Drake
Drake C++ Documentation
AugmentedLagrangianNonsmooth Class Reference

## Detailed Description

Compute the augmented Lagrangian (AL) of a given mathematical program.

min f(x)
s.t h(x) = 0
l <= g(x) <= u
x_lo <= x <= x_up


We first turn it into an equality constrained program with non-negative slack variable s as follows

min f(x)
s.t h(x) = 0
c(x) - s = 0
s >= 0


Depending on the option include_x_bounds, the constraint h(x)=0, c(x)>=0 may or may not include the bounding box constraint x_lo <= x <= x_up.

the (non-smooth) augmented Lagrangian is defined as

L(x, λ, μ) = f(x) − λ₁ᵀh(x) + μ/2 h(x)ᵀh(x)
- λ₂ᵀ(c(x)-s) + μ/2 (c(x)-s)ᵀ(c(x)-s)


where s = max(c(x) - λ₂/μ, 0).

For more details, refer to section 17.4 of Numerical Optimization by Jorge Nocedal and Stephen Wright, Edition 1, 1999 (This formulation isn't presented in Edition 2, but to stay consistent with Edition 2, we use μ/2 as the coefficient of the quadratic penalty term instead of 1/(2μ) in Edition 1). Note that the augmented Lagrangian L(x, λ, μ) is NOT a smooth function of x, since s = max(c(x) - λ₂/μ, 0) is non-smooth at c(x) - λ₂/μ = 0.

#include <drake/solvers/augmented_lagrangian.h>

## Public Member Functions

AugmentedLagrangianNonsmooth (const MathematicalProgram *prog, bool include_x_bounds)

template<typename T >
Eval (const Eigen::Ref< const VectorX< T >> &x, const Eigen::VectorXd &lambda_val, double mu, VectorX< T > *constraint_residue, T *cost) const

const MathematicalProgramprog () const

bool include_x_bounds () const

int lagrangian_size () const

const std::vector< bool > & is_equality () const

const Eigen::VectorXd & x_lo () const

const Eigen::VectorXd & x_up () const

Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable
AugmentedLagrangianNonsmooth (const AugmentedLagrangianNonsmooth &)=default

AugmentedLagrangianNonsmoothoperator= (const AugmentedLagrangianNonsmooth &)=default

AugmentedLagrangianNonsmooth (AugmentedLagrangianNonsmooth &&)=default

AugmentedLagrangianNonsmoothoperator= (AugmentedLagrangianNonsmooth &&)=default

## ◆ AugmentedLagrangianNonsmooth() [1/3]

 AugmentedLagrangianNonsmooth ( const AugmentedLagrangianNonsmooth & )
default

## ◆ AugmentedLagrangianNonsmooth() [2/3]

 AugmentedLagrangianNonsmooth ( AugmentedLagrangianNonsmooth && )
default

## ◆ AugmentedLagrangianNonsmooth() [3/3]

 AugmentedLagrangianNonsmooth ( const MathematicalProgram * prog, bool include_x_bounds )
Parameters
 prog The mathematical program we will evaluate. include_x_bounds. Whether the Lagrangian and the penalty for the bounds x_lo <= x <= x_up are included in the augmented Lagrangian L(x, λ, μ) or not.

## ◆ Eval()

 T Eval ( const Eigen::Ref< const VectorX< T >> & x, const Eigen::VectorXd & lambda_val, double mu, VectorX< T > * constraint_residue, T * cost ) const
Parameters
 x The value of all the decision variables. lambda_val The estimated Lagrangian multipliers. The order of the Lagrangian multiplier is as this: We first call to evaluate all constraints. Then for each row of the constraint, if it is an equality constraint, we append one single Lagrangian multiplier. Otherwise we append the Lagrangian multiplier for the lower and upper bounds (where the lower comes before the upper), if the corresponding bound is not ±∞. The order of evaluating all the constraints is the same as prog.GetAllConstraints() except for prog.bounding_box_constraints(). If include_x_bounds=true, then we aggregate all the bounding_box_constraints() and evaluate them at the end of all constraints. mu μ in the documentation above. The constant for penalty term weight. This should be a strictly positive number. [out] constraint_residue The value of the all the constraints. For an equality constraint c(x)=0 or the inequality constraint c(x)>= 0, the residue is c(x). Depending on include_x_bounds, constraint_residue may or may not contain the residue for bounding box constraints x_lo <= x <= x_up at the end. [out] cost The value of the cost function f(x).
Returns
The evaluated Augmented Lagrangian (AL) L(x, λ, μ).

## ◆ include_x_bounds()

 bool include_x_bounds ( ) const
Returns
Whether the bounding box constraint x_lo <= x <= x_up is included in the augmented Lagrangian L(x, λ, μ).

## ◆ is_equality()

 const std::vector& is_equality ( ) const
Returns
Whether each constraint is equality or not. The order of the constraint is explained in the class documentation.

## ◆ lagrangian_size()

 int lagrangian_size ( ) const
Returns
The size of the Lagrangian multiplier λ.

## ◆ operator=() [1/2]

 AugmentedLagrangianNonsmooth& operator= ( AugmentedLagrangianNonsmooth && )
default

## ◆ operator=() [2/2]

 AugmentedLagrangianNonsmooth& operator= ( const AugmentedLagrangianNonsmooth & )
default

## ◆ prog()

 const MathematicalProgram& prog ( ) const
Returns
The mathematical program for which the augmented Lagrangian is computed.

## ◆ x_lo()

 const Eigen::VectorXd& x_lo ( ) const
Returns
all the lower bounds of x.

## ◆ x_up()

 const Eigen::VectorXd& x_up ( ) const
Returns
all the upper bounds of x.

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