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Numerical Optimization ( norm minim) 
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#1
Apr1610, 10:28 PM

P: 55

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 


#2
Apr1710, 04:10 AM

HW Helper
P: 3,307

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 


#3
Apr1710, 04:11 AM

HW Helper
P: 3,307

hint: think tangents & normals



#4
Apr1710, 09:07 AM

P: 55

Numerical Optimization ( norm minim)
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 


#5
Apr1710, 10:55 PM

HW Helper
P: 3,307

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 


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