We propose an efficient regression method which is highly robust against outliers and stable even in the severely noisy situations. The robustness here comes from the adoption of the minimax concave loss, while the stability comes from separate treatments of the outlier and noise by an introduction of an auxiliary vector modeling the Gaussian noise. We present a necessary and sufficient condition for convexity of the smooth part of the entire cost under a certain assumption, where a general model is used with its potential use for other applications envisioned. We show that the proposed formulation can be solved via reformulation by the forward-backward-based primal-dual method under the convexity condition. The numerical examples show the remarkable robustness of the proposed estimator under highly noisy situations.