Spacing in alignedat
Since you need to choose the alignment point anyway, I propose a different approach, where you state the desired width of the object (default 3em):
\documentclass{article}
\usepackage{amsmath}
\renewcommand{\vec}[1]{\mathbf{#1}}
\newcommand{\splitcase}[2][3em]{%
\makebox[#1][l]{$\displaystyle#2$} &
\settowidth{\dimen0}{$\displaystyle#2$}%
\addtolength{\dimen0}{-1em}%
\hspace*{\dimen0}%
\\
}
\begin{document}
\begin{equation*}
(G^{*} \vec{x}^{*})(\vec{w}^{*})=
\begin{cases}
\splitcase{
-f_{0}^{*}(\vec{x}_{0}^{*})
-(f_{1}^{*} \upsilon_{1}^{*})(\vec{x}_{1}^{*})
-\dots
-(f_{m}^{*} \upsilon_{m}^{*})(\vec{x}_{m}^{*})
}
&\text{if $\vec{u}^{*} \geq \vec{0}$ and
$\vec{x}_{0}^{*}+\dots+\vec{x}_{m}^{*}=\vec{x}^{*}$},
\\
-\infty &\text{otherwise}.
\end{cases}
\end{equation*}
\begin{equation*}
(G^{*} \vec{x}^{*})(\vec{w}^{*})=
\begin{cases}
\splitcase[5em]{
-f_{0}^{*}(\vec{x}_{0}^{*})
-(f_{1}^{*} \upsilon_{1}^{*})(\vec{x}_{1}^{*})
-\dots
-(f_{m}^{*} \upsilon_{m}^{*})(\vec{x}_{m}^{*})
}
&\text{if $\vec{u}^{*} \geq \vec{0}$ and
$\vec{x}_{0}^{*}+\dots+\vec{x}_{m}^{*}=\vec{x}^{*}$},
\\
-\infty &\text{otherwise}.
\end{cases}
\end{equation*}
\end{document}
I wouldn't change much about the setup of the first equation except (a) make sure the -
symbol at the start of the math rows is typeset as a unary symbol, (b) change \begin{alignedat}{3}
to \begin{alignedat}{2}
, and (c) leave a bit more whitespace between rows 2 and 3. Whether to use arrow accents or bold-facing a vector is largely a matter of taste; if you use the arrow method, be sure to leave a bit of space before the superscript *
symbols.
The spacing around the curly brace is unchanged if one uses \begin{cases} ... \end{cases}
instead of \left\{ ... \right.
\documentclass{article}
\usepackage{amsmath} % for 'alignedat' env and '\text' macro
\usepackage{bm} % for '\bm' macro
\begin{document}
\begin{align*}
(G^{*} \vec{x}^{\mkern1.5mu*})(\vec{w}^{\mkern1.5mu*})
&= \left\{
\begin{alignedat}{2}
&{-}f_{0}^{*}(\vec{x}_{0}^{\mkern1.5mu*})
&&-(f_{1}^{*} \upsilon_{1}^{*})(\vec{x}_{1}^{\mkern1.5mu*})
-\dots-(f_{m}^{*} \upsilon_{m}^{*})(\vec{x}_{m}^{\mkern1.5mu*}) \\
&&&\text{if $\vec{u}^{\mkern1.5mu*} \geq \vec{0}$ and
$\vec{x}_{0}^{\mkern1.5mu*}+\dots+\vec{x}_{m}^{\mkern1.5mu*}
=\vec{x}^{\mkern1.5mu*}$}, \\[1ex]
&{-}\infty
&&\text{otherwise}.
\end{alignedat}
\right. \\ % end of first equation
(G^{*} \bm{x}^{*})(\bm{w}^{*})
&= \left\{
\begin{alignedat}{2}
&{-}f_{0}^{*}(\bm{x}_{0}^{*})
&&-(f_{1}^{*} \upsilon_{1}^{*})(\bm{x}_{1}^{*})
-\dots-(f_{m}^{*} \upsilon_{m}^{*})(\bm{x}_{m}^{*}) \\
&&&\text{if $\bm{u}^{*} \geq \bm{0}$ and
$\bm{x}_{0}^{*}+\dots+\bm{x}_{m}^{*}
=\bm{x}^{*}$}, \\[1ex]
&{-}\infty
&&\text{otherwise}.
\end{alignedat}
\right. % end of second equation
(G^{*} \bm{x}^{*})(\bm{w}^{*})
&= \begin{cases}
\begin{alignedat}{2}
&{-}f_{0}^{*}(\bm{x}_{0}^{*})
&&-(f_{1}^{*} \upsilon_{1}^{*})(\bm{x}_{1}^{*})
-\dots-(f_{m}^{*} \upsilon_{m}^{*})(\bm{x}_{m}^{*}) \\
&&&\text{if $\bm{u}^{*} \geq \bm{0}$ and
$\bm{x}_{0}^{*}+\dots+\bm{x}_{m}^{*}
=\bm{x}^{*}$}, \\[1ex]
&{-}\infty
&&\text{otherwise}.
\end{alignedat}
\end{cases}
\end{align*}
\end{document}