Foundations of Data Science

polytope
Ellipsoid containing halfsphere
Separating hyperplane
Figure 10.8: Ellipsoid Algorithm
Next, rotate the coordinate axes so that a is replaced by (1, 0, 0, . . . , 0). Finally, make a
nonsingular linear transformation τ so that τE = B = {x : |x| = 1}, the unit sphere.
The important point is that a nonsingular linear transformation τ multiplies the volumes
of all sets by |det(τ)|, so that Vol(Ê) = Vol(τ(Ê)) . Now, the following lemma answers the
Vol(E) Vol(τ(E))
question raised.
Lemma 10.11 Consider the half-sphere B ′ = {x : x 1 ≥ 0 ; |x| ≤ 1}. The following
ellipsoid Ê contains B′ :
{
( ) 2 (
d + 1
Ê = x
x
∣
1 − 1 ) 2 ( ) (
) }
d 2 − 1
+
x 2
d
d + 1 d 2 2 + x 2 3 + . . . + x 2 d ≤ 1 .
Further,
Vol(Ê) ( ) ( ) d d
2 (d−1)/2
Vol(B) = ≤ 1 − 1
d + 1 d 2 − 1
4d .
Proof: See Exercise (10.24).
10.7 Integer Optimization
The problem of maximizing a linear function subject to linear inequality constraints,
but with the variables constrained to be integers is called integer programming:
Max c · x subject to Ax ≤ b x i integers .
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