I have a parameterised optimization problem: \begin{align} \boldsymbol{S}(p)= &\arg \min_{ \boldsymbol{x}} g( \boldsymbol{x})\\ \text{s.t. } & \boldsymbol{A}(p) \textbf{x} = \boldsymbol{b}(p)\\ \end{align} Where we have the following assumptions:

$ \boldsymbol{g}(x) $ is strongly convex, $ \boldsymbol{A}(p) $ is full row-rank. $ \boldsymbol{A}(p) $ and $ \boldsymbol{b}(p) $ is $ C_1 $ continous.

I now want to proove continuity of $\boldsymbol{S}(p)$.

I think I have a working solution strategy (I am not sure if it is precise enough yet)

1: Due to strict convexity, for a fixed p, a unique solution to the KKT conditions exists (which also is the minimum): \begin{equation}\label{key} \boldsymbol{R}( \boldsymbol{z}) = {\left[\begin{matrix}\nabla_{ \boldsymbol{x}} \boldsymbol{g} (x) + \boldsymbol{A}^T \boldsymbol{\lambda} \\ \boldsymbol{A} \textbf{x} - \boldsymbol{b} \end{matrix} \right]}= \boldsymbol{0 } \end{equation} For $ \boldsymbol{z} ={\left[\begin{matrix}\boldsymbol{x }^T & \boldsymbol{\lambda}^T \end{matrix} \right]} ^T $

Using the implicit function theorem to express $ \frac{\partial \boldsymbol{z}}{\partial p} $ : \begin{equation} \frac{\partial \boldsymbol{R} }{\partial \boldsymbol{z} } \frac{\partial \boldsymbol{z}}{\partial p} + \frac{\partial \boldsymbol{R}}{\partial p} =0 \end{equation} Thus: \begin{equation}\label{SENSITIIVITY} \frac{\partial \boldsymbol{z}}{\partial p} = - \left( \frac{\partial \boldsymbol{R} }{\partial \boldsymbol{z} } \right)^{-1} \frac{\partial \boldsymbol{R}}{\partial p} \end{equation} '

The first term: \begin{equation} \frac{\partial \boldsymbol{R} }{\partial \boldsymbol{z} } = {\left[\begin{matrix}\nabla_x^2 ( g(x)) & \boldsymbol{A}^T \\ \boldsymbol{A} & \boldsymbol{0} \end{matrix} \right]} \end{equation} Is invertible if g is strictly convex, and A full row-rank.

Secondly $ \frac{\partial \boldsymbol{R}}{\partial p} $ exists if $ \boldsymbol{A}(p) $ and $ \boldsymbol{b}(p)$ is $C_1$.

Which appearantly shows continuity of $ \boldsymbol{z} $ (and $ \boldsymbol{x} $).

I am unsure if this is precise enough or if I need extra conditions. Can anyone verify?.(or point out what I am missing if it does not hold) Any references or guidelines would be helpful.

Bonus question: By the above, then it seems that $\boldsymbol{z}$ is also $ C_1 $ continous, or do I need any new conditions to state this?.