The weighted least-norm problem

Question

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 :

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

2. 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 $H(x) = Q$

3. Give the associated Lagrangian "using the multiplier ${\lambda}$ define as a vector of $R^m$

   Answer : I don't know

4. Using the necessary order conditions $1$, show that $x$ is defined as $ x = Q^{-1}*A^T{\lambda}$

   Answer: i don't know

5. 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

Answer 1

$$\begin{array}{ll} \text{minimize} & \mathrm x^{\top} \mathrm Q \, \mathrm x\\ \text{subject to} & \mathrm A \mathrm x = \mathrm b\end{array}$$

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.

Answer 2

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