\(\|\Pi_\Delta(p + \alpha g) - p\|_2 / \alpha\) for maximisation,
with \(\Pi_\Delta\) the Euclidean projection onto the probability
simplex (sort-and-threshold algorithm). It is zero exactly at a
Karush-Kuhn-Tucker point of the log-likelihood on the closed simplex,
interior or on a face, which makes it a solver-independent check of an
estimate.
Usage
kkt_residual(p, grad, alpha = 1)
Arguments
- p
Composition on the simplex.
- grad
Score (gradient of the log-likelihood) at p, same length.
- alpha
Positive step; a non-finite or non-positive value is
replaced by 1.
Value
A non-negative number, or NA when the lengths differ.
Examples
# Interior stationary point: the score is parallel to the ones vector.
kkt_residual(c(0.3, 0.7), c(2, 2))
#> [1] 2.775558e-16
# Vertex with an outward score: still a KKT point.
kkt_residual(c(1, 0), c(1, -5))
#> [1] 0
kkt_residual(c(0.5, 0.5), c(1, -1))
#> [1] 0.7071068
