begin{subequations}label{matrix_and_trace}
    begin{tabularx}{textwidth}{Xp{2cm}X}
        begin{eqnarray}
        &textrm{textbf{Original textit{Matrix} Lifting}}nonumber\
        [-0.5em]
        & max_{bm Y_{textrm{RS}}} frac{1}{2} textrm{Tr} (bm L bm Y_{textrm{RS}}) nonumber\
        [-0.5em]
        & textrm{s.t.}~textrm{diag}(bm Y_{textrm{RS}})  = textcolor{red}{bm u_n} nonumber\
        [-0.5em]
        & kbm Y_{textrm{RS}} - bm J_{n} succeq 0 nonumber\
        [-0.5em]
        &bm Y_{textrm{RS}}  geq 0nonumber\
        [-0.5em]
        &textrm{Tr}(bm J bm Y_{textrm{RS}})= sum_{i=1}^k m_i^2
        end{eqnarray}
        & &
        begin{eqnarray}
        & textrm{textbf{Original textit{Trace} Formulation}}nonumber\
        [-0.5em]
        & max_{bm Y_{textrm{FJ}}}  frac{k-1}{2k} textrm{Tr} (bm L bm Y_{textrm{FJ}})nonumber\
        [-0.5em]
        &textrm{s.t.}~ textrm{diag}(bm Y_{textrm{FJ}}) = bm u_nnonumber\
        [-0.5em]
        &textcolor{red}{bm Y_{textrm{FJ}} succeq 0}nonumber\
        [-0.5em]
        &(Y_{i,j})_{textrm{FJ}}geq frac{-1}{k-1}, i neq j nonumber\
        [-0.5em]
        &textrm{Tr}(bm J bm Y_{textrm{FJ}}) = frac{1}{k-1}(k sum_{i=1}^k m_i^2 - n^2)
        end{eqnarray}
    end{tabularx}
end{subequations}