documentclass[a4paper,12pt]{scrartcl}
usepackage[ngerman]{babel}
usepackage{amsthm}
usepackage{mathtools}
usepackage{aligned-overset}

newcommand{Lagr}{mathcal{L}}
newcommand{matr}[1]{mathbf{#1}}
newcommand{X}{matr{X}} %Matrix von X
newcommand{y}{matr{y}} %y als voller Vektor
newcommand{yct}{ubar{y}^mathbf{T}} %Zentriert
newcommand{betahat}{hat{beta}} %betahat

begin{document}
begin{flalign*}
Lagr(beta_{0},beta) &=minlimits_{beta_{0}, beta} left{ frac{1}{2} sum_{i=1}^Nleft(y_{i}-beta_{0} - sum_{j=1}^pbeta_{j}tilde{x}_{ij}right)^2  +lambdasum_{j=1}^pleft|beta_{j}right| right} &\
&= minlimits_{beta_{0}, beta} left{ frac{1}{2} sum_{i=1}^Nleft(y_{i}-beta_{0} - sum_{j=1}^pbeta_{j}(x_{ij}-bar{x}_{j})right)^2  +lambdasum_{j=1}^pleft|beta_{j}right| right}  &
end{flalign*}
vspace*{-1cm}
begin{flalign*}
&frac{partial Lagr}{partial beta_{0}} overset{!}{=}0 \
&-sum_{i=1}^Nleft(y_{i}-betahat_{0} - sum_{j=1}^pbetahat_{j}(x_{ij}-bar{x}_{j})right) = 0 \
& -sum_{i=1}^{N}y_i + N betahat_{0}+sum_{i=1}^Nleft(sum_{j=1}^pbetahat_{j}x_{ij}right) - Nsum_{j=1}^{p}betahat_{j}bar{x}_{j} =0 \
& -frac{1}{N}sum_{i=1}^{N}y_i +betahat_{0}+  sum_{j=1}^pleft(betahat_{j}frac{1}{N}sum_{i=1}^N x_{ij} right) - sum_{j=1}^pbetahat_{j}bar{x}_{j}=0 \
& -bar{y}+betahat_{0}+sum_{j=1}^pbetahat_{j}bar{x}_{j}-sum_{j=1}^pbetahat_{j}bar{x}_{j}=0\
& betahat_{0} =bar{y}\
intertext{textnormal{Aus der Annahme, dass $y$ zentriert ist $frac{1}{N}sum_{i=1}^{N}tilde{y}_{i}=0$, folgt}}
&betahat_{0}=0  &
  end{flalign*}
end{document}