Optimization Problems

Just a boi trying to solve some optimization problems.

First let’s define our parameters:

• matrices $A \in \mathbb{R}^{m \times n}$; $B \in \mathbb{R}^{m \times n}$; $C \in \mathbb{R}^{p \times n}$; $W \in \mathbb{R}^{m \times m} = \text{diag}{w_1, …, w_m }$ where $w_i > 0$
• vectors $b \in \mathbb{R}^n$; $d \in \mathbb{R}^p$
• scalars $\lambda \in \mathbb{R}+$
• By SVD, $A = U \Sigma V^T$ where $U \in \mathbb{R}^{m \times m}$ is orthonormal; $V \in \mathbb{R}^{n \times n}$ is orthonormal; $\Sigma \in \mathbb{R}^{m \times n}$ is a rectangular diagonal matrix

Problem	$x^*$	$f(x^*)$	Equations	Existence	Uniqueness
$\text{argmin} ||Ax - b||_2^2 : x \in R^n$	$A^\dag b + z : z \in \mathcal{N}(AT)$	$| | A A^\dag b + z - b||_2^2$