Optimization under conditions of indefiniteness

Problems reducible to minimization of the mean loss function (MLF) discussed by Middleton (1962) and Tsypkin (1968) are analyzed as optimization problems under conditions of risk or conditions of indefiniteness, and two classes of optimization problems are distinguished: modeling problems and evaluation problems. The concept of mean indefiniteness measure (MIM) is introduced for dealing with a priori information bearing on the choice of loss function, probabilistic iterative algorithms and adaptive algorithms of stochastic approximation type are compared, and the MIM is shown to satisfy convexity, differentiability, and additivity constraints. The minimum MLF is shown to be smallest in the case of a matched loss function, and the least favorable distribution confers a maximum value on the MLF or Shannon information measure. Applications include control systems and radar.