The Chebyshev or L∞ estimator minimizes the maximum absolute residual and is useful in situations where the error distribution has bounded support. In this paper, we derive the asymptotic distribution of this estimator in cases where the error distribution has bounded and unbounded support. We also consider the asymptotics of set-membership estimators such as the Chebyshev centre and maximum in...

Let X be a non-reflexive real Banach space. Then for each norm | · | from a dense set of equivalent norms on X (in the metric of uniform convergence on the unit ball of X), there exists a three-point set that has no Chebyshev center in (X, | · |). This result strengthens theorems by Davis and Johnson, van Dulst and Singer, and Konyagin.

We characterize inner product spaces in terms of the relationship between the relative Chebyshev center of a set and the absolute Cheby-shev center of an associated set. A consequence of the characterization is that an existing nite algorithm can be adapted to calculate the center of a nite set relative to a nite dimensional aane space. We also discuss the case where the constraint set is nonli...

Keywords: Set-valued mapping Chebyshev center Uniformly convex Locally uniformly convex Chebyshev fixed point a b s t r a c t The existence of a continuous Chebyshev selection for a Hausdorff continuous set-valued mapping is studied in a Banach space with some uniform convexity. As applications, some existence results of Chebyshev fixed point for condensing set-valued mappings are given, and th...

In this paper the problem of constructing the minimum `1 uncertainty model containing a finite set of assigned models is addressed. This problem is relevant to `1 robust control design techniques, in order to minimize the size of the uncertainty set associated to the nominal plant. The problem is formulated as a (conditional) Chebyshev center problem and an algorithm for its solution is propose...

Let G be a strict RS-set (resp. an RS-set) in X and let F be a bounded (resp. totally bounded) subset of X satisfying rG(F )> rX(F ), where rG(F ) is the restricted Chebyshev radius of F with respect to G. It is shown that the restricted Chebyshev center of F with respect to G is strongly unique in the case when X is a real Banach space, and that, under some additional convexity assumptions, th...

The classical column generation approach often shows a very slow convergence. Many different acceleration techniques have beenproposed recently to improve the convergence. Here, we briefly survey these methods and propose a novel algorithm based on the Chebyshev center of the dual polyhedron. The Chebyshev center can be obtained by solving a linear program; consequently, the proposed method can...

In Euclidean spaces, the geometric notions of nearest-points map, farthestpoints map, Chebyshev set, Klee set, and Chebyshev center are well known and well understood. Since early works going back to the 1930s, tremendous theoretical progress has been made, mostly by extending classical results from Euclidean space to Banach space settings. In all these results, the distance between points is i...

4 Some specific cutting-plane methods 12 4.1 Bisection method on R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Center of gravity method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 MVE cutting-plane method . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.4 Chebyshev center cutting-plane method . . . . . . . . . . . . . . . . . . . . . 16 4.5 ...