From 78b364582a1bb9f480490ec75d85c5997a1b2bbd Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Sun, 4 May 2025 14:31:48 +0200 Subject: [PATCH] Fix typo --- .../sections/_optimization.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/src/year2/distributed-autonomous-systems/sections/_optimization.tex b/src/year2/distributed-autonomous-systems/sections/_optimization.tex index bef1619..503cc54 100644 --- a/src/year2/distributed-autonomous-systems/sections/_optimization.tex +++ b/src/year2/distributed-autonomous-systems/sections/_optimization.tex @@ -34,7 +34,7 @@ \item[Convex set] \marginnote{Convex set} 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] @@ -45,7 +45,7 @@ \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: \[ - \forall \z_A, \z_B \in Z: \Big( \exists \alpha \in [0, 1]: l(\alpha \z_A + (1-\alpha) \z_B) \leq \alpha l(\z_A) + (1-\alpha) l(\z_B) \Big) + \forall \z_A, \z_B \in Z: \Big( \forall \alpha \in [0, 1]: l(\alpha \z_A + (1-\alpha) \z_B) \leq \alpha l(\z_A) + (1-\alpha) l(\z_B) \Big) \] \begin{figure}[H] @@ -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: \[ \begin{split} - \forall \z_A, \z_B \in Z, \z_A \neq \z_B: \Big( \exists \alpha \in (0, 1)&: l(\alpha \z_A + (1-\alpha) \z_B) < \\ + \forall \z_A, \z_B \in Z, \z_A \neq \z_B: \Big( \forall \alpha \in (0, 1)&: l(\alpha \z_A + (1-\alpha) \z_B) < \\ &\alpha l(\z_A) + (1-\alpha) l(\z_B) - \frac{1}{2} \mu \alpha (1-\alpha) \Vert \z_A-\z_B \Vert^2 \Big) \end{split} \]