Modern　circuits　often　contain　standard　cells　of　different　row　heights　to　meet　various　design　requirements.　Taller　cells　give　larger　drive　strengths　and　higher　speed　at　the　cost　of　larger　areas　and　power.　Multi-row　height　standard　cells　incur　challenging　issues　for　layout　designs,　especially　the　mixed-cell-height　legalization　problem　with　heterogeneous　cell　structures.　Honoring　the　good　cell　positions　from　global　placement,　we　present　in　this　article　a　robust　modulus-based　matrix　splitting　iteration　method　(RMMSIM)　to　solve　the　mixed-cell-height　legalization　problem.　Fixing　the　cell　ordering　from　global　placement　and　relaxing　the　right-boundary　constraints,　our　proposed　method　first　converts　the　problem　into　an　equivalent　linear　complementarity　problem　(LCP),　and　then　properly　splits　the　matrices　in　the　LCP　so　that　the　RMMSIM　can　solve　the　LCP　optimally.　The　RMMSIM　effectively　explores　the　sparse　characteristic　of　a　circuit,　and　takes　only　linear　time　per　iteration;　as　a　result,　it　can　solve　the　QP　very　efficiently.　Finally,　an　allocation　scheme　for　illegal　cells　is　used　to　align　such　cells　to　placement　sites　on　rows　and　fix　the　placement　of　out-of-right-boundary　cells,　if　any.　Experimental　results　show　the　effectiveness　and　efficiency　of　our　proposed　algorithm.　In　addition,　the　RMMSIM　convergence　and　optimality　are　theoretically　proved　and　empirically　validated.　In　particular,　this　article　provides　a　new　RMMSIM　formulation　for　various　optimization　problems　that　require　solving　large-scale　convex　quadratic　programming　problems　efficiently.　©　2020　Association　for　Computing　Machinery.