Kernel　fuzzy　c-means　(KFCM)　is　a　significantly　improved　version　of　fuzzy　c-means　(FCM)　for　processing　linearly　inseparable　datasets.　However,　for　fuzzification　parameter　m=1,　the　problem　of　KFCM　(kernel　fuzzy　c-means)　cannot　be　solved　by　Lagrangian　optimization.　To　solve　this　problem,　an　equivalent　model,　called　kernel　probabilistic　k-means　(KPKM),　is　proposed　here.　The　novel　model　relates　KFCM　to　kernel　k-means　(KKM)　in　a　unified　mathematic　framework.　Moreover,　the　proposed　KPKM　can　be　addressed　by　the　active　gradient　projection　(AGP)　method,　which　is　a　nonlinear　programming　technique　with　constraints　of　linear　equalities　and　linear　inequalities.　To　accelerate　the　AGP　method,　a　fast　AGP　(FAGP)　algorithm　was　designed.　The　proposed　FAGP　uses　a　maximum-step　strategy　to　estimate　the　step　length,　and　uses　an　iterative　method　to　update　the　projection　matrix.　Experiments　demonstrated　the　effectiveness　of　the　proposed　method　through　a　performance　comparison　of　KPKM　with　KFCM,　KKM,　FCM　and　k-means.　Experiments　showed　that　the　proposed　KPKM　is　able　to　find　nonlinearly　separable　structures　in　synthetic　datasets.　Ten　real　UCI　datasets　were　used　in　this　study,　and　KPKM　had　better　clustering　performance　on　at　least　six　datsets.　The　proposed　fast　AGP　requires　less　running　time　than　the　original　AGP,　and　it　reduced　running　time　by　76-95%　on　real　datasets.