An Inexact Proximal Path-Following Algorithm for Constrained Convex Minimization

Quoc Tran-Dinh, Anastasios Kyrillidis, Volkan Cevher
2014 SIAM Journal on Optimization  
Many scientific and engineering applications feature large-scale non-smooth convex minimization problems over convex sets. In this paper, we address an important instance of this broad class where we assume that the non-smooth objective is equipped with a tractable proximity operator and that the convex constraints afford 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
more » ... lifting problem dimensions. We propose an inexact path-following algorithmic framework and theoretically characterize the worst case convergence as well as computational complexity of this framework, and also analyze its behavior when the proximal subproblems are solved inexactly. To illustrate our framework, we apply its instances to both synthetic and real-world applications and illustrate their accuracy and scalability in large-scale settings. As an added bonus, we describe how our framework can obtain points on the Pareto frontier of regularized problems with self-concordant objectives in a tuning free fashion. x . For our analysis, we also use two simple convex functions ω(t) := t − ln(1 + t) for t ≥ 0 and ω * (t) := −t − ln(1 − t) for t ∈ [0, 1), which are strictly increasing in their domain.
doi:10.1137/130944539 fatcat:x524dg3vbzc3depm2im4khg2la