Given \begin{align} s(p)= &\text{arg min}( g( \boldsymbol{x}) ) \\ \text{subject to }& \boldsymbol{A}(p) \boldsymbol{x} = \boldsymbol{b}(p) \\ &c_1 \le x_i \le c_2 , i \in \{1,2,\cdots, n\} \end{align}.
where $ \boldsymbol{A} \in \mathcal{R} ^{ m \times n}$ is full row rank, with $m<n $. and both $ \boldsymbol{A}(p) $ and $ \boldsymbol{b}(p) $ is continous. Further $g(\boldsymbol{x})$ is continous and strictly convex and the problem is feasible for all $p$.
Does it follow from the maximum theorem that s(p) is a continous vector valued function?
I believe that I should be able to proove using this theorem (i.e., it seems to me that the conditions for the theorem are satisfied). However I struggle with the notation referenced in the theorem.
For $C(\boldsymbol{x})= \{x: \boldsymbol{A}(p) \boldsymbol{x} = \boldsymbol{b}(p) ,c_1 \le x_i \le c_2 , i \in \{1,2,\cdots, n\} \}$
1
Is $C(\boldsymbol{x})$ a compact-valued correspondence such that $C(\boldsymbol{x}) \neq \emptyset$ : Yes since $\boldsymbol{x}$ is always bounded and existence of solution is given by feasibility. 2
Is $C(\boldsymbol{x})$ continous: Yes (implicit function theorem?).
3
Now all assumptions of the theorem seems to be verified, and since we have strict convexity we must have a unique solutions such that s(p) is a continous vector valued function.
So this is may draft, but I feel like I am on shaky grounds. Would be nice if someone could verify.