==== Strong Duality ====

We show [[wp>Slater's condition]] holds for the convex optimization problem:

\begin{eqnarray*}
 &\min_{A,\;\eta,\;\xi\geq 0} &  \lambda_A\|A\|_* + \lambda_{\eta} \|\eta^2\| + \sum_{i=1}^m \xi_i \nonumber \\
 &s.t.  & s(\hat{Y}^i(w)) \geq s( \tilde{Y}^i(w))  + F(\hat{Y}^i(w), \tilde{Y}^i(w))- \xi_i
\end{eqnarray*}


=== Proof ===

Let $A $ and $ \eta$ be any value, and set $\xi_i = \max\left\lbrace 1, s( \tilde{Y}^i(w)) + F(\hat{Y}^i(w), \tilde{Y}^i(w)) - s(\hat{Y}^i(w)) + 1\right\rbrace$.

\begin{eqnarray*}
 & \xi_i \geq s( \tilde{Y}^i(w)) + F(\hat{Y}^i(w), \tilde{Y}^i(w)) - s(\hat{Y}^i(w)) + 1 \\
\Rightarrow & s(\hat{Y}^i(w)) \geq s( \tilde{Y}^i(w)) + F(\hat{Y}^i(w), \tilde{Y}^i(w))- \xi_i + 1 \\
\Rightarrow & s(\hat{Y}^i(w)) > s( \tilde{Y}^i(w)) + F(\hat{Y}^i(w), \tilde{Y}^i(w))- \xi_i
\end{eqnarray*}

and also,
$$ \xi_i \geq 1 > 0$$.

Therefore strong duality holds ■