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Abstract In this paper, we present a sequential semidefinite programming (SSDP) algorithm for nonlinear semidefinite programming. At each iteration, a linear semidefinite programming subproblem and a modified quadratic semidefinite programming subproblem are solved to generate a master search direction. In order to avoid Maratos effect, a second-order correction direction is determined by solving a new quadratic programming. And then a penalty function is used as a merit function for arc search. The superlinear convergence is shown under the strict complementarity and the strong second-order sufficient conditions with the sigma term. Finally, some preliminary numerical results are reported.