An Inequality Constrained Nonlinear Kalman-Bucy Smoother
by Interior Point Likelihood Maximization
Kalman-Bucy smoothers are often used to estimate the
state variables as a function of time
in a system with stochastic dynamics and measurement noise.
This is accomplished using an algorithm for which the
number of numerical operations grows linearly with the number of time points.
All of the randomness in the model is assumed to be Gaussian.
Including other available information, for example a bound
on one of the state variables, is non trivial because
it does not fit into the standard Kalman-Bucy smoother algorithm.
In this paper we present an interior point method
that maximizes the likelihood
with respect to the sequence of state vectors satisfying
The method obtains the same decomposition that is normally
obtained for the unconstrained Kalman-Bucy smoother, hence
the resulting number of operations grows linearly with the number
of time points.
We present two algorithms,
the first is for the affine case and
the second is for the nonlinear case.
Neither algorithm requires the optimization to start
at a feasible sequence of state vector values.
Both the unconstrained affine and unconstrained nonlinear
Kalman-Bucy smoother are special cases of the class of problems
that can be handled by these algorithms.