In　the　paper,　we　consider　the　problem　of　maximizing　the　multilinear　extension　of　a　nonsubmodular　set　function　subject　to　a　k-cardinality　constraint　with　adaptive　rounds　of　evaluation　queries.　We　devise　an　algorithm　which　achieves　a　ratio　of　(　1-　e-(gamma　2)-　epsilon)　and　requires　O　(logn/epsilon(2))　adaptive　rounds　and　O(logn/epsilon(2))　queries,　where　gamma　is　the　continuous　generic　submodularity　ratio　that　compares　favorably　in　flexibility　to　the　traditional　submodularity　ratio　proposed　by　Das　and　Kempe.　The　key　idea　of　our　algorithm　is　originated　from　the　parallel-greedy　algorithm　proposed　by　Chekuri　et　al.,　but　incorporating　with　two　major　changes　to　retain　the　performance　guarantee:　First,　identify　all　good　coordinates　with　the　continuous　generic　submodularity　ratio　and　gradient　values　approximately　as　large　as　the　best　coordinate,　and　increase　along　all　these　coordinates　uniformly;　Second,　increase　xalong　these　coordinates　by　a　dynamical　increment　whose　value　depends　on　gamma.　The　key　difficulty　of　our　algorithm　is　that　when　the　function　is　nonsubmodular,　the　set　of　the　best　coordinate　does　not　decrease　during　iterations;　while　provided　submodularity,　the　decreasing　can　be　ensured　by　the　parallel-greedy　algorithm.　Our　algorithms　slightly　compromise　performance　guarantee　for　the　sake　of　extending　to　constrained　nonsubmodular　maximization　with　parallelism,　provided　that　the　state-of-art　algorithm　for　the　corresponding　submodular　version　attains　an　approximation　ratio　of　(1　-　1/e-　epsilon)　and　requires　O　(logn/epsilon(2))　adaptive　rounds.　(C)　2021　Elsevier　B.V.　All　rights　reserved.