In　this　paper,　a　type　of　cascadic　adaptive　finite　element　method　for　nonlinear　eigenvalue　problem　is　proposed　based　on　the　complementary　approach.　In　this　new　scheme,　we　only　need　to　do　some　smoothing　steps　for　the　linearized　boundary　value　problems　on　a　series　of　adaptive　finite　element　spaces　and　solve　some　nonlinear　eigenvalue　problems　on　a　low　dimensional　space.　Hence　the　efficiency　can　be　improved　since　we　do　not　need　to　solve　the　nonlinear　eigenvalue　problems　on　each　adaptive　space　directly.　Besides,　the　complementary　error　estimate　for　nonlinear　eigenvalue　problem　will　be　introduced　in　our　paper.　This　estimate　can　not　only　provide　an　accurate　error　estimate　for　the　nonlinear　eigenvalue　problem　but　also　provide　the　way　to　refine　mesh　and　control　the　number　of　smoothing　steps　for　the　cascadic　adaptive　algorithm.　Some　numerical　examples　are　presented　to　validate　the　efficiency　of　the　proposed　algorithm　in　this　paper.　(C)　2020　Elsevier　B.V.　All　rights　reserved.