There has been an increased interest in simple LASSO problems arising in sparse signal reconstruction. The scientific community seems to favour first-order methods to solve these problems. In this talk we will address a broader family of problems which include total-variation minimization, L1-analysis, the combinations of those two and finally other Dantzig-selector alternatives. We will argue that such problems challenge first-order methods but can be solved efficiently with specially designed second-order methods. We will provide theoretical analysis and computational evidence to illustrate our findings.