Hierarchical　trees　are　widely　used　in　differentially　private　statistical　releases.　Existing　fast　specialized　algorithms　guarantee　hierarchical　consistency　but　not　non-negativity,　which　may　incur　meaningless　negative　values　due　to　randomness.　Quadratic　programming　can　achieve　optimally　non-negative　consistent　releases,　but　traditional　numerical-based　methods　face　significant　performance　challenges　when　handling　large-scale　hierarchical　trees.　In　this　article,　we　construct　the　element　set　of　Lagrange　multiplier　coefficients　K　that　satisfy　the　first　type　of　KKT　condition　(values　0≥0　and　gradients　=　0=0).　Applying　Generation　Matrix,　we　find　that　meticulously　constructing　an　unconstrained　set　S　⊆　KS⫅K,　we　can　further　find　a　larger　S　⊆　KS⫅K　until　S　=　KS=K,　as　demonstrated　in　our　proof　(we　also　term　this　as　the　Unconstrained　Set　Expansion　Theorem).　Based　on　the　theorem,　we　design　an　efficient　Generation　Matrix-based　optimally　non-negative　consistent　release　algorithm　(GMNC),　which　can　easily　handle　large-scale　hierarchical　trees　with　more　than　100　million　nodes.　Our　experiments　show　that　GMNC　has　an　extremely　high　iteration　efficiency　with　no　more　than　20　(only　2∼8　on　average)　iterations　even　in　the　worst　case.　Compared　to　Interior　Point　Convex　Method,　GMNC　improves　the　data　scale　processing　limit　up　to　33.4　times　in　our　experimental　environment　while　requiring　only　0.7%　of　the　running　time.　©　2004-2012　IEEE.