The formation of synthetic aperture radar (SAR) images is formulated as an inverse problem, a flexible approach suitable for a variety of acquisition systems and signal models. This paper focuses on increasing robustness to data saturation, specifically by optimizing a one-sided quadratic cost function to promote consistency with the received data. We model the SAR acquisition process using a linear function and we present an efficient implementation of this function and its adjoint for use in iterative optimization algorithms. Improved image quality and robustness to saturation are observed in experiments on synthetic images. Preliminary work on controlling azimuth ambiguities and incorporating image models enables saturation-robust reconstruction from satellite SAR data as well.