Fourier ptychographic microscopy (FPM) uses illumination of LED array with angle change to overcome the resolution limitations of low numerical aperture objectives. In an original FPM system, position errors of an LED array severely impact an image reconstruction process. Therefore, accurately correcting the position of an LED array is very important to improve the quality of reconstructed images. To solve this problem, the study proposes a position correction method based on a genetic annealing optimization algorithm. First, the influence of the following three factors’ relative positions on an incident wave vector is analyzed: the LED array, sample, and numerical aperture of objective lens. Next, the genetic annealing optimization algorithm is utilized in estimating global error parameters for LED array error locations. Finally, global error parameters are used to quickly and accurately correct the position of the LED array during the reconstruction process. Both simulation and experimental results indicate that the proposed method can significantly improve the quality of FPM reconstructed images.