peculiar min-max question

Hallo:

In research work on a certain statistical problem the following min/max
problem and the question of a fast algorithm came up:

Given n vectors x_1,...,x_n in R^d and denote by \|\theta\| the
Euclidian norm of \theta\in R^d. I would like to efficiently compute
(or at least approximate)

V_n=min_{\|\theta\|=1} \max_{1\leq i\leq n}|<x_i,\theta>|

where <x_i,\theta> denotes the ususal inner-product.

The brute-force approach works, sort of.... Here is what I tried:
Sample m independend and uniformly distributed \theta_j on the sphere,
compute the m\times n matrix of |<x_i,\theta_j>| and then find the
min/max of that matrix. This is easily done in MATLAB. However, n and
the dimension d (to have m small enough) have to be reasonable small...

Therefore I am looking for a different approach. For me it look's like a
problem in convex geometry or optimization....

Since I am not an expert in these fields of mathematics (me working
mostly in probability theory) I would appreciate any suggestions or remarks.

Many thanks!

Peter

 

Consider the following problem:
Given x_1,..., x_n as stated, compute

W_n = max_{\theta \in P} <\theta, \theta>

where

P = \{ \theta \in R^d | |<\theta, x_i>| \leq 1 , 1 \leq i \leq n \} .

If I am not mistaken, if V_n > 0 then W_n is finite and V_n = 1/W_n;
whereas if V_n = 0 then W_n is infinite.

The problem of finding W_n is a case of quadratic programming. There is
a treatment of it in G. B. Dantzig, _Linear Programming and Extensions_,
2d edn., 1968. This problem (linear constraints, quadratic objective)
can be solved by a variant of the simplex algorithm.