# Thread: minimum value subject to the constraint

1. ## minimum value subject to the constraint

Hi i was wondering if anyone can help me with this problem.. i was thinking of using Lagrangian's multipliers method to solve this., but i alsos get confused with transpose part i.e. (^t)

i need to find an expression in terms of s for the minumum value of (x^t)x subject to the constraint (s^t)x = k

s is a fixed vector in R^n
k is a real constant

and i also need to justify this is a minimum

if anyone can help..that would be great thanks

2. Originally Posted by dopi
Hi i was wondering if anyone can help me with this problem.. i was thinking of using Lagrangian's multipliers method to solve this., but i alsos get confused with transpose part i.e. (^t)

i need to find an expression in terms of s for the minumum value of (x^t)x subject to the constraint (s^t)x = k

s is a fixed vector in R^n
k is a real constant

and i also need to justify this is a minimum
You don't need Lagrange multipliers, it's just simple algebra!

$\displaystyle (\mathbf{s}-\mathbf{x})^{\textsc{t}}(\mathbf{s}-\mathbf{x}) = \mathbf{s}^{\textsc{t}}\mathbf{s} + \mathbf{x}^{\textsc{t}}\mathbf{x} - 2\mathbf{s}^{\textsc{t}}\mathbf{x}$, and therefore $\displaystyle \mathbf{x}^{\textsc{t}}\mathbf{x} = 2\mathbf{s}^{\textsc{t}}\mathbf{x} - \mathbf{s}^{\textsc{t}}\mathbf{s} + (\mathbf{s}-\mathbf{x})^{\textsc{t}}(\mathbf{s}-\mathbf{x})$. But $\displaystyle (\mathbf{s}-\mathbf{x})^{\textsc{t}}(\mathbf{s}-\mathbf{x})\geqslant0$. So the minimum value of $\displaystyle \mathbf{x}^{\textsc{t}}\mathbf{x}$ is $\displaystyle 2k - \mathbf{s}^{\textsc{t}}\mathbf{s}$, and it occurs when $\displaystyle \mathbf{x} = \mathbf{s}$.

3. ## verifying the min

Hi thanks for the reply..however as i am learning about lagrangian multipliers and kuhn tucker methods, i sujested using langrangian method, so then i could use hessian matrix to justify my minimum, by checking the leading minor principles. Like i said when i expand or differentiate equations with the the transpose sign,,..i get confused where to put the t.

But with what you have shown i dont know how to justify the minumum..any suggestions??