i need find an expression in terms of v for the minimum value of (x^t)x subject to the constraint (v^t)x = k
v is a fixed vector in R^n
k is a real constant
I need to use langrange multitpliers to solve this.
I let L(x,lambda) = (x^t)x + lambda*(k - (v^t)x)
so therefore grad L = 2x - lambda*v = 0
so i therefore let x = 1/2*lambda*v and put this into the constraint to get
(v^t)*1/2*v*lambda = k
so 1/2*lamda*(v^t)*v = k in terms of v
To verify this is the min, i differentiated grad L again to get 2 >0 so this is convex and hence a global minimizer.
Can some one please check my solution, i was getting whether to differentiate with respect to v as well as it is a vector, but wasnt too sure?.. if some one can give me some advise that would be great thanks