# Constrained Optimization

• May 19th 2010, 02:05 AM
demode
Constrained Optimization
http://img261.imageshack.us/img261/5851/29531326.gif

(a) The reference they were referring is : "Find a lower triangular matrix L, and a diagonal matrix D such that $A = LDL^T$". I don't see how this is relevant though.

Here's what I've done so far

$L= x+y+z- \lambda(x^2+y^2+z^2-k)$

$L_x = 1-2 \lambda x = 0$
$L_y = 1-2 \lambda y = 0$
$L_z = 1-2 \lambda z = 0$

From this, critical points are $x=y=z = \frac{1}{2 \lambda}$

Since $f_{xx}=f_{yy}=f_{zz}=-2\lambda$
The Hessian matrix will be

$H = \begin{bmatrix} -2 \lambda & 0 & 0 \\ 0& -2 \lambda & 0 \\ 0&0& -2 \lambda \end{bmatrix}$

So what else do I need to do here to solve this? (I can't find any similar examples from my textbook)

Also, can anyone show me how to prove part (b). I don't understand it...