Breast microwave sensing (BMS) has been studied as a potential technique for cancer detection due to the observed microwave properties of malignant and healthy breast tissues. This work presents a novel radar-based image reconstruction algorithm for use in BMS that reframes the radar image reconstruction process as an optimization problem. A gradient descent optimizer was used to create an optimization-based radar reconstruction (ORR) algorithm. Two hundred scans of MRI-derived breast phantoms were performed with a preclinical BMS system. These scans were reconstructed using the ORR, delay-and-sum (DAS), and delay-multiply-and-sum (DMAS) beamformers. The ORR was observed to improve both sensitivity and specificity compared to DAS and DMAS. The estimated sensitivity and specificity of the DAS beamformer were 19% and 44%, respectively, while for ORR, they were 27% and 56%, representing a relative increase of 42% and 27%. The DAS reconstructions also exhibited a hot-spot image artifact, where a localized region of high intensity that did not correspond to any physical phantom feature would be present in an image. This artifact appeared like a tumour response within the image and contributed to the lower specificity of the DAS beamformer. This artifact was not observed in the ORR reconstructions. This work demonstrates the potential of an optimization-based conceptualization of the radar image reconstruction problem in BMS. The ORR algorithm implemented in this work showed improved diagnostic performance and fewer image artifacts compared to the widely employed DAS algorithm.