Numerical Optimization ( norm minim)

sbashrawi

1. The problem statement, all variables and given/known data

Consider the half space defined by H = {x ∈ IRn | aT x +alpha ≥ 0} where a ∈ IRn
and alpha ∈ IR are given. Formulate and solve the optimization problem for finding the point
x in H that has the smallest Euclidean norm.

2. Relevant equations



3. The attempt at a solution
I need help in this problem. I think the problem can be written as

min ||x|| sunbjected to a(transpose) x + a >= 0

am I right

lanedance

consider the set [itex] \left\{ x \in \mathbb{R}^n : ||x||^2 = c\} [/itex] for some constant c, geometrically what does it represent?

now consider the half space, the boundary of which is a plane. how does the plane intersect the above set, in particular for the minimum value of c. This should lead to a simple solution

lanedance

hint: think tangents & normals

sbashrawi

Here is my work:

f(x) = ||x|| ^2
subjeted to c(x) = a^{T} x +[tex]\alpha[/tex][tex]\geq[/tex]0

so L ( x,[tex]\lambda[/tex]) = f(x) - [tex]\lambda[/tex] c(x)
gradiant L(x,[tex]\lambda[/tex]) = 2x - [tex]\lambda[/tex] grad(c(x)) = 0
grad c(x)= a
so 2x - [tex]\lambda[/tex]a = 0
this gives that x = [tex]\frac{}{}[1/2]\lambda[/tex]a
and
c(x) = 0
gives : [tex]\lambda[/tex] = -2 [tex]\alpha[/tex][tex] / ||a||^2

imlpies

x = - \alpha[/tex] a / ||a||^2

lanedance

Lagrange multipliers ok,, though bit hard to read what is a vector

the answer makes sense to me as the boundary plane will have a as its normal, and the answer is both on the boundary plane & parallel to a