# What Is a Simple and Explicit Dual of $h \left( x \right) = {\left\| x - d \right\|}_{2}^{2} + \sum_{i = 1}^{4} {\left\| {A}_{i} X \right\|}_{2}$

What is a simple and explicit dual of $h(x) =||x-d||_2^2 +\Sigma^{4}_{i=1}||A_iX||_2$

Where $A_i\in \mathbb{R}^{5x5}$ matrices.

Since the primal has no constraints I have tried to come up with the following two constraints and derive the dual:

1) $y=x-d$

$inf_{x,y}||y||_2^2 + \Sigma^{4}_{i=1}||A_iX||_2 + \lambda(y-x+d)$

2) $y_i= A_ix$

$inf_{x,y}||x-d||_2^2 + \Sigma^{4}_{i=1}||y_i||_2 + \Sigma^{4}_{i=1}\lambda_i(y_i-A_ix)$

In both situations I am stuck and not sure how to continue.

Here's how I'd rewrite it: \begin{array}{ll} \text{minimize} & \|x-d\|_2^2 + \sum_i t_i \\ \text{subject to} & \|A_ix\|_2 \leq t_i, ~ i=1,2,3,4 \end{array} Alternatively, \begin{array}{ll} \text{minimize} & \|x-d\|_2^2 + \sum_i t_i \\ \text{subject to} & (A_ix,t_i) \in \mathcal{K}_2, ~ i=1,2,3,4 \end{array} Where $\mathcal{K}_2$ is the second-order cone of appropriate size. The Lagrangian is $$L(x,y_1,z_1,y_2,z_2,y_3,z_3,y_4,z_4) = \|x-d\|_2^2 + \sum_i t_i - \sum_i \langle (y_i,z_i), (A_ix,t_i) \rangle = \|x-d\|_2^2 + \sum_i t_i(1-z_i) - \langle x, \sum_i A_i^T y_i \rangle$$ Where each Lagrange multiplier $(y_i,z_i)$ must also lie in $\mathcal{K}_2$; that is, $\|y_i\|_2\leq z_i$. The dual function, therefore, is $$g(y_1,z_1,y_2,z_2,y_3,z_3,y_4,z_4)=\inf_{x,t_1,t_2,t_3,t_4} L(x,y_1,z_1,y_2,z_2,y_3,z_3,y_4,z_4)$$ The optimality conditions are $$2(x-d) = \sum_i A_i^T y_i, \quad z_1=z_2=z_3=z_4=1$$ The $t,z$ terms are all zero, leaving us with the terms involving $x$. We're supposed to eliminate $x$, but I'm going to start by eliminating the $y_i$ sum, simplifying, and then putting it back in: \begin{aligned} g(\dots) &= \|x-d\|_2^2 - \langle x, 2(x-d) \rangle \\ &= x^Tx - 2d^T x + d^T d - 2x^Tx + 2d^Tx \\ &= \|d\|_2^2 -\|x\|_2^2 = \|d\|-2^2 - \tfrac{1}{4} \|\sum_i A_i^T y_i + 2d \|_2^2 \end{aligned} How convenient! ;-) So the dual is \begin{array}{ll} \text{maximize} &\|d\|_2^2 - \tfrac{1}{4} \|\sum_i A_i^T y_i + 2d \|_2^2 \\ \text{subject to} & \|y_i\|_2 \leq z_i, ~ i=1,2,3,4 \\ & z_1=z_2=z_3=z_4=1 \end{array} or, eliminating the $z_i$ variables, \begin{array}{ll} \text{maximize} &\|d\|_2^2 - \tfrac{1}{4} \|\sum_i A_i^T y_i + 2d \|_2^2 \\ \text{subject to} & \|y_i\|_2 \leq 1, ~ i=1,2,3,4 \end{array}