Many scientific and engineering applications feature nonsmooth convex
minimization problems over convex sets. In this paper, we address an important
instance of this broad class where we assume that the nonsmooth objective is
equipped with a tractable proximity operator and that the convex constraint set
affords a self-concordant barrier. We provide a new joint treatment of proximal
and self-concordant barrier concepts and illustrate that such problems can be
efficiently solved, without the need of lifting the problem dimensions, as in
disciplined convex optimization approach. We propose an inexact path-following
algorithmic framework and theoretically characterize the worst-case analytical
complexity of this framework when the proximal subproblems are solved
inexactly. To show the merits of our framework, we apply its instances to both
synthetic and real-world applications, where it shows advantages over standard
interior point methods. As a by-product, we describe how our framework can
obtain points on the Pareto frontier of regularized problems with
self-concordant objectives in a tuning free fashion.

通过使用可计算的接近算子和自共轭障碍的凸约束集，提出了一种新的关于近端和自共轭障碍概念的混合处理方法，并提出了一种非精确路径跟踪算法框架，可以有效地解决非光滑凸最小化问题，同时可以以无调谐自由的方式获得自共轭目标正则化问题的 Pareto 前沿点。