documentclass{scrartcl}

usepackage{amsmath}
usepackage{amssymb}

usepackage{xcolor}

begin{document}

noindent
We so obtain the dual program
%
$$
boxed{
begin{array}{rl}
min & nu \
text{s.t.} & mathrm{tr}(X) = 1 \
    & langle X,mathbf 1rangle=0 \
    & langle X,E^{ij}ranglele nu,quadtext{for all $ijin E$}
end{array}}qquad textcolor{lightgray}{(Xinmathbf S^n_+,nu text{ free})}$$
%
By strong duality, we know that $nu>0$ in the optimal point, and we can re-scale the problem by $Xmapsto X/nu$.
%
$$
boxed{
begin{array}{rl}
max & mathrm{tr}(X) \
text{s.t.} & langle X,mathbf 1rangle=0 \
    & langle X,E^{ij}ranglele 1,quadtext{for all $ijin E$}
end{array}}qquad textcolor{lightgray}{(Xinmathbf S^n_+)}$$
%
The optimal value of this program is $1/a(G)$. We can again decompose $X$ into $ZZ^T$ with $Z=(z_1,...,z_n)$ to rewrite the program as
%
$$
boxed{
begin{array}{rl}
max & sum_{i=1}^n | z_i|^2 \
text{s.t.} & sum_{i=0}^n z_i=0 \
    & |z_i-z_j|le 1,quadtext{for all $ijin E$}
end{array}}qquad textcolor{lightgray}{(z_iinmathbb R^ntext{ free})}$$
%

end{document}

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fbox{
begin{minipage}{0.5textwidth}
[
  begin{array}{rl}
  min & nu \
  text{s.t.} & mathrm{tr}(X) = 1 \
      & langle X,mathbf 1rangle=0 \
      & langle X,E^{ij}ranglele nu,quadtext{for all $ijin E$}
  end{array}
]
end{minipage}%
}%
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begin{minipage}{0.2textwidth}
  (displaystyle qquad textcolor{lightgray}{(Xinmathbf S^n_+,nu text{ free})} )
end{minipage}