Unified Methods for Exploiting Piecewise Linear Structure in Convex Optimization
We develop methods for rapidly identifying important components of a convex optimization problem for the purpose of achieving fast convergence times. By considering a novel problem formulation-the minimization of a sum of piecewise functions-we describe a principled and general mechanism for exploiting piecewise linear structure in convex optimization. This result leads to a theoretically justified working set algorithm and a novel screening test, which generalize and improve upon many prior results on exploiting structure in convex optimization. In empirical comparisons, we study the scalability of our methods. We find that screening scales surprisingly poorly with the size of the problem, while our working set algorithm convincingly outperforms alternative approaches.
The authors propose new algorithms for minimizing composite objectives comprised of strongly convex and piecewise (?) components. Apart from numerous claims about importance of the paper contribution, the authors propose a “working set” algorithm and a piecewise screening test, along with an experimental study which seems to show good performance in some application scenarios. 
The authors seem to be unaware of the literature about bundle methods (see [Hiriart-Urruty, Lemaréchal, 1993] for the references). The proposed “working set” algorithm looks like a bundle family method. Note that the “basic” bundle algorithm – the Kelley method – also often shows fast convergence. Nevertheless, it minimax complexity is disastrous (although in the strongly convex situation it exhibits dimension-independent convergence rates, which are suboptimal). Same as a majority of work on bundle algorithms, the authors provide some bounds for the model gap, but do not seem being interested in providing any guaranties. My feeling is that to legitimate this type of methods (which “often” exhibit “fast convergence”), the minimax analysis is necessary (cf. methods of bundle-level family, [Lemarechal et al, 1995, Lan 2015].  Hiriart-Urruty, J. B., & Lemaréchal, C. (1993). Convex analysis and minimization algorithms II: Advanced theory and bundle methods, vol. 306 of Grundlehren der mathematischen Wissenschaften. Lemaréchal, C., Nemirovskii, A., & Nesterov, Y. (1995). New variants of bundle methods. Mathematical programming, 69(1-3), 111-147. Lan, G. (2015). Bundle-level type methods uniformly optimal for smooth and nonsmooth convex optimization. Mathematical Programming, 149(1-2), 1-45.