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Sequentially localizable functionals. Booth, Raymond Sydney

Abstract

A standard way of finding the unique zero on (0, 1) of a continuous decreasing functions with f(0) f(1)< 0, is to test the sign of f(1/2), then the sign of f(1/4) (if f(1/2)< 0) or the sign of f(3/4) (if f(1/2) >0), etc. In this way, the zero of f is localized in n steps to an interval of length 2⁻ⁿ. The unique maximum of a unimodal function on [0,1] can be similarly localized, but the unique maximum of a unimodal function on the unit square cannot. We start by generalizing these problems: let A be a compact subset of Eⁿ, let F be a set of real-valued functions on A, and for each f in F let S(f) be a point in the Cartesian product Aᵏ; S(f) is called a functional on F. Examples of such functionals are zeros, extrema, inflexion points, saddle points, etc., as well as sets of these. A test function T is a function of m real variables, which takes up only a finite number (≥2) of distinct values. An abscissa set Xᵢ is an ordered m-tuple (xlᵢ,...xmᵢ) ,, with each Xji in A. A sequential strategy is a way of selecting abscissa sets X₁, X₂,...,where the knowledge of T(f (xlᵢ, . ., f (xmᵢ)) is used to determine Xi+i. The N-th set of indeterminacy for S(f) is the largest subset of in which S(f) can lies consistent with the results of the first N-1 tests. A functional S(f) is sequentially localizable if a test function T and a sequential strategy exist, such that for every f in F the sets of indeterminacy shrink to a point (which must then be S(f) itself). First, several conditions are given to ensure the sequential localizability of a functional, these are presented in terms of certain topologies induced on Aᵏ and in terms of contraction maps. It is then shown that if a functional is localizable, there exists an optimal strategy under which the sets of indeterminacy converge fastest; further, the speed of localization is always exponential. Next, the concept of a random strategy and of random localizability is introduced, and it is shown that in many cases random localizability and sequential localizability are equivalent. Also, the speed of the former is not too much worse than the speed of the latter. Finally, optimal and near-optimal strategies are worked out for some functionals of interest.

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