The paper considers structured convex optimization where constraints are given in a stage-wise manner. The paper studies a non-convex reformulation for this problem and proposes new algorithms to ensure convergence to global minimizers for both non-degenerate and degenerate cases. The reformulation is proven effective in theory and experiments. The author feedback phase has clarified several aspects, resulting in a consensus on weak acceptance. We hope the detailed feedback with improvement suggestions from the 4 reviews will be implemented for the camera ready version, in particular about the clarity and readability of the paper.