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
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
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
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