The　problem　of　recovering　a　low-rank　matrix　from　partial　entries,　known　as　low-rank　matrix　completion,　has　been　extensively　investigated　in　recent　years.　It　can　be　viewed　as　a　special　case　of　the　affine　constrained　rank　minimization　problem　which　is　NP-hard　in　general　and　is　computationally　hard　to　solve　in　practice.　One　widely　studied　approach　is　to　replace　the　matrix　rank　function　by　its　nuclear-norm,　which　leads　to　the　convex　nuclear-norm　minimization　problem　solved　efficiently　by　many　popular　convex　optimization　algorithms.　In　this　paper,　we　propose　a　new　nonconvex　approach　to　better　approximate　the　rank　function.　The　new　approximation　function　is　actually　the　Moreau　envelope　of　the　rank　function　(MER)　which　has　an　explicit　expression.　The　new　approximation　problem　of　low-rank　matrix　completion　based　on　MER　can　be　converted　to　an　optimization　problem　with　two　variables.　We　then　adapt　the　proximal　alternating　minimization　algorithm　to　solve　it.　The　convergence　(rate)　of　the　proposed　algorithm　is　proved　and　its　accelerated　version　is　also　developed.　Numerical　experiments　on　completion　of　low-rank　random　matrices　and　standard　image　inpainting　problems　have　shown　that　our　algorithms　have　better　performance　than　some　state-of-art　methods.　©　2018　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature