_dense () const |
| | Getter for dense version of A.
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| const Eigen::VectorXd & | b () const |
| | Getter for b.
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| EvalType | eval_type () const |
| | Getter for eval type.
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| 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.
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| | LorentzConeConstraint (const LorentzConeConstraint &)=delete |
| LorentzConeConstraint & | operator= (const LorentzConeConstraint &)=delete |
| | LorentzConeConstraint (LorentzConeConstraint &&)=delete |
| LorentzConeConstraint & | operator= (LorentzConeConstraint &&)=delete |
| 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.
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| | 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.
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| 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.
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| 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.
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| | Constraint (const Constraint &)=delete |
| Constraint & | operator= (const Constraint &)=delete |
| | Constraint (Constraint &&)=delete |
| Constraint & | operator= (Constraint &&)=delete |
| virtual | ~EvaluatorBase () |
| void | Eval (const Eigen::Ref< const Eigen::VectorXd > &x, Eigen::VectorXd *y) const |
| | Evaluates the expression.
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| void | Eval (const Eigen::Ref< const AutoDiffVecXd > &x, AutoDiffVecXd *y) const |
| | Evaluates the expression.
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| void | Eval (const Eigen::Ref< const VectorX< symbolic::Variable > > &x, VectorX< symbolic::Expression > *y) const |
| | Evaluates the expression.
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| void | set_description (const std::string &description) |
| | Set a human-friendly description for the evaluator.
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| const std::string & | get_description () const |
| | Getter for a human-friendly description for the evaluator.
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| 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.
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| std::ostream & | Display (std::ostream &os) const |
| | Formats this evaluator into the given stream, without displaying the decision variables it is bound to.
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| std::string | ToLatex (const VectorX< symbolic::Variable > &vars, int precision=3) const |
| | Returns a LaTeX string describing this evaluator.
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| int | num_vars () const |
| | Getter for the number of variables, namely the number of rows in x, as used in Eval(x, y).
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| int | num_outputs () const |
| | Getter for the number of outputs, namely the number of rows in y, as used in Eval(x, y).
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| 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) .
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| 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.
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| bool | is_thread_safe () const |
| | Returns whether it is safe to call Eval in parallel.
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| | EvaluatorBase (const EvaluatorBase &)=delete |
| EvaluatorBase & | operator= (const EvaluatorBase &)=delete |
| | EvaluatorBase (EvaluatorBase &&)=delete |
| EvaluatorBase & | operator= (EvaluatorBase &&)=delete |
