Edge-preserving image smoothing plays a vital role in the field of computational imaging. It is a valuable technique that has applications in various tasks. However, different tasks have specific requirements for edge preservation. Existing filters do not take into account the task-dependent smoothing behavior, resulting in visually distracting artifacts. We propose a flexible edge-preserving image filter based on a nonconvex Welsch penalty. Compared with the convex models, our model can better handle complex data and capture nonlinear relationships, thus providing better results. We combine deep unsupervised learning and graduated nonconvexity to solve our nonconvex objective function, where the main network structure is designed as a Swin transformer complemented with the locally enhanced feed-forward network. Experimental results show that the proposed method achieves excellent performance in various applications, including image smoothing, high dynamic range tone mapping, detail enhancement, and edge extraction.