Lagrangian　Relaxation　(LR)　and　General　Mixed　Integer　programming　(MIP)　are　two　main　approaches　for　solving　Unit　Commitment　(UC)　problems.　This　paper　compares　the　LR　and　the　state　of　art　general　MIP　method　for　solving　UC　problems　based　on　performance　analysis　and　numerical　testing.　In　this　paper　we　have　rigorously　proved　that　UC　is　indeed　an　NP　complete　problem,　and　therefore　it　is　impossible　to　develop　an　algorithm　with　polynomial　computation　time　to　solve　it.　In　comparison　with　the　general　MIP　methods,　the　LR　methodology　is　more　scaleable　and　efficient　to　obtain　near　optimal　schedules　for　large　scale　and　hard　UC　problems　at　the　cost　of　a　small　percentage　of　deviation　from　the　optimal　solution.　In　particular,　solving　hydro　generation　sub-problems　within　the　LR　framework　can　take　advantages　of　both　LR　and　general　MIP　methods　and　provide　a　synergetic　combination　of　both　approaches.