A　primal-dual　algorithm　is　proposed　for　computing　the　distance　from　a　point　to　an　order　simplex.　An　advantage　of　the　algorithm　is　that,　for　any　initial　active　set,　it　can　adjust　the　active　set　to　improve　both　primal　and　dual　feasibility　until　the　optimal　active　set　is　found.　We　verify　that　the　algorithm　takes　only　O(n)　elementary　arithmetic　operations,　where　n　is　the　problem　dimension.　Numerical　results　demonstrate　the　efficiency　of　the　primal-dual　algorithm　compared　with　the　primal　feasible　algorithm　and　the　dual　feasible　algorithm.　The　primal-dual　algorithm　proves　very　useful　in　projected　gradient　algorithms　applied　to　general　order　simplex　constrained　problems　since　a　series　of　projection　subproblems　are　requested　there　and　the　primal-dual　algorithm　makes　warm　starts　possible.