Linear algebraic equations based methods in the time difference of arrival (TDOA) localization are akind of excellent algorithms. In consideration of the constraints on parameters to be estimated, the first-order error in matrix and the second-order noise in vector, a sequential-quadratic-programming-based localization algorithm isproposed in this paper. An optimal solution of the linear equation model was obtained in the algorithm. Meanwhile, with the rank 1 constraint ignored, the problem was rewritten to a series of convex functions and then a semi-definite relaxation (SDR) solution was yielded additionally. However, the solution can only be used as an initial value for other TDOA localization algorithms due to the insufficient accuracy of the SDR solution. Final localization experiments demonstrate that the proposed algorithm has a higher positioning accuracy compared with other TDOA localization algorithms based on linear model. At the same time the result can be closer to the Cramer-Rao Lower Bound, especially at the low signal to noise ratio regime.