We demonstrate a single-shot near on-axis digital holographic microscope that uses a constrained optimization approach for retrieval of the complex object function in the hologram plane. The recovered complex object is back-propagated from the hologram plane to image plane using the Fresnel back-propagation algorithm. A numerical aberration compensation algorithm is employed for correcting the aberrations in the object beam. The reference beam angle is calculated automatically using the modulation property of Fourier transform without any additional recording. We demonstrate this approach using a United States Air Force (USAF) resolution target as an object on our digital holographic microscope. We also demonstrate this approach by recovering the quantitative phase images of live yeast cells, red blood cells and dynamics of live dividing yeast cells.