Dual extremal problems and their applications to minimax estimation problems

A characterization of essentially strictly convex functions on reflexive Banach spaces

“…In the sequel we will need Theorem A below that completes an earlier result by Soloviov [So,Theorem 1.2.1] (see also [Za,Theorem 9.3.2]). Its proof easily follows from [BBC,Theorems 5.4 and 5.6] and the above considerations.…”

Minimax estimation methods under ellipsoidal constraints

“…Note that this proof significantly relies upon the uniqueness (up to a constant factor) of the eigenvector corresponding to the maximal eigenvalue Φ(t) . Sufficiency of this condition has been noted for a similar problem in [25]. Unfortunately, in the general case equality (4.5) does not hold.…”

“…The dual optimization method [22][23][24][25]28] is applicable in a wide range of minimax optimization problems. The essence of this method is to find the best estimate corresponding to the least favorable characteristics of the observation model.…”

“…Let us show that the pair (û,t) forms a saddle point for function g(u, t) = Φ(t)u, u , u ∈ S, t ∈ T 0 , where S is the unit sphere in R p (we identify symmetric points of S) and T 0 = {t ∈ T : im[C] ⊆ im[H(t)]}. To do so, we only need to use Corollary 1.3.4 from [25] whose conditions in this case are as follows:…”

“…Duality theory and convex analysis methods have become the foundation for minimax estimation and control algorithms [2,[19][20][21]. In this work, we use the dual optimization method [22][23][24][25][26][27] to solve a multidimensional optimization minimax estimation problem. This approach has proven effective in case when the dimensionality of the vector of unknown parameters or uncertain characteristics is significantly less than the number of observations.…”