On the Theoretical Justification of Adaptive Optimization

Results in support of a class of steady-state adaptive optimization algorithms are presented. It is first proved that a local dynamic model with suitable functional form and "correct" parameter values has the same optimum steady state as the true process, provided that the true process optimum occurs at a point where the gradient of the performance measure vanishes. The functional form needed is dictated by the structure of the performance measure. Subsequently, it is shown that if the suitable local model parameters are identified on line using an estimator with forgetting factor, the model parameter estimates cannot converge to values far from the above mentioned "correct" values. Thus an adaptive steady-state optimizer based on a suitable dynamic model cannot converge to a steady state far from the true optimum.