Generalized　fuzzy　c-means　(GFCM)　is　an　extension　of　fuzzy　c-means　using　L-p-norm　distances.　However,　existing　methods　cannot　solve　GFCM　with　m　=　1.　To　solve　this　problem,　we　define　a　new　kind　of　clustering　models,　called　L　p-norm　probabilistic　K-means　(L-p-PKM).　Theoretically,　L　p-PKM　is　equivalent　to　GFCM　at　m　=　1,　and　can　have　nonlinear　programming　solutions　based　on　an　efficient　active　gradient　projection　(AGP)　method,　namely,　inverse　recursion　maximum-step　active　gradient　projection　(IRMSAGP).　On　synthetic　and　UCI　datasets,　experimental　results　show　that　L　p-PKM　performs　better　than　GFCM　(m　＞　1)　in　terms　of　initialization　robustness,　p-influence,　and　clustering　performance,　and　the　proposed　IRMSAGP　also　achieves　better　performance　than　the　traditional　AGP　in　terms　of　convergence　speed.