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Fix typo <noupdate>

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Tian Cheng (Riccardo) Xia
2025-05-04 14:31:48 +02:00
parent f7dfdd101d
commit 78b364582a
1 changed files with 3 additions and 3 deletions
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6
src/year2/distributed-autonomous-systems/sections/_optimization.tex
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@ -34,7 +34,7 @@
\item[Convex set] \marginnote{Convex set} \item[Convex set] \marginnote{Convex set}
A set $Z \subseteq \mathbb{R}^d$ is convex if it holds that: A set $Z \subseteq \mathbb{R}^d$ is convex if it holds that:
\[ \[
\forall \z_A, \z_B \in Z: \Big( \exists \alpha \in [0, 1]: (\alpha \z_A + (1-\alpha)\z_B) \in Z \Big) \forall \z_A, \z_B \in Z: \Big( \forall \alpha \in [0, 1]: (\alpha \z_A + (1-\alpha)\z_B) \in Z \Big)
\] \]
\begin{figure}[H] \begin{figure}[H]
@ -45,7 +45,7 @@
\item[Convex function] \marginnote{Convex function} \item[Convex function] \marginnote{Convex function}
Given a convex set $Z \subseteq \mathbb{R}^d$, a function $l: Z \rightarrow \mathbb{R}$ is convex if it holds that: Given a convex set $Z \subseteq \mathbb{R}^d$, a function $l: Z \rightarrow \mathbb{R}$ is convex if it holds that:
\[ \[
\forall \z_A, \z_B \in Z: \Big( \exists \alpha \in [0, 1]: l(\alpha \z_A + (1-\alpha) \z_B) \leq \alpha l(\z_A) + (1-\alpha) l(\z_B) \Big) \forall \z_A, \z_B \in Z: \Big( \forall \alpha \in [0, 1]: l(\alpha \z_A + (1-\alpha) \z_B) \leq \alpha l(\z_A) + (1-\alpha) l(\z_B) \Big)
\] \]
\begin{figure}[H] \begin{figure}[H]
@ -67,7 +67,7 @@
Given a convex set $Z \subseteq \mathbb{R}^d$, a function $l: Z \rightarrow \mathbb{R}$ is strongly convex with parameter $\mu > 0$ if it holds that: Given a convex set $Z \subseteq \mathbb{R}^d$, a function $l: Z \rightarrow \mathbb{R}$ is strongly convex with parameter $\mu > 0$ if it holds that:
\[ \[
\begin{split} \begin{split}
\forall \z_A, \z_B \in Z, \z_A \neq \z_B: \Big( \exists \alpha \in (0, 1)&: l(\alpha \z_A + (1-\alpha) \z_B) < \\ \forall \z_A, \z_B \in Z, \z_A \neq \z_B: \Big( \forall \alpha \in (0, 1)&: l(\alpha \z_A + (1-\alpha) \z_B) < \\
&\alpha l(\z_A) + (1-\alpha) l(\z_B) - \frac{1}{2} \mu \alpha (1-\alpha) \Vert \z_A-\z_B \Vert^2 \Big) &\alpha l(\z_A) + (1-\alpha) l(\z_B) - \frac{1}{2} \mu \alpha (1-\alpha) \Vert \z_A-\z_B \Vert^2 \Big)
\end{split} \end{split}
\] \]