A tricky optimization problem: writing the dual (using Lagrangian) I am trying to find the dual problem of the following:$$\min_{θ_v\in\mathbb R^l,b_v\in\mathbb R,ξ_{i,v}\in\mathbb R,λ_{i,k,j}\in\mathbb R}\frac1n\sum_{i=1}^n\left(\sum_{v=1}^mξ_{i,v}+\sum_{k=1}^m\sum_{j=k+1}^mλ_{i,k,j}\right)+\frac12\sum_{v=1}^m\|θ_v\|^2$$such that 
$$|{\langle\theta_k, x_i^{(k)} \rangle + b_k - \langle\theta_j, x_i^{(j)} \rangle -b_j}| \leq \epsilon + \lambda_{i,k,j}$$
$$y_i (\langle\theta_{v}, x_i^{(v)} \rangle + b_v ) \geq  1 - \xi_{i,v}$$
$$ \xi_{i,j} \geq 0 $$
$$\lambda_{i,k,j}\geq 0$$
for $i=1, \dots, n$, $k = 1, \dots, m$ and $j = k+1, \dots, m$
We can replace the first condition by
$$({\langle\theta_k, x_i^{(k)} \rangle + b_k - \langle\theta_j, x_i^{(j)} \rangle -b_j})^2 \leq \epsilon + \lambda_{i,k,j}$$
if it helps. We can even look at a very similar problem if the previous one is too hard 
$$\min\limits_{\theta_v\in \mathbb R^l, b_v \in \mathbb R, \xi_{i,v} \in \mathbb R, \lambda_{i,k,j} \in \mathbb R} \frac{1}{n} \sum\limits_{i=1}^{n}  \left(\sum\limits_{v=1}^{m} \xi_{i,v} +  \sum\limits_{k=1}^{m} \sum\limits_{j=k+1}^{m} \lambda_{i,k,j}^2 \right) + \frac{1}{2} \sum\limits_{v=1}^{m} ||\theta_v||^2$$
such that 
$${\langle\theta_k, x_i^{(k)} \rangle + b_k - \langle\theta_j, x_i^{(j)} \rangle -b_j} =  \lambda_{i,k,j}$$
$$y_i (\langle\theta_{v}, x_i^{(v)} \rangle + b_v ) \geq  1 - \xi_{i,v}$$
$$ \xi_{i,j} \geq 0 $$
$$\lambda_{i,k,j}\geq 0$$
for $i=1, \dots, n$, $k = 1, \dots, m$ and $j = k+1, \dots, m$
 A: $\newcommand{\R}{\mathbb{R}}\newcommand{\Minimize}{\operatorname*{Minimize}}$Let us define the operator $[{}\cdot{}]_+:\R^n\to\R$ as 
$$
[z]_+ = \begin{cases}
z, &\text{if } z \geq 0
\\
0, &\text{otherwise}
\end{cases}
$$
Then, the optimization problem
$$
\Minimize_{x} f(x) + [g(x)]_+,
$$
is equivalent to 
\begin{align}
&\Minimize_{x} f(x) + y
\\
&\text{subject to }y \geq 0, g(x) \leq y.
\end{align}
One can be seen as the dual of the other.
That said, your optimization problem can be written as (I multiplied the cost by $n$ for convenience, but you can undo that)
\begin{align}
&\Minimize_{\theta_\nu, b_\nu} \sum_{i,\nu} 
\left[1 - y_i (\langle \theta_\nu, x_i^{(\nu)}\rangle + b_\nu)\right]_+
\\
&+
\sum_{i,j,k} 
\left[
|\langle \theta_k, x_i^{(k)}\rangle+b_k - 
\langle \theta_j, x_i^{(j)}\rangle+b_j| - \epsilon
\right]_+ + \tfrac{n}{2}\sum_\nu \|\theta_\nu\|^2.
\end{align}
You can in fact think of your problem as the dual of this one (in fact, one if the dual of the other as the cost function is continuous).
