1,970 research outputs found
Optimizing condition numbers
In this paper we study the problem of minimizing condition numbers over a compact convex subset of the cone of symmetric positive semidefinite matrices. We show that the condition number is a Clarke regular strongly pseudoconvex function. We prove that a global solution of the problem can be approximated by an exact or an inexact solution of a nonsmooth convex program. This asymptotic analysis provides a valuable tool for designing an implementable algorithm for solving the problem of minimizing condition numbers
Bregman distances and Chebyshev sets
A closed set of a Euclidean space is said to be Chebyshev if every point in
the space has one and only one closest point in the set. Although the situation
is not settled in infinite-dimensional Hilbert spaces, in 1932 Bunt showed that
in Euclidean spaces a closed set is Chebyshev if and only if the set is convex.
In this paper, from the more general perspective of Bregman distances, we show
that if every point in the space has a unique nearest point in a closed set,
then the set is convex. We provide two approaches: one is by nonsmooth
analysis; the other by maximal monotone operator theory. Subdifferentiability
properties of Bregman nearest distance functions are also given
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