Let \( X:T\rightrightarrows \mathbb{R}^n \) be lower semicontinuous and let \( f:\text{graph}(X)\rightarrow\mathbb{R} \) be lower semicontinuous. Then, the extended real optimal-value function $$ \varphi(t) := \sup_{ x\in X(t) } \ f(t,x) $$ is lower semicontinuous.

Let \( X:T\rightrightarrows \mathbb{R}^n \) be lower semicontinuous and let \( f:\text{graph}(X)\rightarrow\mathbb{R} \) be upper semicontinuous. Then, the extended real optimal-value function $$ \varphi(t) := \inf_{x\in X(t)} \ f(t,x) $$ is upper semicontinuous.

Let \( X:T\rightrightarrows \mathbb{R}^n \) be upper semicontinuous with nonempty compact values and let \( f:\text{graph}(X)\rightarrow\mathbb{R} \) be upper semicontinuous. Then, the extended real optimal-value function $$ \varphi(t) := \max_{ x\in X(t) } \ f(t,x) $$ is upper semicontinuous.

Let \( X:T\rightrightarrows \mathbb{R}^n \) be upper semicontinuous with nonempty compact values and let \( f:\text{graph}(X)\rightarrow\mathbb{R} \) be lower semicontinuous. Then, the extended real optimal-value function $$ \varphi(t) := \min_{x\in X(t)} \ f(t,x) $$ is lower semicontinuous.

(Berge Maximum Theorem) Let \( X:T\rightrightarrows \mathbb{R}^n \) be continuous with nonempty compact values and let \( f:\text{graph}(X)\rightarrow\mathbb{R} \) be continuous. Then, the optimal-value function $$ \varphi(t) := \max_{ x\in X(t) } \ f(t,x) $$ is continuous. Moreover, the solution set mapping is upper semicontinuous.

(Convex Optimization - Lower Semicontinuity) Let \( T\subseteq\mathbb{R}^m \) be a given set and let \( S(t) \) denote the solution set of the convex problem $$ \begin{align*} \min_{x} \quad & f(x) \\ \text{s.t.} \quad & g_i(x) \le t_i, \quad \text{for all }i = 1,\dotsc,m. \end{align*} $$ Assume that \( f \) and \( g_i \) are convex and weakly analytic for all \( i=1,\dotsc,m \) and that \( S(t) \neq \emptyset \) holds for all \( t\in T \). Then, the solution set mapping \( S \) is lower semicontinuous on \( T \).

@book{Bank_1982,
  title = {Non-Linear Parametric Optimization},
  isbn = {9783034863285},
  doi = {10.1007/978-3-0348-6328-5},
  publisher = {Birkhäuser Basel},
  author = {Bank, B. and Guddat, J. and Klatte, D. and Kummer, B. and Tammer, K.},
  year = {1982}
}

@book{aliprantis2006,
  title={Infinite dimensional analysis: a hitchhiker's guide},
  author={Aliprantis, Charalambos D and Border, Kim C},
  year={2006},
  publisher={Springer},
  doi={10.1007/3-540-29587-9}
}