A Kalman-Tracking Filter Approach to Nonlinear Programming
Computers & Mathematics with Applications
ISSN or ISBN
The problem considered herein is that of finding the minimum of a nonlinear function f(θ) when the gradient and Hessian matrix are unknown or cannot be easily computed. The function f(θ) may also be observed in the presence of stochastic noise. Typical nonlinear programming algorithms solve this problem deterministically by employing function approximations which do not account for approximation errors. In the presence of noise the algorithm may either terminate prematurely or may not converge at all. In this paper we show that this type of nonlinear programming problem can be associated with a tracking problem commonly arising in aerospace applications. In this new domain, finding the minimum of a function is equivalent to tracking a point θ∗ (the minimum) in the parameter space, based on noisy measurements of its position [function evaluations f(θ)]. A new nonlinear programming algorithm based on a two-level Kalman filter is presented which accounts for both modeling and approximation errors. The lower-level Kalman filter performs function approximations, while the upper-level filter accounts for the tracking of the parameter. Simulations show that the algorithm performs similarly to Newton's method.
Ramos, Jose A., "A Kalman-Tracking Filter Approach to Nonlinear Programming" (1990). CEC Faculty Articles. 422.