The weighted least-norm problem We want to minimize the function $f$ : x ∈ $\mathbb R^n$ 
with $f(x) := 1/2x^TQx $ under the constraint of equal $h(x)=b-Ax=0$
$Q$ is a positive definite symmetric matrix ($Q>0$) of size $n\times n$ .$A$ is a real matrix of dimension $m \times n$ and $\text{Rank } (A) = m$ and $b$ belongs to ${\mathbb R}^m$
Some questions that I would like to answer : 


*

*Giving associated canonical optimization problem:
Answer:  $(P1) : \min |Ax − b| $

*Gradients of $f(x)$ and $h(x)$
Answer: 
$f(x) := 1/2x^TQx $
where Q is symmetric. Then it is easy to see that:
$\nabla f(x) = Qx$ and $H(x) = Q$

*Give the associated Lagrangian "using the multiplier ${\lambda}$ define as a vector of $R^m$
Answer :  I don't know

*Using the necessary order conditions $1$, show that $x$ is defined as $ x = Q^{-1}*A^T{\lambda}$  
Answer: i don't know

*Since  $Q > 0 $ and  $\text{Rank }(A) = m $, we have  $ Q^{-1}*A^T{\lambda}$  invertible ,Deduce by using the equality constraint that ${\lambda}_{opt}=(AQ^{-1}*A^T)^{-1} b$
Answer:
I don't know
 A: $$\begin{array}{ll} \text{minimize} & \mathrm x^{\top} \mathrm Q \, \mathrm x\\ \text{subject to} & \mathrm A \mathrm x = \mathrm b\end{array}$$
where $\mathrm Q \in \mathbb R^{n \times n}$ is symmetric and positive definite, $\mathrm A \in \mathbb R^{m \times n}$ has full row rank and $\mathrm b \in \mathbb R^m$.
From the symmetry of $\mathrm Q$, we conclude it has a spectral decomposition $\mathrm Q = \mathrm V \Lambda \mathrm V^{\top}$. From the positive definiteness of $\mathrm Q$, we conclude we can factor it as follows
$$\mathrm Q = \mathrm V \Lambda \mathrm V^{\top} = \mathrm V \Lambda^{\frac 12} \Lambda^{\frac 12} \mathrm V^{\top} = \left( \Lambda^{\frac 12} \mathrm V^{\top} \right)^{\top} \left( \Lambda^{\frac 12} \mathrm V^{\top} \right)$$
Let $\mathrm y := \Lambda^{\frac 12} \mathrm V^{\top} \mathrm x$. Hence, we have a least-norm problem
$$\begin{array}{ll} \text{minimize} & \mathrm y^{\top} \mathrm y\\ \text{subject to} & \mathrm A \mathrm V \Lambda^{-\frac 12} \mathrm y = \mathrm b\end{array}$$
After some work, we arrive at the least-norm solution
$$\mathrm x_{\text{LN}} := \color{blue}{\mathrm Q^{-1} \mathrm A^{\top} \left( \mathrm A \mathrm Q^{-1} \mathrm A^{\top} \right)^{-1} \mathrm b}$$
Take a look at Lieven Vandenberghe's lecture notes on the least-norm problem.
A: I have expressed myself sorry already by using the necessary conditions order 1 shows that x is defined by : $x = Q^-1*A^T{\lambda}$ and calculate the hessian of the Lagrangian in $x$optimal ${\lambda} optimal $. Deduce that we have a strict local minimum. And I would like to solve the following problem : $min \dfrac 1 {2} ||x||_2^2$  with the constraint Ax=b 
$||x||_2^2$ is given by ${\sum}x_k $=c avec $a_k>0 $. 
From the preceding problem give the expression, of x optimal in the form of A and B (square-square solution)
Someone can you help me it's urgent, thanks you very much paul-henri
