Lagrange dual of quadratic optimization problem with quadratic equality constraints

Question

What is the Lagrange dual of the following optimization problem in $w \in \mathbb{R}^2$?

$$\begin{array}{ll} \text{minimize} & w^T Q \, w\\ \text{subject to} & w_1^2 = 1\\ & w_2^2 = 1\end{array}$$

Answer 1

The standard representation of the constraints is$$\omega_1^2=1\iff \omega^T\begin{bmatrix}1&0\\0&0\end{bmatrix}\omega=1\\\omega_2^2=1\iff \omega^T\begin{bmatrix}0&0\\0&1\end{bmatrix}\omega=1$$therefore

$$L(\omega,\lambda,\nu){=\omega^TQ\omega+\omega^T\cdot\left(\nu_1\begin{bmatrix}1&0\\0&0\end{bmatrix}+\nu_2\begin{bmatrix}0&0\\0&1\end{bmatrix}\right)\cdot\omega\\=\omega^T\cdot\left(Q+\nu_1\begin{bmatrix}1&0\\0&0\end{bmatrix}+\nu_2\begin{bmatrix}0&0\\0&1\end{bmatrix}\right)\cdot\omega}$$and the dual function would be:$$g(\lambda,\nu)=\min_{\omega}L(\omega,\lambda,\nu)$$which yields to the following dual problem:$${\max_{\lambda,\nu} g(\lambda,\nu)\\s.t.\\\lambda\ge 0}$$