Modern　circuits　often　contain　standard　cells　of　different　row　heights　to　meet　various　design　requirements.　Higher　cells　give　larger　drive　strengths　at　the　costs　of　larger　areas　and　power.　Multi-row-height　standard　cells　incur　challenging　issues　to　layout　designs,　especially　the　mixed-cell-height　legalization　problem　due　to　the　heterogeneous　cell　structures.　Honoring　the　good　cell　positions　from　global　placement,　we　present　in　this　paper　a　fast　and　near-optimal　algorithm　to　solve　the　legalization　problem.　Fixing　the　cell　ordering　from　global　placement　and　relaxing　the　right　boundary　constraints,　we　first　convert　the　problem　into　a　linear　complementarity　problem　(LCP).　With　the　converted　LCP,　we　split　its　matrices　to　meet　the　convergence　requirement　of　a　modulus-based　matrix　splitting　iteration　method　(MMSIM),　and　then　apply　the　MMSIM　to　solve　the　LCP.　This　MMSIM　method　guarantees　the　optimality　if　no　cells　are　placed　beyond　the　right　boundary　of　a　chip.　Finally,　a　Tetris-like　allocation　approach　is　used　to　align　cells　to　placement　sites　on　rows　and　fix　the　placement　of　out-of-right-boundary　cells,　if　any.　Experimental　results　show　that　our　proposed　algorithm　can　achieve　the　best　cell　displacement　and　wirelength　among　all　published　methods　in　reasonable　runtimes.　The　MMSIM　optimality　is　theoretically　proven　and　empirically　validated.　In　particular,　our　formulation　provides　new　generic　solutions　and　research　directions　for　various　optimization　problems　that　require　solving　large-scale　quadratic　programs　efficiently.