Homework 2

System optimization and scheduling |

Name : | 杜彪 | Class ID : | 3034 | Student ID : | 3113033039 | Department : | CS | Email: | 1012298204@qq.com | | |

Solution 2
Solution of problem 1.6.4 :
It suffices to show that the subspace spanned by is the same as the subspace spanned by, for .We will prove this by induction. Clearly, when k = 1 the statement is true. Assume it is true for k-1 < n-1, i.e.

Where denotes the subspace spanned by the vectors. Assume. Since and minimize f over the manifold, from our assumption we have that
.
The fact that yields . (1)
If, then from formulation (1) and the inductive hypothesis it follows that (2)
We know that is orthogonal to .
Therefore formulation (2) is possible only if which contradicts our assumption. Hence.If , then formulation (1) and our inductive hypothesis again imply formulation (2) which is not possible.
So the vectorsare linearly independent.
Combined with formulation (1) and linear independence of the vectors we can get that .

Solution of problem 2.1.12 :
(a) Assume that z is a fixed vector in. Then the problem is equal to find a vector of the simplex X, which is at a minimum distance from z; that is
Minimize f(x) = ||z-x||2
Subject to x ∈X, that is subject to = r
Suppose, H = In = and A = , we can write the problem as
Minimize f(x) =
Subject to Ax = r
We can easily get that
,
where, so .
(b) The original problem equals to minimize
.
Subject to x = r
Let H =, , A = .
Then we will obtain that
,

Solution of problem 3.1.9:
(a) Supposeandrespectively are the lengths of the tangent lines from the vertices of the triangle to the circle, and let be the radius of the circle. So the area of the triangle is, and by the theorem of Heron it can also be