Heat　conduction　optimization　with　arbitrary　boundary　conditions　is　a　challenging　problem　that　lacks　a　universal　optimization　criterion.　In　the　present　work,　the　concept　of　generalized　entransy　dissipation　(GED)　is　proposed　through　transforming　heat　conduction　optimization　problems　with　arbitrary　boundaries　into　their　homogeneous　counterparts.　It　is　demonstrated　that　minimizing　GED　leads　to　optimal　thermal　performance　of　heat　conduction　problems　with　arbitrary　boundary　conditions.　In　addition,　GED-based　continuous　optimization　problems　are　convex,　guaranteeing　the　uniqueness　and　global　optimality　of　the　solution　and　benefitting　numerical　calculations.　Two　typical　problems　with　complex　boundary　conditions　are　studied　by　applying　the　minimum　principle　of　GED,　and　the　results　are　compared　with　other　optimization　objectives.　The　numerical　results　show　that　GED　achieves　better　thermal　performance　than　entropy　generation-(EG)　and　entransy　dissipation-(ED)　based　optimizations.　For　the　optimization　of　boundary　average　temperature　under　the　given　input　heat　flux　of　200　W,　GED　achieves　the　best　result,　where　the　optimized　average　temperature　is　48.8　K　and　27.5　K　lower　compared　with　EG　and　ED　optimizations,　respectively.　In　general,　GED　offers　a　reasonable　and　easy　to　implement　optimization　principle　for　heat　conduction　processes　with　arbitrary　boundaries　and　may　provide　new　insights　for　heat　conduction　optimization.