Barrier¶

ipctk.barrier(d: float, dhat: float) → float¶

Function that grows to infinity as d approaches 0 from the right.

\[b(d) = -(d-\hat{d})^2\ln\left(\frac{d}{\hat{d}}\right)\]

Parameters:¶

d: float¶

distance

dhat: float¶

activation distance of the barrier

Returns:¶

The value of the barrier function at d.

ipctk.barrier_gradient(d: float, dhat: float) → float¶

Derivative of the barrier function.

\[b'(d) = (\hat{d}-d) \left( 2\ln\left( \frac{d}{\hat{d}} \right) - \frac{\hat{d}}{d} + 1\right)\]

Parameters:¶

d: float¶

distance

dhat: float¶

activation distance of the barrier

Returns:¶

The derivative of the barrier wrt d.

ipctk.barrier_hessian(d: float, dhat: float) → float¶

Second derivative of the barrier function.

\[b''(d) = \left( \frac{\hat{d}}{d} + 2 \right) \frac{\hat{d}}{d} - 2\ln\left( \frac{d}{\hat{d}} \right) - 3\]

Parameters:¶

d: float¶

distance

dhat: float¶

activation distance of the barrier

Returns:¶

The second derivative of the barrier wrt d.