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Wikipedia / Chebyshev center

In geometry, the Chebyshev center having non-empty interior can mean two different things:

  • it can be the center of the minimal-radius ball enclosing, or

  • it can be the center of a largest inscribed.

a triangle where one side is much shorter than the two others; the enclosed kind of Chebyshev center will be in the vicinity of the midpoints of the two long sides, whereas the inscribed kind of Chebyshev center will be in the vicinity of the short side, where the triangle is thickest. This article is mostly about the enclosing kind of Chebyshev center.

In the field of parameter estimation minimizes the worst possible estimation error for x (e.g. best worst case). This is equivalent to the enclosing kind of Chebyshev center.

Mathematical representation

can be computed by solving:

with respect to the Euclidean norm, or alternatively by solving:

Despite these properties, finding the Chebyshev center may be a hard numerical optimization problem. For example, in the second representation above, the inner maximization is non-convex if the set Q is not convex.

Properties

In inner product spaces without loss of generality.

is the tetrahedron formed by the convex hull norm yields

Relaxed Chebyshev center

ellipsoids.

with

, we can write the inner maximization problem of the Chebyshev center as:

is the trace operator and

is the set of positive semi-definite matrices, and changing the order of the min max to max min (see the references for more details), the optimization problem can be formulated as:

with

This last convex optimization problem is known as the relaxed Chebyshev center (RCC). The RCC has the following important properties:

  • The RCC is an upper bound for the exact Chebyshev center.

  • The RCC is unique.

  • The RCC is feasible.

Constrained least squares

It can be shown that the well-known constrained least squares (CLS) problem is a relaxed version of the Chebyshev center.

The original CLS problem can be formulated as:

with

It can be shown that this problem is equivalent to the following optimization problem:

with

One can see that this problem is a relaxation of the Chebyshev center (though different than the RCC described above).

RCC vs. CLS

. This means that the CLS estimate is the solution of a looser relaxation than that of the RCC. Hence the CLS is an upper bound for the RCC, which is an upper bound for the real Chebyshev center.

Modeling constraints

is defined affects its relaxed versions. This of course affects the quality of the RCC and CLS estimators. As a simple example consider the linear box constraints:

which can alternatively be written as

It turns out that the first representation results with an upper bound estimator for the second one, hence using it may dramatically decrease the quality of the calculated estimator.

This simple example shows us that great care should be given to the formulation of constraints when relaxation of the feasibility region is used.

Linear programming problem

The inscribed form of the Chebyshev center problem can be formulated as a linear programming is an intersection of finitely many half-spaces. Given a polytope.

See also

References