The solution of the least squares semidefinite programming problem with equality and inequality constraints was studied. Under the Slater constraint specification, the optimum solutions for the dual and original problems are the same. Therefore, the least squares semidefinite programming problem was transformed into its corresponding dual problem, and the original problem was solved by solving the dual problem. For the solution of the dual problem of the least squares semidefinite programming problem, the quadratic model was constructed, and Cauchy point was obtained by minimizing the quadratic model along the negative gradient direction. On this basis, the positive constraint set and the non-active constraint set were divided by using positive constraint technique. Then, the L-BFGS technique was applied to accelerate the free variables to obtain the optimum solution of the dual problem. Finally, the global convergence of the algorithm was proved theoretically, and a preliminary numerical experiment was carried out to compare the algorithm with the smoothed Newton method. The results show that the algorithm has certain advantages in computing time.