Adaptive Optimisation and D-optimum Experimental Design

We consider the situation where one has to maximise a function (; x) with respect to x 2 IR q , when is unknown and estimated by least squares through observations y k = f > (x k) + k , with k some random error. Classical applications are regulation and extremum control problems. The approach we adopt corresponds to maximizing the sum of the current estimated objective and a penalisation for poor estimation: x k+1 maximises (^ k ; x) + (k =k) d k (x), with ^ k the estimated value of at step k and d k the penalisation function. Suucient conditions for strong consistency of ^ k and for almost sure convergence of (1=k) P k i=1 (; x i) to the maximum value of (; x) are derived in the case where d k () is the variance function used in the sequential construction of D-optimum designs. A classical sequential scheme from adaptive control is shown not to satisfy these conditions, and numerical simulations connrm that it indeed has convergence problems.