InequalityPenaltyFunction

Overview¶

InequalityPenaltyFunction provides the scalar ramp-like mapping used to turn inequality violations into smooth optimization penalties.

Authors: Frank Dellaert, et al. (see THANKS for the full author list)

See LICENSE for the license information

try:
    import google.colab
    %pip install --quiet gtsam-develop
except ImportError:
    pass

Key Concepts¶

RampFunction implements the exact ramp max(x,0) behavior.
SmoothRampPoly2 and SmoothRampPoly3 provide polynomial smooth approximations near the boundary.
SoftPlusFunction provides a smooth log-exp approximation.
These functions influence solver behavior near active inequality constraints.

Mathematical Formulation¶

These functions shape how inequality violations are penalized by replacing $r=\max(g,0)$ with either the exact ramp or a smooth approximation:

RampFunction: exact, non-smooth at 0.
SmoothRampPoly2, SmoothRampPoly3: polynomial smoothings near the boundary.
SoftPlusFunction: smooth log-exp form.

They are used inside inequality penalty factors to improve optimization behavior near active-set transitions.

Key User API¶

double operator()(double x, OptionalJacobian<1,1> H={})
function() to retrieve a callable functor
RampFunction::Ramp(x, H)
SmoothRampPoly2(epsilon), SmoothRampPoly3(epsilon)
SoftPlusFunction(k)

Concise C++ Example¶

#include <gtsam/constrained/InequalityPenaltyFunction.h>

using namespace gtsam;

SmoothRampPoly2 phi(2.0);
Matrix H;
double r = phi(1.0, H);  // smooth ramp value and derivative

SoftPlusFunction softPlus(0.5);
double rs = softPlus(2.0);