Given \begin{align} s(\theta)= &\text{arg min}( g( \boldsymbol{x}) ) \\ \text{subject to }& \boldsymbol{A}(\theta) \boldsymbol{x} = \boldsymbol{b}(\theta) \\ &c_1 \le x_i \le c_2 , i \in \{1,2,\cdots, n\} \end{align}.
where $ \theta\in \Theta$ , $ \boldsymbol{A} \in \mathcal{R} ^{ m \times n}$ is full row rank, with $m<n $, $ c_1 $ and $ c_2 $ are constants, both $ \boldsymbol{A}(\theta) $ and $ \boldsymbol{b}(\theta) $ is continuous and the problem is feasible for all $ \theta \in \Theta $. Further $g(\boldsymbol{x})$ is continuous and strictly convex.
Does it follow from the Maximum theorem that $s(\theta)$ is a continuous vector valued function?
The theorem is short and seems rather straight forward, however, I am in need a second opinion verifying that what I believe to hold is valid. (I still somewhat struggle with the concepts).
Below follows my simple draft
Let $\boldsymbol{x} \in \mathbb{R}^n$, and $\theta \in \Theta \in \mathbb{R}^1$
Let the constraint set be denoted
$ C(\theta)= \{ \boldsymbol{x}: \boldsymbol{A}(\theta) \boldsymbol{x} = \boldsymbol{b}(\theta) , c_1 \le x_i \le c_2 , i \in \{1,2,\cdots, n\} \} $
Now:
$ \theta \mapsto C(\theta) $ is compact (affine constraints) such that $ C(\theta) \neq \emptyset$ (assumption of feasibility) $\forall \theta \in \Theta $. Therefore $ C:\mathbb{R}^1 \rightrightarrows \mathbb{R}^n$ $ is a compact valued correspondence.
With $\boldsymbol{A}(\theta)$ full row rank, given continous parameters, $ C(\theta) $ is continous (implicit function theorem). Since also $g( \boldsymbol{x}) $ is continuous, by the maximum theorem (in this case we use the opposite version, i.e., minimum) , $ s(\theta) $ is continuous (i.e. both upper and lower hemicontinuous). Now due to strict convexity any solution is single-(vector) valued and $ s(\theta) $ is a continuous vector valued function.
Am I missing something?
