Modern　circuit　designs　often　contain　standard　cells　of　different　row　heights　to　meet　various　design　requirements.　Due　to　the　higher　interference　among　heterogeneous　cell　structures,　the　legalization　problem　for　mixed-cell-height　standard　cells　becomes　more　challenging.　In　this　paper,　we　present　an　analytical　legalization　algorithm　for　mixed-cell-height　standard　cells　to　simultaneously　minimize　the　average　and　the　maximum　cell　movements.　We　formulate　it　as　a　mixed　integer　quadratic　programming　problem　(MIQP),　which　allows　cell　spreading　concurrently　in　both　the　horizontal　and　vertical　directions.　By　relaxing　its　discrete　constraints　to　linear　ones,　we　convert　the　MIQP　into　a　quadratic　programming　problem　(QP).　To　solve　the　QP　efficiently,　we　further　reformulate　it　as　a　linear　complementarity　problem　(LCP),　and　solve　the　LCP　by　a　modulus-based　matrix　splitting　iteration　method　(MMSIM).　To　guarantee　the　convergence　of　the　MMSIM　and　the　equivalence　between　the　QP　and　the　LCP,　we　use　a　series　of　operations　to　ensure　that　its　induced　objective　matrix　is　symmetric　positive　definite　and　its　constraint　matrix　is　of　full　row　rank.　Experimental　results　demonstrate　the　effectiveness　of　our　algorithm　in　reducing　both　the　average　and　the　maximum　cell　movements　for　mixed-cell-height　legalization.　©　2019　Association　for　Computing　Machinery.