We adapt Nesterov's fast gradient method to the monotone fused LASSO model. The LASSO model is a special case of the fused LASSO model. The LASSO technique improves prediction accuracy and reduces the number of predictors, while the fused LASSO procedure also encourages flatness of the regression predictors. The monotone fused LASSO model describes regression with monotonic constraints better than the fused LASSO model. We give closed-form solutions for each iteration and prove the boundedness of the optimal solution set, and we show that the algorithm converges in polynomial time with respect to the input data size. Numerical examples are provided and discussed. Our approach can easily be adapted to related problems, such as the monotone regression and the fused LASSO model.