Ill-posed inverse problems are fundamental in many domains, ranging from astrophysics to medical imaging. Emerging diffusion models provide a powerful prior for solving these problems. Existing maximum-a-posteriori (MAP) or posterior sampling approaches, however, rely on different computational approximations, leading to inaccurate or suboptimal samples. To address this issue, we introduce a new approach to solving MAP problems with diffusion model priors using a dual ascent optimization framework. Our framework achieves better image quality as measured by various metrics for image restoration problems, it is more robust to high levels of measurement noise, it is faster, and it estimates solutions that represent the observations more faithfully than the state of the art.