Are linear algorithms optimal in guaranteed-estimation problems? (Q1913073)

The article is devoted to the investigation of the optimality of guaranteed-estimation algorithms that are linear (affine) in the measured signal. The question whether linear algorithms are optimal has been discussed in many papers on approximation theory, the theory of guaranteed and stochastic estimation, and optimal algorithm theory. This is associated, in particular, with the fact that effective numerical procedures for finding linear estimates exist. The author examines the question whether linear (affine) estimates are optimal, for an abstract setting of the problem of guaranteed estimation of the value of a linear continuous functional in normed linear spaces without any assumptions about the symmetry of the problem constraints about zero. Optimality in the class of estimates provided by affine continuous functionals is proven under certain assumptions. The second part of the article discusses the optimality of linear algorithms in problems of guaranteed-estimation of a vector functional for various types of metrics in a finite-dimensional estimates space.