Numerical Optimization ( norm minim)

by sbashrawi
Tags: minim, norm, numerical, optimization
 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
 HW Helper P: 3,307 consider the set $\left\{ x \in \mathbb{R}^n : ||x||^2 = c\}$ 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
 HW Helper P: 3,307 hint: think tangents & normals
 P: 55 Numerical Optimization ( norm minim) Here is my work: f(x) = ||x|| ^2 subjeted to c(x) = a^{T} x +$$\alpha$$$$\geq$$0 so L ( x,$$\lambda$$) = f(x) - $$\lambda$$ c(x) gradiant L(x,$$\lambda$$) = 2x - $$\lambda$$ grad(c(x)) = 0 grad c(x)= a so 2x - $$\lambda$$a = 0 this gives that x = $$\frac{}{}[1/2]\lambda$$a and c(x) = 0 gives : $$\lambda$$ = -2 $$\alpha$$$$/ ||a||^2 imlpies x = - \alpha$$ a / ||a||^2
 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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