Langrange multipliers - check my soluton please

[dopey9]

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.

My solution

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.

Prohlem

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

 

[mighty2000]

Your solution looks correct. To verify that it is a minimum, you can also use the second derivative test. Differentiate the gradient again with respect to x and set it equal to 2. If the second derivative is positive, then it is indeed a minimum.

In this case, since the second derivative is a constant (2), it is positive and therefore confirms that your solution is a minimum.

Regarding differentiating with respect to v, since v is a fixed vector, it is treated as a constant in this problem and therefore does not need to be differentiated with respect to.

Overall, your solution and approach using Lagrange multipliers is correct. Well done!